Dirichlet Forms and Symmetric Markov Processes, Second Edition (De Gruyter Studies in Mathematics)

.D Ex .e ˛ A //

:p .x/ D H 0C 1.x/ .D P . < 1//: x A A A

.4:1:9/

pA˛ (resp. pA ) is an ˛-excessive (resp. excessive) function and consequently a nearly Borel measurable finely continuous function according to Theorem A.2.5. Keeping this in mind we proceed to the proof of the following theorem. Theorem 4.1.1. If N is exceptional, then N is contained in a properly exceptional set B. B can be taken to be Borel. Proof. If N is exceptional, then there is a nearly Borel set B0 N such that pB0 .x/ D 0 m-a.e. By virtue of Lemma 4.1.5, pB0 vanishes q.e., that is, except on some nearly Borel exceptional set B1 . Apply the same argument to the function

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4 Analysis by symmetric Hunt processes

pB0 [B1 . In this way we get a sequence ¹Bk º1 of nearly Borel exceptional sets kD0 such that p.[nkD1 Bk / .x/ D 0; 8x 2 X n BnC1 . We thus get pB .x/ D 0; 8x 2 S X n B, for the nearly Borel exceptional set B D 1 kD0 Bk . On the other hand, the proof of Theorem A.2.3 shows that pB .x/ D Px .X t or X t  2 B; 9t  0/; x 2 X n B. Hence, X n B is M-invariant and B is properly exceptional. We can prove the second assertion by replacing the sets Bk in the above with larger Borel sets possessing the same properties. We can simplify the description of q.e. fine continuity as follows: Lemma 4.1.6. A function u is finely continuous q.e. if and only if there exists a properly exceptional Borel set B such that u is Borel measurable and finely continuous on X n B. Proof. We prove this when u is bounded. Let N be the set appearing in the definition of q.e. fine continuity for u: Then limn!1 nRn u.x/ D u.x/; 8x 2 X n N . We extend u to X by setting u.x/ D 0; x 2 N . Choose Borel u1 and u2 such that u1  u  u2 on X and u1 D u2 m-a.e. Then by the m-symmetry of Rn , we see that Rn u1 D Rn u2 m-a.e. and consequently q.e. on account of Lemma 4.1.5. Let Bn be the corresponding exceptional set. Then there exists by virtue of Theorem 4.1.1 a properly exceptional Borel set B containing N and all Bn ; n D 1; 2; : : : . On X n B we have u.x/ D limn!1 nRn u1 .x/, which yields the Borel measurability of u. The next lemma will be useful. Lemma 4.1.7.1 Let ˛  0 and let ¹un º be a decreasing sequence of ˛-excessive functions with limit u and suppose that u D 0 m-a.e. Then u D 0 q.e. Proof. Let " > 0 and let K be a compact subset of ¹u  "º. Since HK˛ un .x/  un .x/, we get HK˛ u.x/  u.x/ provided that u.x/ is finite. Hence u.x/ D 0 implies that ˛ 0 D u.x/  HK˛ u.x/  "pK .x/ ˛ .x/ D 0. Since u D 0 m-a.e., K is exceptional. By Theorem A.2.4, we and pK can conclude that the nearly Borel set ¹u > 0º is exceptional.

It is clear that any polar set A is exceptional. We claim that the converse statement holds if and only if the process M satisfies an absolute continuity condition. 1 This

is a variant of Blumenthal–Getoor [1; VI (3.2)].

4.1

Smallness of sets and symmetry

157

Theorem 4.1.2. The following two conditions are equivalent: (i) A set is polar if and only if it is exceptional, (ii) R˛ .x;  / is absolutely continuous with respect to m for each ˛ > 0 and x 2 X. Proof. (i))(ii): Suppose E is Borel and m.E/ D 0, then, in view of Lemma 4.1.1, R˛ .x; E/ D 0 for m-a.e. x 2 X and consequently q.e. x 2 X by virtue of Lemma 4.1.5. Let N be the corresponding Borel exceptional set. N is then polar on account of (i) and R˛ .x; E/ D lim t #0 Ex .R˛ .X t ; E/I X t 62 N / D 0, 8x 2 X . (ii))(i): If B is a Borel exceptional set, then u.x/ D Px .B < 1/ D 0 m-a.e. u being excessive, we have by condition (ii) that u.x/ D lim˛!1 ˛R˛ u.x/ D 0, 8x 2 X. The last theorem in this section concerns the semi-polar set. We need an amplified version of (4.1.8): for any nearly Borel sets A and B .g; HA˛ HB˛ R˛ h/ D .HB˛ HA˛ R˛ g; h/;

g; h 2 B C :

.4:1:10/

In fact, this follows from (4.1.8) and the symmetry of R˛ , for .g; HA˛ HB˛ R˛ h/ D lim ˇ.g; HA˛ Rˇ HB˛ R˛ h/ ˇ !1

D lim ˇ.g; HA˛ R˛ .I  .ˇ  ˛/Rˇ /HB˛ R˛ h/ ˇ !1

D lim ˇ.HB˛ R˛ .I  .ˇ  ˛/Rˇ /HA˛ R˛ g; h/ ˇ !1

D lim ˇ.HB˛ Rˇ HA˛ R˛ g; h/ D .HB˛ HA˛ R˛ g; h/: ˇ !1

As an easy consequence of (4.1.10), we get ˛ ˛ .x/ D pK .x/ m-a.e. HG˛ pK

.4:1:11/

for any compact set K and an open set G K. Theorem 4.1.3. Any semi-polar set is exceptional. Proof. Let K be an arbitrary compact thin set. It suffices to prove that K is exceptional, because then the general statement of the theorem follows in the same way as in the proof of Lemma 4.1.4. First note that   ˛ Px lim pK .X t / D 1; K < 1 D Px .K < 1/ m-a.e. x 2 X: .4:1:12/ t " K

158

4 Analysis by symmetric Hunt processes

T To see this, let ¹Gn º be open sets such that Gn GN nC1 and n Gn D K. From (4.1.11), we see Gn C K ı  Gn D K . On account of the thinness of K, we have Px .K < 1; Gn < K ; 8n/ D Px .K < 1/ m-a.e. x 2 X:

.4:1:13/

On the other hand, we get from (A.2.9) of A.2 that Px .limn!1 Gn D K / D 1 for any x 2 X n K and hence for m-a.e. x 2 X as was observed in the proof of ˛ ˛ Lemma 4.1.3. pK being ˛-excessive, we know that ¹e ˛ Gn pK .X Gn /º1 nD1 is ˛ ˛ G n p .X a bounded supermartingale and limn!1 e / exists P -a.s., which

x Gn K ˛ .X Gn / Px -a.s. for m-a.e. x 2 X . Hence by (4.1.11) equals e  K limn!1 pK ˛ ˛ Ex .e ˛ K / D lim Ex .e ˛ Gn pK .X Gn // D Ex .e ˛ K lim pK .X Gn //: n!1

n!1

Combining this with (4.1.13), we are led to Px .K < 1/ D Px .Gn < K ; 8n; ˛ limn!1 Gn D K ; limn!1 pK .X Gn / D 1; K < 1/ m-a.e. x 2 X , which means (4.1.12). ˛ Now T1set Bn D ¹x 2 X W pK .x/  1  1=nº. Since each Bn is finely closed and nD1 Bn is empty, (4.1.12) implies that Bn < K Px -a.s. on ¹K < 1º for m-a.e. x 2 X. Hence, HB˛n HK˛ f D HK˛ f m-a.e., 8f 2 B C .X/, and (4.1.10) together with (4.1.8) leads us to .HK˛ g; h/ D .HK˛ HB˛n g; h/:

.4:1:14/

Take a strictly positive L1 -function h; then (4.1.14) means T that the measure .E/ D .HK˛ 1E ; h/ is concentrated on each Bn . Since 1 nD1 Bn is empty, ˛ .x/ D HK˛ 1.x/ D 0 m-a.e., proving that K is ex D 0 and consequently pK ceptional. The following diagram indicates the relations among small sets in the m-symmetric Hunt process M: polar *  (Theorem 4.1.2)        exceptional -

properly  exceptional (Theorem 4.1.1)

?

H

HH H j H

(Theorem 4.1.3)

semi-polar ?

m-negligible  of potential zero (Lemma 4.1.1)

4.1

Smallness of sets and symmetry

159

Example 4.1.1 (Spatially homogeneous Markov process). Let a transition function ¹p t ; t > 0º on .Rd ; B.Rd // be given by (1.4.20) with a family ¹ t ; t > 0º of probability measures satisfying (1.4.17) and (1.4.19). From the expression Z p t f .x/ D f .x C y/ t .dy/; f 2 C1 .Rd /; .4:1:15/ Rd

it is easy to see that ¹p t ; t > 0º has the Feller property. Let M D .; M; X t ; Px / be the associated Hunt process on .Rd ; B.Rd // according to Theorem A.2.2. We may assume that .!/ D 1; 8! 2 , because Px . < t / D 1  p t 1.x/ D 0, 8x 2 Rd ; 8t > 0: M possesses the spatial homogeneity: Px .X t1 2 E1 ; : : : ; X tk 2 Ek / D P0 .X t1 C x 2 E1 ; : : : ; X tk C x 2 Ek /; t1 <    < tk ; E1 2 B.Rd /; : : : ; Ek 2 B.Rd /, that is, the law of Px is obtained by the spatial shift from the single stochastic process .; M; X t ; P0 / starting at the origin 0. The latter is called a Lévy process, which possesses, together with the regularity (M.6) of sample paths, the property of stationary independent increments. The Hunt process on Rd with the transition function being given by the symmetric stable convolution semigroup  t of index ˛ (see Example 1.5.2) is called the symmetric stable process. In particular, the (standard) Brownian motion on Rd is the Hunt process with  t being given by (1.4.25). This will be seen to be a diffusion at the end of §4.5. M is symmetric with respect to the Lebesgue measure if and only if ¹ t º satisfies (1.4.18). If, moreover,  t is absolutely continuous with respect to the Lebesgue measure (for instance if M is the Brownian motion or the symmetric stable process), then the notion of an exceptional set is merely a synonym for a polar set by Theorem 4.1.2. In this case, any Borel polar set is also properly exceptional. However,  t is not absolutely continuous in general. In the one dimensional case, for example, it is known that  t is purely discontinuous if and only if S D 0 and J is purely discontinuous and bounded in the Lévy–Khinchin formula (1.4.21).2 Thus, the role of the present notion of the exceptional set can not be covered by the traditional notion of the polar set in general (see Theorem 4.2.4). Theorem 4.1.3 is violated by some non-symmetric Markov process. Take the deterministic process M of uniform motion to the right: M has the transition function p t .x; A/ D ı.xCt / .A/; t > 0; x 2 R1 ; A 2 B.R1 / .4:1:16/ Then every singleton is thin but not exceptional. 2 K.

Itô [3; §0.5].

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4 Analysis by symmetric Hunt processes

4.2

Identification of potential theoretic notions

The transition function ¹p t ; t > 0º of the m-symmetric Hunt process M on X clearly satisfies the continuity condition (1.4.15). Hence, ¹p t ; t > 0º uniquely determines a strongly continuous Markovian semigroup ¹T t ; t > 0º on L2 .XI m/ and a Dirichlet form E on L2 .XI m/ as in Lemma 1.4.3 and the paragraph immediately following it respectively. We call them respectively the semigroup and the Dirichlet form on L2 .XI m/ of the process M. In this section, several probabilistic notions which have appeared in the preceding sections relevant to M are identified with those analytical notions in Chapter 2 relevant to E. First of all, we identify the (1-order) hitting probability pA1 defined by (4.1.9) with the (1-)equilibrium potential eA of Lemma 2.1.1 when A is open. Lemma 4.2.1. If A is an open set with finite capacity, then pA1 is a version of eA . Proof. As T t is the extension to L2 .XI m/ of p t on Bb \ L2 .XI m/, p t f is a version of T t f when f is a universally measurable non-negative function in L2 .XI m/. In particular, if f is ˛-excessive with respect to ¹p t ; t > 0º, then it is also ˛-excessive with respect to ¹T t ; t > 0º in the sense of §2.2. Since pA1 is 1-excessive with respect to ¹p t ; t > 0º and pA1 .x/ D 1; 8x 2 A, it suffices to show, in view of Lemma 2.1.1 and Lemma 2.3.2 that pA1  eA

m-a.e.

.4:2:1/

Take a Borel modification eA of eA such that eA .x/ D 1; 8x 2 A, and let Y t .!/ D e t eA .X t .!//; t  0; ! 2 . Then .Y t ; F tR0 ; Phm / t 0 is a supermartingale for any non-negative Borel function h such that X h.x/m.dx/ D 1. Indeed, the Markov property of M implies, for 0  s < t; Ehm .e t eA .X t /jFs0 / D e s e .t s/ p t s eA .Xs /; Phm -a.s., which in turn is not greater than e s eA .Xs /, Phm -a.s., because the set ¹x 2 XI e .t s/ p t s eA .x/ > eA .x/º is m-negligible. Let D be a finite subset of .0; 1/ with min D D a and max D D b and put .DI A/ D min¹t 2 DI X t 2 Aº. If the set in the braces is empty, we set .DI A/ D b. The optional sampling theorem3 applied to ¹Y t ; F t0 ; Phm º t 2D yields Ehm .e  .DIA/ I .DI A/ < b/  Ehm .Y .DIA/ /  Ehm .Ya /  .h; eA /: Letting D increase to a countable dense subset of .0; b/ and then b tend to infinity, we arrive at .h; pA1 /  .h; eA / which means (4.2.1). 3 Cf.

C. Dellacherie and P. A. Meyer [1; V 11].

4.2 Identification of potential theoretic notions

161

The next theorem justifies the usage of the term “q.e.” in the either sense of §2.1 or §4.2. Theorem 4.2.1. (i) Let ¹An º be a decreasing sequence of open sets with finite capacity. limn!1 Cap.An / D 0 if and only if lim p 1 .x/ n!1 An

D 0 q.e.

.4:2:2/

(ii) Assume that any compact set is of finite capacity. Then a set N  X is exceptional if and only if Cap.N / D 0. Proof. (i) Suppose that Cap.An / ! 0, n ! 1. Since Cap.An / D E1 .eAn ; eAn /  .eAn ; eAn /, we have from Lemma 4.2.1 that limn!1 pA1 n .x/ D 0 m-a.e. which can be strengthen to (4.2.2) by virtue of Lemma 4.1.7. Conversely, by Lemma 4.2.1 and the same reasoning as in the proof of Lemma 2.1.2 (iii), (4.2.2) leads us to limn!1 Cap.An / D limn!1 E1 .pA1 n ; pA1 n / D 0. (ii) Suppose Cap.N / D 0. Then there exist open sets An T N satisfying 1 conditions in (i) and, consequently, pB .x/ D 0 q.e. for B D 1 nD1 An . N /. Hence, N is exceptional. Conversely, assume that N is a compact exceptional set. Let ¹Gn º be a sequence open sets such that Gn G nC1 and T1 of decreasing, relatively compact 1 1 G D N . Then lim p .x/ D pN .x/ D 0 m-a.e. because of (A.2.9) n n!1 nD1 Gn and m-negligibility of N . We have then Cap.N /  limn!1 Cap.Gn / which vanishes as in the proof of the “if” part of (i). Now it suffices to use (2.1.6) to complete the proof. Our next concern is the notions of the continuity of functions introduced in §2.1 and §4.1. Theorem 4.2.2. If u is quasi continuous, then u is finely continuous q.e. More specifically, there exists a properly exceptional set N such that u is nearly Borel measurable on X n N and for any x 2 X n N ,   Px u.X t / is right continuous and lim u.X t 0 / D u.X t  /; 8t 2 Œ0; / D 1; t 0 "t





Px lim u.X t 0 / D u.X  /; X  2 X D Px .X  2 X/: t 0 "

(4.2.3) (4.2.4)

Proof. There exists a decreasing sequence ¹An º1 nD1 of open sets such that limn!1 Cap.An / D 0 and ujX nAn is continuous for each n. Then we have the relation (4.2.2) by Theorem 4.2.1. Denote by N0 the exceptional set in (4.2.2).

162

4 Analysis by symmetric Hunt processes

Due T to Theorem 4.1.1, we can find a properly exceptional set N containing N0 [ . 1 nD1 An /. The property (4.2.3) and (4.2.4) clearly hold for x 2 X n N because   Px lim An D 1 D 1; x 2 X n N: .4:2:5/ n!1

Therefore, u is finely continuous on X n N . In the rest of this section and in the most parts of the subsequent sections of Chapter 4 as well, we need the following assumption (4.2.6) for the process M: the Dirichlet form E on L2 .XI m/ of M is regular:

.4:2:6/

This condition on M is not very restrictive because we show in Chapter 7 that, given any regular Dirichlet form E, there exists uniquely in a certain sense (see Theorem 4.2.7) an m-symmetric Hunt process whose Dirichlet form is E. Notice that the assumption in Theorem 4.2.1 (ii) is fulfilled under (4.2.6). We can easily give a partial converse to Theorem 4.2.2. Lemma 4.2.2. (i) If u is finely continuous q.e. and u 2 F , then u is quasi continuous. (ii) If u is quasi continuous and u 2 F , then u has the property .4:2:3/ with time interval Œ0; / being strengthened to Œ0; 1/. Proof. (i) Let u 2 F be finely continuous q.e. By Theorem 2.1.3, u admits a quasi continuous version e u: e u is then finely continuous q.e. by Theorem 4.2.2. Since u De u m-a.e., we conclude from Lemma 4.1.5 that u D e u q.e. As u differs from a quasi continuous function e u only by a set of zero capacity, u is quasi continuous. (ii) Theorem 2.1.3 specifically means that the function u coincides q.e. with a quasi continuous function in the restricted sense. The restriction that u 2 F in the first assertion of the above lemma will be removed in §4.6. The second assertion will be extended to u 2 Fe in transient case at the end of the next section. The next theorem asserts that the process M is properly associated with its Dirichlet form E in a sense. Theorem 4.2.3. For any non-negative universally measurable function u 2 L2 .XI m/, (i) p t u is a quasi continuous version of T t u; t > 0, (ii) R˛ u is a quasi continuous version of G˛ u; ˛ > 0. We prove this theorem utilizing the following variant of the monotone lemma.

4.2 Identification of potential theoretic notions

163

Lemma 4.2.3. Fix p  1: Let H be a family of non-negative functions on X such that (H.1) u1 ; u2 2 H; c1 u1 C c2 u2  0 for constants c1 ; c2 ) c1 u1 C c2 u2 2 H , (H.2) un 2 H; un " u 2 Lp .XI m/ ) u 2 H , (H.3) 8A open, 9un 2 H; un " 1A . Then H contains all non-negative Borel functions in Lp .XI m/. Proof. Fix a relatively compact open set E  X and put D D ¹A  E W A is Borel and 1A 2 H º. Then D is a Dynkin class containing all open subsets of E and D D B.E/.4 The rest of the proof is clear. Proof of Theorem 4:2:3. We already noticed that p t u is a version of T t u. Since Z 1 Z 1 ˛t .R˛ u; v/ D e .p t u; v/dt D e ˛t .T t u; v/dt 0

0

D .G˛ u; v/;

8v 2 C0 .X/;

R˛ u is a version of G˛ u 2 F . Furthermore, R˛ u is ˛-excessive and, consequently, finely continuous on X. Hence, we get (ii) by virtue of Lemma 4.2.2. We next prove (i) for non-negative u 2 C0 .X/. Then R˛ p t u is a quasi continuous version of G˛ T t u and lim˛!1 ˛R˛ p t u.x/ D lim˛!1 p t .˛R˛ /u.x/ D p t u.x/; 8x 2 X. On the other hand, ˛G˛ T t u is E1 -convergent to T t u 2 F on account of Lemma 1.3.3. Therefore, p t u is a quasi continuous version of T t u by virtue of Theorem 2.1.4 (ii). Let us put H D ¹u 2 L2 .XI m/ W u is non-negative Borel and satisfies (i)º. We have just proved H C0C .X/ and, hence, H satisfies (H.3) of Lemma 4.2.3. (H.1) is trivially satisfied. To check (H.2), let un 2 H increase to u 2 L2 .XI m/. Then T t un is E1 -convergent to T t u by (i) of Lemma 1.3.3. Since p t un is a quasi continuous version of T t un and convergent to p t u pointwise, we get u 2 H by Theorem 2.1.4 (ii) again. Lemma 4.2.3 implies that (i) holds for u Borel. The extension to the universally measurable u is clear. As a result of the regularity of the transition function embodied in Theorem 4.2.3, we can prove the following: Theorem 4.2.4. The equivalent two conditions of Theorem 4:1:2 are also equivalent to the following one: p t .x;  / is absolutely continuous with respect to m for each t > 0 and x 2 X. 4 Cf.

K. Itô [3].

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4 Analysis by symmetric Hunt processes

Proof. Since this condition implies the second one of Theorem 4.1.2, it is sufficient to derive this property of p t from the first condition (i) of Theorem 4.1.2. Assume that any exceptional set is polar. If A is an m-negligible Borel set, then p t .x; A/ D 0 m-a.e. by Lemma 4.1.1. Since p t .x; A/ is quasi continuous as a function of x 2 X by Theorem 4.2.3, it vanishes q.e. by Lemma 2.1.4. Hence, there exists a Borel polar set N such that p t .x; A/ D 0; 8x 2 X n N . Therefore, p2t .x; A/ D Ex .p t .X t ; A/I X t … N / D 0; 8x 2 X, and we obtain Theorem 4.2.4. We now amplify the first lemma of this section as follows. Let eB be the (1-)equilibrium potential of a set B introduced in Theorem 2.1.5. 1 Theorem 4.2.5. Let B be a nearly Borel set of finite capacity. Then pB is a quasi continuous version of eB .

Proof. For a nearly Borel set B of finite capacity, choose decreasing open sets An B such that limn!1 Cap.An / D Cap.B/. Then eAn is E1 -convergent to a function e0 2 F . Since e0 2 LB and E1 .e0 ; e0 / D Cap.B/, we have that e0 D eB . Hence 1 pB  eB m-a.e. .4:2:7/ 1 because pB  pA1 n ; n D 1; 2; : : : : 1 We first conclude from inequality (4.2.7) and Lemma 2.3.2 that pB 2 F. 1 1 Lemma 4.2.2 then implies that pB is quasi continuous. Furthermore, pB D 1 1 is a quasi continuous version q.e. on B because of Theorem 4.1.3. Therefore pB of eB by Theorem 2.3.1 (iii).

We state here a 0-order version of Theorem 4.2.3 (ii). We let  Z 1 u.X t /dt : Ru.x/ D Ex

.4:2:8/

0

Theorem 4.2.6. Assume in addition to (4.2.6) that .F R; E/ is transient. For any non-negative universally measurable function u with X uRud m < 1; Ru is a quasi continuous version of Gu .2 Fe /: Proof. In view of the proof of Theorem 1.5.3, there are non-negative functions un 2 L2 .XI m/ such that un " u and Gun 2 Fe are E-convergent to Gu 2 Fe . Moreover, by virtue of the proof of Theorem 1.5.4, Gun is the E-limit of G˛ un 2 F as ˛ # 0. Since Ru is the increasing limit of R˛ un as ˛ # 0 and n ! 1, Theorem 4.2.6 follows from Theorem 4.2.3 (ii) combined with the 0order version of Theorem 2.1.4 and Theorem 2.1.6.

4.2 Identification of potential theoretic notions

165

We say that M satisfies the absolute continuity condition if its transition function fulfills that p t .x;  / is absolutely continuous with respect to m for each t > 0 and x 2 X;

.4:2:9/

which just appeared in Theorem 4.2.4. This condition and its consequences stated below are often useful in strengthening “quasi everywhere” statements into “everywhere” ones. Exercise 4.2.1. Assume that ¹p t º is strongly Feller: p t .Bb .X/  Cb .X/. Show that M then satisfies the absolute continuity condition. Lemma 4.2.4. Assume the absolute continuity condition. There exists then a nonnegative jointly measurable function r˛ .x; y/; ˛ > 0; x; y 2 X, such that Z r˛ .x; y/f .y/m.dy/; x 2 X; f 2 Bb .X/; (4.2.10) R˛ f .x/ D X

(4.2.11) r˛ .x; y/ is ˛-excessive in x (and in y); r˛ .x; y/ D r˛ .y; x/; Z r˛ .x; y/ D rˇ .x; y/ C .ˇ  ˛/ r˛ .x; z/rˇ .z; y/m.dz/; ˇ > ˛ > 0: X

(4.2.12) Proof. We can find a non-negative jointly measurable function, say v˛x .y/, satisfying (4.2.10). Fixing ˛ > 0 and x 2 X, we regard v˛x .y/ as a function of y. By the symmetry of p t , we have e ˛t .p t v˛x ; f / D e ˛t .v˛x ; p t f / D e ˛t R˛ p t f .x/;

.4:2:13/

which is not greater than R˛ f .x/ D .v˛x ; f / for any non-negative f and hence e ˛t p t v˛x .y/  v˛x .y/ for m-a.e. y 2 X: By virtue of the absolute continuity of p t , we see then that e ˛t p t v˛x .y/ is, for every fixed y 2 X, non-decreasing as t # 0. It suffices to let r˛ .x; y/ D lim e ˛t p t v˛x .y/: t #0

Indeed r˛ .x; y/ is then ˛-excessive in y and, by letting t # 0 in (4.2.13), we get (4.2.10) for f 2 CbC .X/. Further r˛ .x; y/ is an increasing limit as t # 0 R y y of e ˛t X p t .y; dz/v˛x .z/ D R˛ g t .x/ where g t .z/ D e ˛t p t .y; dz/=m.dz/. y Since R˛ g t .x/ is ˛-excessive in x, so is r˛ .x; y/. The proof of (4.2.11) and (4.2.12) is now easy.

166

4 Analysis by symmetric Hunt processes

Exercise 4.2.2. Assume the absolute continuity condition and consider the resolvent density appearing in the lemma above. Show that, for a positive Radon measure  on X,  is of finite energy integral, i.e.,  2 S0 , if and only if Z r1 .x; y/.dx/.dy/ < 1: X X

Show also that in this case the function Z R˛ .x/ D r˛ .x; y/.dy/

.4:2:14/

X

is a quasi continuous and ˛-excessive version of the ˛-potential U˛  of ; ˛ > 0. We shall be also concerned with a condition on M slightly weaker than (4.2.9): p t .x;  / is absolutely continuous with respect to m for each t > 0 and x 2 X n N ;

.4:2:15/

where N is a certain Borel properly exceptional set. In other words, this condition says that there exists ˇa Borel properly exceptional set N  X such that the restricted Hunt process MˇX nN satisfies the absolute continuity condition (4.2.9). Theorem 4.2.7. Suppose the Dirichlet form E satisfies the Sobolev type inequality: for some q > 2 and a positive constant S; 2 kukL q .X Im/  S E1 .u; u/;

u2F:

.4:2:16/

Then M satisfies (4.2.15) for some Borel properly exceptional set N: Proof. Let ¹T t I t > 0º be the semigroup on L2 .XI m/ associated with the Dirichlet form E: It is known5 that the Sobolev type inequality (4.2.16) is equivalent to the inequality for the semigroup kT t f kL1 .X Im/  C e t t =2 kf kL1 .X Im/ ;

t > 0; f 2 L1 .XI m/; .4:2:17/

2q holding for a positive constant C and  D q2 : C Choose a countable subfamily H0 of C0 .X/ such that, for any open set A  X; there exists ¹fn º  H0 with fn " 1A ; n ! 1: On account of Theorem 2.1.2 and Theorem 4.2.3, we can find a regular nest ¹Fk º such that

¹p t f W f 2 H0 ; t 2 QC º  C1 .¹Fk º/: 5 Cf.

E. A. Carlen, S. Kusuoka and D. W. Stroock [1].

4.2 Identification of potential theoretic notions

167

F : Suppose (4.2.16), then we see from (4.2.17) that, for We let N D X n [1 kD1 k each x 2 X n N; the inequality Z t =2 p t f .x/  C e t f .x/m.dx/ .4:2:18/ X

holds true for any t 2 QC and any f 2 H0 : (4.2.18) extends to any t > 0; and then to any non-negative Borel f 2 L1 .XI m/ by Lemma 4.2.3, yielding the absolute continuity of p t .x;  / with respect to m for each t > 0 and x 2 X n N: So far we have fixed an m-symmetric Hunt process M on X whose Dirichlet form E on L2 .XI m/ is regular. Theorem 4.2.3 enables us to formulate a statement concerning the uniqueness of such a process. Let us say that two m-symmetric Hunt processes are equivalent if they possess a common properly exceptional set outside which their transition functions coincide. Theorem 4.2.8. Let M1 and M2 be two m-symmetric Hunt processes on X possessing a common regular Dirichlet space on L2 .XI m/. Then M1 and M2 are equivalent. .i /

Proof. Denote by p t the transition function of Mi ; i D 1; 2: By virtue of Theorem 4.2.3 and Lemma 2.1.5, there exists a set B0 of capacity zero such that .1/

.2/

p t u.x/ D p t u.x/;

8x 2 X n B0

.4:2:19/

for every t 2 QC and u 2 C1 .X/; C1 .X/ being a countable uniformly dense subset of C0 .X/. (4.2.19) then holds for every t > 0 by right continuity. Applying Theorem 4.1.1 and Theorem 4.2.1 to M1 and M2 alternatively, we can find an increasing sequence ¹Bn º of Borel sets containing B0 such that SB2n1 (resp. B2n ) is properly exceptional relative to M1 (resp. M2 ). Then B D 1 nD1 Bn is a common .1/ .2/ properly exceptional set outside which p t D p t ; 8t > 0. Example 4.2.1. Let M be the spatially homogeneous symmetric Markov process on Rd (see Example 4.1.1). Then the Dirichlet form of M on L2 .Rd / is given by (1.4.24) and M satisfies the condition (4.2.6) according to Example 1.4.1. The d -dimensional Brownian motion is a special example of M and its Dirichlet form is . 12 D; H 1 .Rd //, H 1 .Rd / being the Sobolev space of order 1. The Dirichlet form E of the transient symmetric stable process M of index ˛ is given by (1.4.24) with .x/ D cjxj˛ ; 0 < ˛  2; ˛ < d; c > 0. On account of Theorem 2.1.6 and Example 2.2.2, a set N  Rd is of zero capacity with respect to E1 if and only if the set N is of zero capacity based on the Riesz

168

4 Analysis by symmetric Hunt processes

kernel R.˛/ . Therefore Theorem 4.2.1 and Example 4.1.1 imply that a set N is of zero capacity based on the Riesz kernel if and only if N is polar with respect to the stable process M. Thus, the Brownian polar set is just the set of Newtonian capacity zero when d  3. When d D 2, the Brownian polar set is just the set of logarithmic capacity zero in view of Example 2.2.3.6 Since the Brownian paths X t are continuous (see the paragraph after Theorem 4.5.4), it follows from Theorem 4.2.2 that any quasi continuous function u on Rd has the following property: Px .u.X t / is continuous in t 2 Œ0; 1// D 1;

8x 2 Rd n N;

N being a certain Borel polar set depending on u. Theorem 4.2.4 is violated by the uniform motion to the right: it has the transition function (4.1.16) which is not absolutely continuous with respect to Lebesgue measure. But the resolvent R˛ .x; dy/ has the density ´ e ˛.xy/ x < y R˛ .x; y/ D 0 x  y:

4.3

Orthogonal projections and hitting distributions

We still consider the m-symmetric Hunt process M on X satisfying condition (4.2.6). Just as in §2.3 we set for a fixed nearly Borel set B  X FX nB D ¹u 2 F W e u D 0 q.e. on Bº;

.4:3:1/

e u being a quasi continuous version of u: FX nB is a closed subspace of the Hilbert space .F ; E˛ / for each ˛ > 0. Denote by HB˛ its orthogonal complement: F D FX nB ˚ HB˛ :

.4:3:2/

The orthogonal projection on the space HB˛ is denoted by PHB˛ . The ˛-order hitting distribution HB˛ has been defined by (4.1.5). In dealing with a function v defined and non-negative q.e. on X , we always assume that v is extended to be a non-negative universally measurable function on X. Then the operation HB˛ v.x/ makes sense for every x 2 X and HB˛ v.x/ D Ex .e ˛ B v.X B //. The key theorem of this section is the following: Theorem 4.3.1. For each u 2 F and ˛ > 0; HB˛ je uj.x/ is finite q.e. and HB˛e u is a quasi continuous version of PHB˛ u. 6 This

goes back to S. Kakutani [1] and J. L. Doob [1].

4.3 Orthogonal projections and hitting distributions

169

First we shall prove a lemma. Lemma 4.3.1. Let u 2 F be an ˛-excessive function with respect to ¹p t ; t > 0º. Denote by uB the ˛-reduced function of u on B defined in 2:3. Then HB˛ u is a quasi continuous version of uB . Proof. As we saw in the preceding section, u is quasi continuous and ˛-excessive with respect to ¹T t ; t > 0º. It suffices to show HB˛ u  uB HB˛ u.x/

m-a.e.;

(4.3.3)

D u.x/ q.e. on B:

(4.3.4)

In fact HB˛ u being ˛-excessive and finely continuous, we see from (4.3.3) and Lemma 2.3.2 that HB˛ u 2 F . Then Lemma 4.2.2 and (4.3.4) lead us to HB˛ u D HB˛ u D u D e u q.e. on B. Thus, we can conclude HB˛ u D uB by Theorem 2.3.1. Since (4.3.4) follows immediately from Theorem A.2.6 and Theorem 4.1.3, it only remains to verify the inequality (4.3.3). To prove (4.3.3) we take a non-negative Borel quasi continuous version e uB of uB . We claim that there exists a properly exceptional set N such that

A

uB .x/  e uB .x/; e ˛t p t e e uB .x/ D u.x/;

8t > 0; 8x 2 X n N;

(4.3.5)

8x 2 B n N;

(4.3.6)

uB .X t / is right continuous in t  0/ D 1; Px .e

x 2 X n N:

(4.3.7)

uB is quasi continuous Since uB is ˛-excessive with respect to ¹T t ; t > 0º and p t e by Theorem 4.2.3, we see that (4.3.5) holds q.e. x 2 X for each t > 0. By (2.3.4), (4.3.6) is true for q.e. x 2 B. Furthermore, Theorem 4.2.2 asserts the validity of (4.3.7) for q.e. x 2 X. According to Theorem 4.1.1, we can choose a properly exceptional set N such that the inequality (4.3.5) for every rational t > 0, the equality (4.3.6) and the relation (4.3.7) hold for all x 2 X n N . An application of Fatou’s lemma then yields (4.3.5) for every t > 0. Fix x 2 X n N . Then (4.3.5) and (4.3.7) imply that the stochastic process ˛t .e e uB .X t /; F t ; Px / is a right continuous non-negative supermartingale. The optional sampling theorem then gives HB˛e uB .x/ D Ex .e ˛ B e uB .X B //  e uB .x/. ˛ uB .x/ D HB˛ u.x/ from (4.3.6) and (4.3.7). We On the other hand, we have HB e get HB˛ u  e uB q.e. proving (4.3.3). Proof of Theorem 4:3:1. Suppose first that u 2 F is ˛-excessive with respect to ¹p t ; t > 0º. Then the reduced function uB of u on B coincides with PHB˛ u as we saw already in §2.3. Hence HB˛ u is a quasi continuous version of PHB˛ u by virtue of Lemma 4.3.1.

170

4 Analysis by symmetric Hunt processes

In dealing with a general non-negative u 2 F , we may assume that u is bounded because, otherwise, the approximation of u by the truncations u ^ n works. For u can be then expressed as a difference of two bounded ˛-exceseach ˇ > 0; Rˇ e u / is a quasi continuous version of sive functions. Hence, HB˛ .Rˇ e PHB˛ .Gˇ u/. On the other hand, since ˇGˇ u is E1 -convergent to u as ˇ ! 1 on account of Lemma 1.3.3, so is PHB˛ .ˇGˇ u/ to PHB˛ u. Furthermore, by choosing a properly exceptional set N such that the relation (4.2.3) holds for e u and u /.x/ D HB˛e u.x/; x 2 X n N . Due to x 2 X n N , we can get limˇ !1 HB˛ .ˇRˇ e Theorem 2.1.4, we conclude that HB˛e u is a quasi continuous version of PHB˛ u. It is not difficult to derive from Theorem 4.3.1 its 0-order counterpart. We assume the transience of .E; F /. Then its extended Dirichlet space Fe is a Hilbert space and admits, for a nearly Borel set B  X, the orthogonal decomposition Fe D Fe;X nB ˚ HB ; Fe;X nB D ¹u 2 Fe W e u D 0 q.e. on Bº:

(4.3.8)

The orthogonal projection on HB is denoted by PHB . The (0-order) hitting distribution is defined by HB .x; E/ D Px .X B 2 E I B < 1/: Then HB v.x/ D Ex .v.X B / I B < 1/: Theorem 4.3.2. Assume the transience of .E; F /. For each u 2 Fe ; HB je uj.x/ is u is a quasi continuous modification of PHB u. finite q.e. and HBe We prepare a lemma. Lemma 4.3.2. Assume the transience of .E; F /. If u 2 Fe is bounded, then HBe u is a quasi continuous element of Fe and there exist un 2 F and ˛n # 0 such that kHB˛ne un  HBe ukE ! 0;

kun  ukE ! 0;

n ! 1;

.4:3:9/

and un ; HB˛ne un / < 1: sup ˛n .HB˛ne

.4:3:10/

n

Proof. Suppose u 2 Fe satisfies juj  M for some constant M . Take un 2 F which is E-convergent to u. On account of the 0-order version of Theorem 1.4.2 and Theorem 2.1.4, we may assume that jun j  M and e un ! e u q.e.

4.3 Orthogonal projections and hitting distributions

171

Put Cn D .un ; un / and choose ˛n # 0 such that ˛n Cn  1. By virtue of Theorem 4.3.1, un ; HB˛ne un /  E˛n .un ; un /  E.un ; un / C 1: E˛n .HB˛ne Therefore both E.HB˛ne un ; HB˛ne un / and ˛n .HB˛ne un ; HB˛ne un / are uniformly un is E-conbounded. In particular, the Cesàro mean of a subsequence of HB˛ne ˛n vergent. But HB e un converges to HBe u q.e. by the bounded convergence theorem. Hence by the 0-order version of Theorem 2.1.4, we get the lemma by replacing un by the above Cesàro mean. Proof of Theorem 4:3:2. Suppose first that u 2 Fe is bounded. It suffices to show that u De u q.e. on B; HBe u; v/ D 0; E.HBe

8v 2 Fe;X nB :

(4.3.11) (4.3.12)

(4.3.11) is immediate from Theorem 4.1.3. To see (4.3.12), take un 2 F of Lemma 4.3.2 and observe that E˛n .HB˛ne un ; v/ D 0;

8v 2 FX nB :

Because of (4.3.10), q ˇ ˇ q q ˛n ˛n ˇ˛n .H ˛ne ˇ ˛n ˛n .HB e un ; HB e un / .v; v/ ! 0; B un ; v/ 

n ! 1;

and we get E.HBe u; v/ D 0;

8v 2 FX nB :

.4:3:13/

By Lemma 4.3.2 again, any v 2 Fe;X nB admits a sequence vn 2 F and ˛n # 0 such that vn  HB˛n vn 2 FX nB is E-convergent to v  HB v D v. Hence (4.3.13) extends to (4.3.12). For a general non-negative u 2 Fe ; un D u ^ n 2 Fe is E-convergent, HBe un 2 HB is E-convergent and HBe un " HBe u pointwise. Hence HBe u is finite q.e. and a quasi continuous modification of PHB u. We can now prove the 0-order counterpart of Theorem 4.2.5. The (0-order) hitting probability of a set B is given by pB .x/ D Px .B < 1/:

.4:3:14/

Theorem 4.3.3. Assume the transience of .E; F /. Let B be a nearly Borel set of .0/ finite 0-order capacity and eB be its 0-order equilibrium potential. Then pB is .0/

a quasi continuous modification of eB .

172

4 Analysis by symmetric Hunt processes .0/

.0/

Proof. By virtue of the 0-order version of Theorem 2.1.5, eB D PHB eB . Hence .0/ .0/ eB is a quasi continuous modification of eB by the preceding theorem. Since HBe .0/

.0/

eB is right continuous along the sample path (Theoe eB D 1 q.e. on B and e rem 4.2.2), .0/

eB .X B / D 1; B < 1/ D Px .B < 1/ q.e. Px .e .0/

Consequently HBe eB D pB q.e. Corollary 4.3.1. Assume the transience of .E; F /. (i) For a decreasing sequence ¹An º of open sets, lim Cap.0/ .An / D 0 if and only if

n!1

lim pAn .x/ D 0 q.e.

n!1

(ii) If u is quasi continuous and u 2 Fe , then u satisfies (4.2.3) with Œ0; / being replaced by Œ0; 1/. Proof. (i) follows from Theorem 4.3.3 and the transience inequality (1.5.8). (ii) In view of the 0-order version of Theorem 2.1.3, u is equal q.e. to a quasi continuous function in the restricted sense relative to Cap.0/ . We can then use (i) in a similar way as in the proof of Theorem 4.2.2 to get the strengthened version of (4.2.3). Theorem 4.3.2 will be extended to the general (not necessarily transient) case in §4.6.

4.4

Parts of forms and processes

We continue with the setting of the preceding section; M is an m-symmetric Hunt process on X whose Dirichlet form E on L2 .XI m/ is regular. X nB For ˛ > 0 and a nearly Borel set B  X , let us introduce the kernel R˛ by   Z B X nB ˛s e 1E .Xs /ds ; x 2 X; E 2 B.X/: .4:4:1/ R˛ .x; E/ D Ex 0

X nB

As we already saw in §4.1, ¹R˛ ; ˛ > 0º is an m-symmetric resolvent kernel on .X; B  .X// and the following Dynkin formula holds: R˛ f .x/ D R˛X nB f .x/ C HB˛ R˛ f .x/;

x 2 X; f 2 B C .X/:

.4:4:2/

4.4

Parts of forms and processes

173

For ˛ > 0 and f 2 B C .X/\L2 .XI m/, this formula expresses the orthogonal decomposition (4.3.2) of R˛ f 2 F by virtue of Theorem 4.3.1 and Theorem 4.2.3. X nB R0C ; R0C and HB0C are denoted by R; RX nB and HB respectively. Thus Rf .x/ D RX nB f .x/ C HB Rf .x/;

x 2 X; f 2 B C .X/: .4:4:3/ R If .E; F / is transient, then, for f 2 B C .X/ such that X f Rf d m < 1, formula (4.4.3) represents the orthogonal decomposition (4.3.8) of Rf 2 Fe on account of Theorem 4.3.2 and Theorem 4.2.6. In particular we have arrived at the following theorem. X nB

Theorem 4.4.1. (i) If ˛ > 0 and f 2 B C .X/\L2 .XI m/, then R˛ X nB continuous, R˛ f 2 FX nB and E˛ .R˛X nB f; v/ D .f; v/;

8v 2 FX nB :

f is quasi

.4:4:4/

R (ii) Suppose that .E; F / is transient. If f 2 B C .X/ satisfies X f Rf d m < 1, then RX nB f is quasi continuous, RX nB f 2 Fe;X nB and E.RX nB f; v/ D .f; v/;

8v 2 Fe;X nB :

.4:4:5/

The restriction of the Dirichlet form E to FX nB FX nB is denoted by .FX nB ; E/: FX nB is not necessarily dense in L2 .XI m/, but clearly .FX nB ; E/ is a Dirichlet form on L2 .XI m/ in the wide sense (see §1.4). According to Theorem 1.4.3, this is uniquely related to a Markovian resolvent on L2 .XI m/ which is not necessarily strongly continuous. Theorem 4.4.1 means that this resolvent is determined just X nB by ¹R˛ ; ˛ > 0º. Let us denote the set X n B by G. The set FG is a subspace of L2G .XI m/ D ¹u 2 L2 .XI m/ W u D 0 m-a.e. on Bº which can be identified with L2 .GI m/ in an obvious manner. Hence, the form .FG ; E/ can also be considered as a Dirichlet form on L2 .GI m/ in the wide sense. We denote this by EG and call it the part of the Dirichlet form E on the nearly Borel set G. The part MG of the Markov process M on the nearly Borel G has been introduced in §A.2 and, indeed, MG is a Markov process on .G; B.G// with an m-symmetric transition function p G t on .G; B.G// defined by (4.1.2) for A D B. But MG is not normal unless G is finely open. Theorem 4.4.2. Assume that m-almost all points of G are irregular for B D X nG: m.G \ B r / D 0:

.4:4:6/

174

4 Analysis by symmetric Hunt processes

Then EG is a Dirichlet form on L2 .GI m/. EG is associated with MG in the sense that the strongly continuous semigroup ¹T tG ; t > 0º on L2 .GI m/ corresponding to EG is determined by the transition function ¹p G t ; t > 0º of MG . r G Proof. Since lim t #0 p G t u.x/ D u.x/; 8x 2 G n B for any u 2 C0 .X/, ¹p t ; t > 0º determines a strongly continuous semigroup ¹T tG ; t > 0º on L2 .GI m/ on account of Lemma 1.4.3. Theorem 4.4.1 reads as follows:

R˛G f 2 FG

and

EG;˛ .R˛G f; v/ D .f; v/L2 .GIm/ ;

8v 2 FG ;

for any f 2 B C .X/ \ L2 .GI m/. Since the resolvent of MG is the restriction to .G; B.G// of ¹R˛G ; ˛ > 0º, this implies that EG is the Dirichlet form on L2 .GI m/ associated with the strongly continuous semigroup ¹T tG ; t > 0º. Condition (4.4.6) is satisfied if G is finely open q.e. (§4.6) and in particular G is open. When G is open, the part MG of M on G is a Hunt process on .G; B.G// (see A.2). The part EG of the form E on G also has a nicer property: Theorem 4.4.3. Let G be an open set. (i) The part EG of the Dirichlet form E on G is a regular Dirichlet form on L2 .GI m/. More specifically, for any special standard core C of E, the space CG defined by .2:3:12/ is a core of EG . (ii) A subset of G is of capacity zero with respect to EG if and only if it is of capacity zero with respect to E. A function is quasi continuous on G with respect to EG if and only if it is quasi continuous with respect to E. G C 2 (iii) p G t f is a quasi continuous version of T t f for any f 2 B .G/\L .GI m/.

Proof. (i) Let C be a special standard core of E. Clearly CG is uniformly dense in C0 .G/. CG is also E1 -dense in FG by Lemma 2.3.4. Hence EG is regular. (ii) Denote by CapG .A/ the 1-capacity of an open set A  G with respect to EG . Let ¹An º be a sequence of decreasing open subsets of G. It is enough to show the equivalence CapG .An / # 0 , Cap.An / # 0. The implication “)” is clear from the obvious inequality CapG .A/  Cap.A/ for open A  G. Since EG is the Dirichlet form of the process MG , the proof of Lemma 4.2.1 applies and we see that the 1-equilibrium potential eAG with respect to EG possesses as its version the (1-)hitting probability pAG .x/ D Ex .e  A I A < G / with respect to the process MG . If Cap.An / # 0, then pAn .x/ # 0 m-a.e. and, thus, pAGn .x/ # 0 m-a.e. Hence, CapG .An / D EG;1 .pAGn ; pAGn / ! 0, n ! 1. (iii) follows from (i), Theorem 4.4.2 and Theorem 4.2.3.

4.4

Parts of forms and processes

175

Theorem 4.4.4. Suppose .E; F / is transient and let G be a nearly Borel set satisfying .4:4:6/. Then the part EG of E on G is transient and its extended Dirichlet space equals .Fe;G ; E/. Proof. Let g be a reference function of the transient Dirichlet space .E; F /. Obviously EG is then transient with a reference function gG the restriction of g to G. By virtue of Theorem 4.4.1 (ii), we have for any f RD h  gG with non-negative 1 bounded measurable function h on G that RG f .D 0 p G t f dt / 2 Fe;G and E.RG f; v/ D .f; v/;

8v 2 Fe;G :

Therefore Theorem 1.5.5 implies that .Fe;G ; E/ is the extended Dirichlet space of EG . Exercise 4.4.1. Let E be a countable set with discrete topology and M D .X t ; Px / be an m-symmetric Hunt process on E associated with the regular Dirichlet form .E; F 0 / of Example 1.2.5. Denote by i the first exit time of M from i 2 E: i D inf¹t > 0 W X t … iº: Show that i is exponentially distributed under Pi with Ei Œi  D 

mi ; qi i m0i

i 2 E;

and, for each i 2 E; Pi .Xi D j / D 

qij ; qi i

j ¤ i;

Pi .Xi D / D 

X qij ki D1C : qi i qi i j ¤i

For the sake of later use, we finally consider a general right process M on X which is m-symmetric. The associated Dirichlet form on L2 .XI m/ is denoted by .E; F / whose regularity is not assumed however. The part MG of M on a nearly Borel set G has the resolvent ¹R˛G ; ˛ > 0º defined by (4.4.1) for B D X n G: We show that the form E still serves to describe MG : We use the term ‘q.e.’ to mean ‘except for an exceptional set’ for M in the sense of §4.1. Theorem 4.4.5. (i) Assume that G is either an open set or the fine interior of a nearly Borel finely closed set. The transition function of MG is then m-symmetric and generates a strongly continuous Markovian semigroup on L2 .GI m/: The bG / of MG on L2 .GI m/ satisfies the following: F bG  Dirichlet form .b EG; f b b b F and E G D E on F G  F G : S (ii) Let ¹Gn º be an increasing sequence of subsets of X as in (i) such that 1 nD1 Gn S1 b D X q.e. Then nD1 F Gn is E1 -dense in F .

176

4 Analysis by symmetric Hunt processes

Proof. It is enough to show for a set G as in (i) that R˛G .C0 .X//  F ;

E˛ .R˛G f; R˛G g/ D .R˛G f; g/;

f; g 2 C0 .X/:

.4:4:7/

Indeed this implies the m-symmetry of the transition function of MG ; which in turn generates a strongly continuous Markovian semigroup on L2 .GI m/ as Theorem 4.4.2. The right-hand side of (4.4.7) equals b E G;˛ .R˛G f;R˛G g/ and R˛G .C0 .X// b G yielding the conclusion in (i). Furthermore, for a sequence ¹Gn º is dense in F as in (ii) and for any f 2 C0 .X/, E˛ .R˛ f  R˛Gn f; R˛ f  R˛Gn f / D .f; R˛ f /  .f; R˛Gn f / ! 0;

n ! 1;

because R˛Gn f ! R˛ f q.e. as n ! 1; due to the quasi-left-continuity of X: This means that (ii) is true. We first prove (4.4.7) with G replaced by a nearly Borel finely closed set F . We let E D X n F and, for a fixed f 2 C0 .X/,  Z 1 Rt E ˛t  0 1E .Xs /ds R˛ f .x/ D Ex e f .X t /dt : 0

E

We have then R˛F f .x/ D lim !1 R˛ f .x/ because E is finely open and E D Rt inf¹t > 0 W 0 1E .Xs /ds > 0º: Consequently E.R˛F f; R˛F f / D lim

lim ˇ.R˛E f  ˇRˇ R˛E f; R˛F f /:

ˇ !1  !1

.4:4:8/

This identity holds in the sense that, if the limit in the right-hand side is finite, then R˛F f 2 F and the limit coincides with the left-hand side. To compute the right-hand side, we utilize the generalized resolvent equation Rˇ f  R˛E f C .ˇ  ˛/Rˇ R˛E f  Rˇ .1E  R˛E f / D 0:

.4:4:9/

This equation for ˇ D ˛ implies that R˛E f 2 F ; E˛ .R˛E f; v/ C .R˛E f; v/1E m D .f; v/:

.4:4:10/

E

In particular, R˛ is m-symmetric. Keeping this in mind, we can rewrite the righthand side of (4.4.8) as lim

lim ˇ.Rˇ f  ˛Rˇ R˛E f  Rˇ .1E  R˛E f /; R˛F f /

ˇ !1  !1

D .f; R˛F f /  ˛.R˛F f; R˛F f /  lim

lim .f; R˛E .1E  ˇRˇ R˛F f //:

ˇ !1  !1

4.4

Parts of forms and processes

177

If we introduce the random functional  t by ² ³ Z s 1E .Xv /dv > t ;  t D inf s > 0 W 0

Z

then R˛E .1E

 g/.x/ D Ex

1

e

˛ t  t

 g.X t /dt ;

0

which tends to Ex .e ˛0 g.X0 // as ! 1 provided that g is bounded and finely continuous. Since 0 D E and Px .X0 2 E r / D 1 by Lemma A.2.7 and further R˛F f .x/ D 0 for x 2 E r , we get that lim

lim .f; R˛E .1E  ˇRˇ R˛F f // D Ef m .e ˛0 R˛F f .X0 // D 0:

ˇ !1  !1

Thus we arrive at (4.4.7) for G replaced by a nearly Borel finely closed set F . In the same way, we also have the relation 0

E˛ .R˛F f; R˛F f / D .f; R˛F f / if F and F 0 are nearly Borel finely closed and F  F 0 . Assume now that G is the fine interior of a nearly Borel finely closed set F : G D ¹x 2 X W Px .E > 0/ D 1º; E D F c : Then G is a nearly Borel finely open set and B.D G c / D E r ; which readily implies that Px .B D E / D 1; x 2 X; in view of Lemma A.2.7. Therefore (4.4.7) for G and that for F are the same. Take finally any open set G and a sequence ¹Fn º of closed sets increasing to G. Denoting R˛Fn f by R˛n f , we get, for m < n; E˛ .R˛n f  R˛m f; R˛n f  R˛m f / D .f; R˛n f /  .f; R˛m f /, which means that ¹R˛n f º is E˛ -Cauchy. Since limn!1 R˛n f .x/ D R˛G f .x/ by the quasi-left-continuity of X, we see that (4.4.7) is valid for the open set G: Exercise 4.4.2. Show the equation (4.4.9). Example 4.4.1. Let M be the d -dimensional Brownian motion. According to Example 1.4.1, the Dirichlet form E of M is given by ( 12 D; H 1 .Rd //. Let D be a domain in Rd . According to the identity (2.3.19) in Example 2.3.1, the part on D of the Dirichlet form ( 12 D; H 1 .Rd // is nothing but the form ( 12 D; H01 .D// on L2 .D/. By virtue of Theorem 4.4.3, the form ( 12 D; H01 .D// is the Dirichlet form of the part of M on the domain D, which is often called the absorbing Brownian motion on D. We can also conclude from (2.3.18) and Theorem 4.3.1 that a non-negative u D HR˛ d nDe u. function u 2 H 1 .Rd / is ˛-harmonic on D if and only if e

178

4 Analysis by symmetric Hunt processes

Example 4.4.2. Consider the half-space D  Rd , i.e., D D ¹x D .x 0 ; xd / W x 0 2 Rd 1 ; xd > 0º. We set @D D ¹x D .x 0 ; xd / W x 0 2 Rd 1 ; xd D 0º and D D D [ @D. The Sobolev space ( 12 D; H 1 .D// is a Dirichlet space on L2 .D/ which is not regular (see Example 1.2.3). However if we regard this as a Dirichlet space on L2 .D/ rather than on L2 .D/, then this is regular as was seen in Example 1.5.3 and Example 1.6.2. b D .X b t ; Px / The associated diffusion M x2D on D is the reflecting Brownian b motion; i.e., M is obtained from the Brownian motion M D .X t ; Px /x2Rd on Rd by setting b t D .X t0 ; jX t.d / j/; where X t D .X t0 ; X t.d / /: .4:4:11/ X b is expressed by the b˛ of M To verify this assertion, observe that the resolvent R resolvent R˛ of M as follows: b˛ f .x/ D R˛ f1 .x/ C R˛ f2 .x/; R

x 2 D; f 2 C01 .D/:

.4:4:12/

Here f1 is the extension of f by setting f1 .x/ D 0; x 2 Rd n D: f2 is defined by f2 .x/ D 0 for x 2 D and f2 .x 0 ; xd / D f .x 0 ; xd / for xd  0. It is clear b˛ f 2 H 1 .D/; .˛  .1=2//R b˛ f .x/ D f .x/; x 2 D, and from this that R b limxd #0 .@=@xd /R˛ f .x/ D 0. An integration by parts then gives 1 b˛ f; v/ D .f; v/; b˛ f; v/ C ˛.R D .R 2

v 2 H 1 .D/;

.4:4:13/

where the domain of integration in each term is the half-space D. Equation (4.4.13) b (on L2 .D/) is just ( 1 D; H 1 .D/). implies that the Dirichlet space of M 2 1 1 Since . 2 D; H .D/) is regular on L2 .D/, each function u 2 H 1 .D/ admits a modification e u on D which is quasi continuous on D, or, equivalently, finely b Then we continuous q.e. on D with respect to the reflecting Brownian motion M. can see in the same way as in Example 2.3.1 that H01 .D/ D ¹u 2 H 1 .D/ W e u D 0 q.e. on @Dº:

.4:4:14/

Compare this with (1.2.16). We further see by Theorem 4.3.1 that u 2 H 1 .D/ is ˛ e u. ˛-harmonic if and only if e u D H@D

4.5 Continuity, killing, and jumps of sample paths Just as in the preceding two sections we consider the m-symmetric Hunt process M on X whose Dirichlet space .E; F / on L2 .XI m/ is regular. Recall that the Dirichlet space .E; F / is said to possess the local property if E.u; v/ vanishes whenever u; v 2 F have disjoint compact supports.

179

4.5 Continuity, killing, and jumps of sample paths

Lemma 4.5.1. The following two conditions are equivalent to each other: (i) .E; F / possesses the local property. (ii) For each relatively compact open set A and ˛ > 0, the hitting distribution HX˛nA .x;  / is concentrated on the boundary @A for q.e. x 2 A. Proof. Denote by P the projection operator on the space u D 0 q.e. on X n Aº FA D ¹u 2 F W e in the Hilbert space .F ; E˛ /. By virtue of Theorem 4.3.1, we have for non-negative u 2 F that u is a quasi continuous version of Pu: e u  HX˛nAe

.4:5:1/

Suppose that E possesses the local property. Take a non-negative u 2 F \ C0 .X/ such that suppŒu  X n A. Since Pv D v and suppŒv  A for any v 2 FA , we get E˛ .Pu; v/ D E˛ .u; v/ D 0; 8v 2 FA , which together with (4.5.1) implies HX˛nAe u.x/ D 0 q.e. on A. From this and Lemma 1.4.2, we conclude that HX˛nA .x; X n @A/ D 0 q.e. x 2 A. Conversely, assume condition (ii) and take u; v 2 F with disjoint compact supports. Without loss of generality, we may suppose that u is non-negative. Let A be a relatively compact open set such that suppŒv  A; suppŒu  X n A. Then E.u; v/ D E˛ .u; v/ D E˛ .Pu; v/ which vanishes because Pu D 0 in view of (4.5.1) and the assumption (ii). We call a Hunt process on X a diffusion if Px .X t is continuous in t 2 Œ0; // D 1

.4:5:2/

for every x 2 X. In the paragraph preceding to Theorem 4.2.7, we have defined the equivalence of two m-symmetric Hunt processes on X. Theorem 4.5.1. The following conditions are equivalent to each other: (i) .E; F / possesses the local property. (ii) M is of continuous sample paths for q.e. starting point, i.e., there exists a properly exceptional set N such that .4:5:2/ holds for every x 2 X n N . (iii) There exists an m-symmetric diffusion on X which is equivalent to M. Proof. Obviously, (ii) implies the second condition of Lemma 4.5.1. Furthermore, (ii) is equivalent to (iii) in view of Theorems A.2.8 and A.2.9. Hence, it only remains to prove the implication (i))(ii).

180

4 Analysis by symmetric Hunt processes

Assume the local property of .E; F / and let O be a countable open base of X . We may assume that each A 2 O is relatively compact. By virtue of Lemma 4.5.1 and Theorem 4.1.1, there exists a properly exceptional set N  X such that HX˛nA .x; X n @A/ D 0; 8x 2 A n N;

.4:5:3/

for every A 2 O; ˛ > 0 being fixed. We notice that the relation (4.5.3) then holds for every ˛ > 0 because of ˇ

HX nA .x; X n @A/ D HX˛nA .x; X n @A/ C .˛  ˇ/ Z  RˇA .x; dy/HX˛nA .y; X n @A/; AnN

x 2 A:

Let us put d D ¹! 2  W 9t 2 .0; .!//; X t  .!/ ¤ X t .!/º. On account of the sample path property (M.6), [ [ ¹! 2  W 0 < PX nA .s !/ < 1; X PXnA .s !/ .s !/ 2 X n @Aº: d D A2O s2QC

For each x 2 X n N , the Px -measure of the event inside the braces equals Ex .PXs .X nA < 1; X XnA 2 X n @A/I Xs 2 A/ D Ex .HX0C .Xs ; X n @A/I Xs 2 A n N / D 0: nA Hence (4.5.2) holds. Having completely characterized the sample continuity up to the life time  by means of the local property of the Dirichlet form, our next task is to study the location X  .2 X / of the sample path right before the life time . When X  2 X with positive probability, we say that the killing takes place inside X. In the following we occasionally denote the integral of a function v with a positive measure  by h; vi or hv; i: Z h; vi D hv; i D v.x/.dx/: .4:5:4/ X

Lemma 4.5.2. (i) There exists a unique positive Radon measure k on X charging no exceptional set such that for any u 2 Fe Z 1 lim u.x/2 .1  p t 1.x//m.dx/ D he u2 ; ki; .4:5:5/ t #0 t X e u being any quasi continuous version of u.

4.5 Continuity, killing, and jumps of sample paths

181

(ii) For any f; h 2 B C .X/, and t > 0, Z Ehm .f .X  /I   t/ D

0

t

hf k; ps hids:

.4:5:6/

(iii) 1K  k is of finite energy integral for each compact K. For ˛ > 0 and f 2 C0C .X/, the function Ex .e ˛ f .X  // is a quasi continuous version of the potential U˛ .f  k/. Proof. We first note that the formula Z 1 .u.x/  u.y//2 p t .x; dy/m.dx/ 2t X X Z 1 C u.x/2 .1  p t 1.x//m.dx/ " E.u; u/; t X .4:5:7/ for t # 0, holds for any Borel function u 2 Fe . In fact, the left-hand side equals the approximating form E .t / .u; u/ when u 2 F . (4.5.7) then extends to any u 2 Fe as in Theorem 1.5.2 (ii) where the resolvent rather than the semigroup was used. Hence we can subtract a sequence tn # 0 such that the measure .1=tn /.1  p tn 1.x//m.dx/ converges vaguely to a positive Radon measure k on X and .4:5:8/ he u2 ; ki  E.u; u/; u 2 F \ C0 .X/: In particular, 1K  k is of finite energy integral for any compact set K and, hence, k is smooth (see §2.2). k charges no exceptional set on account of Theorem 4.2.1. Using Theorem 2.1.7, we easily see that the inequality (4.5.8) holds for any u 2 Fe . We can then show that the relation (4.5.5) holds for any u 2 Fe as long as the limit on the left-hand side is taken along the above sequence tn . Indeed, p choosing for any " > 0 a function v 2 F \ C0 .X/ such that E.u  v; u  v/ < ", it follows from (4.5.7) and (4.5.8) that ˇs ˇ q ˇ 1 ˇ 2 2 ˇ ˇ hu ; .1  p 1/  mi he u ; ki t n ˇ ˇ t n ˇ ˇs q ˇ ˇ 1 2 2 ˇ hv ; .1  p tn 1/  mi  hv ; kiˇˇ C 2" ˇ tn which is not greater then 3" for sufficiently small tn . Next let us show the relation (4.5.6) for the measure k determined above. We O hO 2 C C .X/. Notice that h 2 first do this for f 2 C0C .X/ and h D ˛R˛ h; 0

182

4 Analysis by symmetric Hunt processes

D.A/ \ L1 .XI m/; A being the infinitesimal generator of the L2 -semigroup T t determined by p t . We then have Ehm .f .X  /I   t/ D lim Ehm

=tn   ŒtX

n!1

f .X.k1/tn /  1¹.k1/tn < k tn º



kD1

D lim

ŒtX =tn 

n!1

Ehm .f .X.k1/tn /.1  p tn 1.X.k1/tn ///

kD1

D lim

ŒtX =tn 

.p.k1/tn h; f .1  p tn 1//;

n!1 kD1

tn being the preceding sequence tending to zero. Choose ı > 0 such that kps h  hkL2  2skAhkL2 for any s < ı. For tn < ı ˇ ˇ ŒtX Z ˇ ˇ =tn  1 t ˇ .p.k1/tn h; f .1  p tn 1//  .ps h; f .1  p tn 1//ds ˇˇ ˇ tn 0 kD1



ŒtX =tn  kD1

C

1 tn

1 tn Z

Z

k tn

.k1/tn

j.p.k1/tn h  ps h; f .1  p tn 1//jds

t

t Œt =tn 

j.ps h; f .1  p tn 1//jds

 2t kAhkL2  kf .1  pı 1/kL2 C khkL2 kf .1  pı 1/kL2 ; which decreases to zero as ı # 0. Therefore, Ehm .f .X  /I   t / D limn!1 t1n  Rt Rt 0 .ps h; f .1  p tn 1//ds D 0 hps h  f; kids proving (4.5.6) for the present f and h. Letting ˛ tend to infinity and then using the monotone lemma, we know that (4.5.6) holds for any f; h 2 B C .X/. Taking the Laplace transform of both sides of (4.5.6), we readily see that the function w.x/ D Ex .e ˛ f .X  // is a version of the potential U˛ .f  k/ for f 2 C0C .X/. Since w is ˛-excessive, it is quasi continuous by Lemma 4.2.2, completing the proof of the third statement (iii). In particular, the Radon measure k is uniquely determined by the process M independently of the choice of ¹tn º. Hence, the proof of (i) is also complete. We now identify the measure k in the above theorem with the killing measure (associated with the form E) appearing in the Beurling–Deny formula.

4.5 Continuity, killing, and jumps of sample paths

183

Lemma 4.5.3. The measure k in Lemma 4:5:2 coincides with the killing measure in Theorem 3:2:1. Proof. Denote by e k the killing measure in the statement of Theorem 3.2.1. In accordance with the construction of e k there, consider a sequence of relatively comk was constructed in pact open sets Gl increasing to X. The difficulty lies in that e two steps and, in fact, e k is a vague limit of kl determined by (3.2.9). However, a similar reasoning to the first part of the proof of Lemma 4.5.2 applies yielding that lim he u2 ; kl i D he u2 ; e ki .4:5:9/ l!1

for any u 2 F with compact support. Let E .l/ and M.l/ be the parts on the open set Gl of the Dirichlet form E and the Hunt process M respectively. By virtue of Theorem 4.4.3, we can apply Lemma 4.5.2 to M.l/ and E .l/ getting a positive Radon measure kl0 on Gl such that for any u 2 FGl 1 lim t #0 t

Z 2

Gl

u.x/ .1 

.l/ p t 1.x//m.dx/

Z D

Gl

e u.x/2 kl0 .dx/

.4:5:10/

and for any f; h 2 C0C .X/ and sufficiently large l Z Ehm .f .Xl  /I l  t/ D

t 0

hf  kl0 ; ps.l/ hids;

.4:5:11/

.l/

where l D X nGl ^  and p t is the transition function of M.l/ . Comparing (4.5.10) with (3.2.9), we have kl0 .E/  kl .E/; E  Gl . Hence, the right-hand side of (4.5.11) is not less than Z 0

Z

t

kl ; ps.l/ hids

hf 



t 0

hf  kl ; ps.l0 / hids;

l > l0 :

Since the left-hand side of (4.5.11) tends to Ehm .f .X  /I   t /, we get from (4.5.6) and (4.5.9) the inequality 1 t

Z 0

t

1 hf  k; ps hids  t

Z 0

t

hf  e k; ps.l0 / hids:

k; hi. The After letting l0 ! 1 and then t # 0, we arrive at hf  k; hi  hf  e converse inequality is obvious and, consequently, k D e k.

184

4 Analysis by symmetric Hunt processes

Finally, we are concerned with the jumping measure J appearing in the Beurling–Deny formula: J is the unique positive Radon measure on X  X n d such that for any u; v 2 F \ C0 .X/ with disjoint support, Z 2 u.x/v.y/J.dxdy/ D E.u; v/: .4:5:12/ Lemma 4.5.4. (i) J charges no part of X  X n d whose projection on the factor X is exceptional. (ii) The formula (3.2.1) extends to any u; v 2 Fe : Z E.u; v/ D E .c/ .u; v/ C .e u.x/  e u.y//.e v .x/  e v .y//J.dxdy/ X X nd

Z C

X

e u.x/e v .x/k.dx/;

(4.5.13)

where e u and e v denote quasi continuous versions and E .c/ is a symmetric form .c/ with DŒE  D Fe satisfying for any u; v 2 Fe of compact support E .c/ .u; v/ D 0 if v is constant on a neighbourhood of suppŒu: .4:5:14/ Proof. Consider a relatively compact open set G and a non-negative v 2 F \ C0 .X/ with R suppŒv  X n G. Then (4.5.12) implies that the measure Jv .dx/ D 2  1G .x/ v.y/J.dxdy/ is of finite energy integral with respect to the part EG of the Dirichlet form E on the set G. Z p ju.x/jJv .dx/  C E.u; u/; 8u 2 FG \ C0 .G/: .4:5:15/ G

Hence Jv charges no set of zero capacity with respect to EG and, consequently, no exceptional set in view of Theorem 4.4.3. This together with Lemma 4.5.2 implies that the integrals in (4.5.13) are independent of the choice of the versions e u and e v . Let us define E .c/ by the equation (4.5.13). By virtue of Theorem 2.1.7 and Theorem 3.2.1, E .c/ then becomes a symmetric form. Its property (3.2.2) extends to (4.5.14) by working with the perturbed form .E g ; Feg / of Lemma 1.6.7 as in the proof of Theorem 3.1.2 and utilizing the expression v D .v  v0 / C v0 with v0 2 F \ C0 .X/ such that v0 D v D constant on suppŒu. We can derive the following identity from formula (4.5.12) and Theorem 4.4.1. For G; v and Jv as in the above proof v  H˛X nG v D U˛G Jv ;

˛ > 0;

.4:5:16/

4.5 Continuity, killing, and jumps of sample paths

185

where U˛G Jv is the ˛-potential of Jv with respect to the form EG . In particular, for any non-negative Borel h vanishing outside G Z ˛G v.XG // D R˛G h.x/v.y/J.dxdy/; .4:5:17/ Ehm .e G being the first leaving time from G W G D X nG ^ &. This formula is strengthened in the next lemma. Lemma 4.5.5. Let G be a relatively compact open set. (i) For any h; f; g 2 BbC .X/ such that suppŒh  G; suppŒf   G and suppŒg  X n G, Ehm .f .XG  /g.XG /I G  t/ Z t Z D2 psG h.x/f .x/g.y/J.dxdy/ ds:

.4:5:18/

0

(ii) The function Ex .e ˛G f .XG  /g.XG // on G is a quasi continuous version of the potential U˛G .fJg / for ˛ > 0 and f; g 2 C0C .X/ with suppŒf   G, suppŒg  X n G. Proof. The proof proceeds similar to the proof of Lemma 4.5.2. In fact, when h satisfies the condition in the proof of Lemma 4.5.2, we see that the left-hand side of Rt PŒt =ı G h; f pı g/ D limı#0 1ı 0 .psG h; f pı g/ds, (4.5.18) equals limı#0 kD1 .p.k1/ı 2 2 2 2 2 where we use the estimate kf  pı gkL 2  .f ; pı g /!ı#0 .f ; g / D 0. The last limit is equal to the right-hand side of (4.5.18) because the preceding lemma implies Z 1 2 e u.x/e v .y/J.dxdy/ D E.u; v/ D lim .u; ps v/ s#0 s for any u; v 2 F with compact disjoint support. Summing up Lemmata 4.5.2–4.5.5, we can formulate a refinement of Theorem 3.2.1 as follows: Theorem 4.5.2. The Dirichlet form E.u; v/; u; v 2 Fe , admits a unique representation .4:5:13/ with E .c/ ; J and k possessing the following properties: The local part E .c/ is a symmetric form with DŒE .c/  D Fe and has the strong local property .4:5:14/.

186

4 Analysis by symmetric Hunt processes

The jumping measure J is a unique positive Radon measure on X  X n d satisfying .4:5:12/. J indicates the jumps of the sample paths in the sense of the identity .4:5:18/. The killing measure k is a unique positive Radon measure satisfying .4:5:5/. k indicates the killing of the sample paths inside X in the sense of the identity .4:5:6/. Combining this theorem with Lemma 4.5.2 (iii) and Theorem 4.5.1, we get Theorem 4.5.3. The following conditions are equivalent to each other: (i) E possesses the strong local property .4:5:14/. (ii) E possesses the local property and k D 0. (iii) M is of continuous sample paths and with no killing inside for q.e. starting points, i.e., there exists a properly exceptional set N such that .4:5:2/ and Px .X  2 X;  < 1/ D 0

.4:5:19/

hold for every x 2 X n N . (iv) M admits an equivalent m-symmetric diffusion on X satisfying .4:5:19/ for every x 2 X. Let us define the resurrected form E res by ´ R E res .u; v/ D E.u; v/  X e u.x/e v .x/k.dx/ DŒE res  D Fe : By (4.5.13), E res is a symmetric form expressible as Z res .c/ u.x/  e u.y//.e v .x/  e v .y//J.dxdy/; E .u; v/ D E .u; v/ C .e

.4:5:20/

u; v 2 Fe :

.4:5:21/ On the other hand, the relation (4.5.7) implies that every normal contraction operates on E res and Z 1 1 2 res e u.x/2 k.dx/  E.u; u/ lim Em ..u.X t /  u.X0 // / D E .u; u/ C 2 X t #0 2t .4:5:22/ 7 for any Borel u 2 Fe . This formula plays an important role in the next chapter. E is called conservative if the associated semigroup T t on L2 .XI m/ satisfies T t 1 D 1; 8t > 0. 7 Recall

our convention that u./ D 0.

4.5 Continuity, killing, and jumps of sample paths

187

Exercise 4.5.1. Prove the equivalence of the next conditions. (i) E is conservative. (ii) Px . < 1/ D 0 for q.e. x 2 X:

(4.5.23)

(iii) M admits an equivalent m-symmetric Hunt process on X which is conservative in the sense that (4.5.23) holds for every x 2 X . If M satisfies the extra condition of absolute continuity (4.2.9), then any ˛-excessive function vanishing m-a.e. is identically zero .˛  0/. This combined with Exercise 4.2.2 enables us to strengthen several assertions obtained so far into “everywhere” assertions as follows. Theorem 4.5.4. Assume the absolute continuity condition for M. (i) For ˛ > 0 and f 2 C0C .X/ Ex .e ˛ f .X  // D R˛ .f  k/.x/;

x 2 X;

.4:5:24/

where the right side is defined by .4:2:14/. (ii) .E; F / possesses the local property if and only if M is a diffusion. (iii) .E; F / possesses the strong local property if and only if M is a diffusion and Px .X  2 X;  < 1/ D 0;

8x 2 X:

.4:5:25/

(iv) .E; F / is conservative if and only if M is conservative. In particular, the d -dimensional Brownian motion formulated in Example 4.1.1 and stated in Example 4.2.1 is a conservative diffusion on Rd in virtue of this theorem. Example 4.5.1 (one-dimensional diffusion). Let I D .r1 ; r2 / be a one-dimensional interval and let m and k be positive Radon measures on I with suppŒm D I . We consider an m-symmetric Hunt process M on I whose Dirichlet space on L2 .XI m/ is given by .F 0 ; E/ of Example 1.2.2. As we saw in Example 2.1.2, no non-empty subset of I is of capacity zero. Accordingly any non-empty subset of I is neither exceptional nor polar with respect to the process M (Theorem 4.2.1). In particular M satisfies the absolute continuity condition (4.2.9) in virtue of Theorem 4.2.4. From the expression (2.1.19) of the 1-equilibrium potential p.x/ of a one-point set ¹yº, we know that p.x/ is strictly positive. Hence Theorem 4.2.5 implies that Px .y < 1/ > 0;

8x; y 2 I;

.4:5:26/

namely, any two points have the communication with respect to the process M.

188

4 Analysis by symmetric Hunt processes

Since E possesses the local property, we see by either Theorem 4.5.1 or Theorem 4.5.4 that M is a diffusion process on I with possible killing inside I indicated by the killing measure k. When k D 0 and m D Lebesgue measure on I , then .F 0 ; E/ D .H01 .I /; 12 D/ and, consequently, M is the absorbing Brownian motion on the interval I in view of Example 4.4.1. Finally we give two examples of multidimensional symmetric diffusions satisfying the absolute continuity condition (4.2.9). Example 4.5.2 (a diffusion associated with a second order differential operator of divergence form). The Dirichlet form associated with a second order differential P operator di;j D1 @x@ .aij .x/ @x@ / has been considered already in §1.6, §3.1 and i

j

§3.3. We consider a family ¹aij .x/º1i;j d of Borel functions on Rd satisfying aij .x/ D aj i .x/; 1  i; j  d , and d d d X X 1X 2 i  aij .x/i j   i2 ;  iD1

i;j D1

8 2 Rd ; 8x 2 Rd

.4:5:27/

F D H 1 .Rd /;

.4:5:28/

iD1

for some constant  > 1. Then E.u; v/ D

d Z X i;j D1

Rd

aij .x/

@u @v dx; @xi @xj

is a regular Dirichlet form on the space L2 .Rd / based on the Lebesgue measure. By invoking the well known works on PDE,8 one can see that the above Dirichlet form admits a Markovian resolvent ¹R˛ º˛>0 on L2 .Rd / which further satisfies the following properties:9 R˛ .L2 .Rd / \ Lp .Rd //  C1 .Rd /; R˛ .C1 .Rd //

p > d;

is a dense subset of C1 .Rd /:

(4.5.29) (4.5.30)

By (4.5.30) and the Hille–Yosida theorem,10 ¹R˛ º˛>0 produces a strongly continuous Markovian semigroup on C1 .Rd /, and accordingly E admits a Feller transition function p t and an associated Hunt process M on Rd (Theorem A.2.2). Since R˛ .x;  / is absolutely continuous with respect to the Lebesgue measure in view of (4.5.29), p t .x;  / has the same property by Theorem 4.2.4. 8 Cf.

G. Stampacchia [2]. H. Kunita [2], M. Tomisaki [2]. 10 Cf. K. Yosida [1]. 9 Cf.

4.6

189

Quasi notions, fine notions and global properties

Obviously E has the local property. As we shall see in Example 5.7.1, E is also conservative. Hence Theorem 4.5.4 implies that M is a conservative diffusion on Rd . Example 4.5.3 (reflecting Brownian motion on a Lipschitz domain). Let D be a bounded domain of Rd which is assumed to be Lipschitz in the sense that each boundary point x admits a neighbourhood U such that D \ U is expressible as in Example 1.6.1by a Lipschitz continuous function f . According to Example 1.6.1, the form E D 12 D; H 1 .D/ is then a regular Dirichlet form on L2 .D/. It is further known thatE admits  a transition function p t on D which is strong Feller in the sense that p t Bb .D/  C.D/ and lim t #0 p t u.x/ D u.x/, 8x 2 D, 8u 2 C.D/.11 There exists a Hunt process M on D with the absolutely continuous transition function p t (Theorem A.2.2). Clearly E possesses the local property. It is also conservative (see Example 1.6.1 and Lemma 1.6.5). Hence M is a conservative diffusion on D by virtue of Theorem 4.5.4. M is called the reflecting Brownian motion on D.

4.6 Quasi notions, fine notions and global properties We maintain the preceding setting: M is an m-symmetric Hunt process whose Dirichlet form E on L2 .XI m/ is regular. In §4.2, we have studied the relation of quasi continuity and q.e. fine continuity of functions. We can now complete this relationship. For two subsets E1 ; E2 of X, we write E1  E2

q.e.

.4:6:1/

if E1 n E2 is exceptional or, equivalently, Cap.E1 n E2 / D 0 (see Theorem 4.2.1). Thus E1 D E2 q.e. iff the symmetric difference E1 E2 is exceptional. In this case, E1 and E2 are said to be q.e. equivalent. Obviously the notion of quasi open sets and quasi closed sets introduced in §2.1 has the class property in the sense of q.e. equivalence. A set E is called q.e. finely open if E is q.e. equivalent to a nearly Borel finely open set. Theorem 4.6.1. (i) A set E is quasi open if and only if E is q.e. finely open. (ii) A numerical function u is quasi continuous if and only if u is finite q.e. and u is q.e. finely continuous. 11 Cf.

R. F. Bass and P. Hsu [2].

190

4 Analysis by symmetric Hunt processes

Proof. (i) If E is quasi open, there is a nest ¹Fk º such that E \ Fk is relatively for every k. Let Ak D X nFk and choose a properly exceptional set N open on FkT containing 1 kD1 Ak such that (4.2.5) holds. Then E n N is Borel and finely open. e Conversely suppose that E is q.e. equivalent to a nearly Borel finely open set E. 2 Take a strictly positive bounded Borel function f 2 L .XI m/ and let Z  Xne E s e f .Xs /ds ; x 2 X: .4:6:2/ v.x/ D Ex 0

Then v is quasi continuous by Theorem 4.4.1 and E D v 1 .0; 1/ q.e. Hence E is quasi open. (ii) Suppose u is finite q.e. and q.e. finely continuous, then, for any open I  R, u1 .I / is q.e. finely open and consequently quasi open by (i). Hence u is quasi continuous. The converse was proven in Theorem 4.2.2. The second subject of this section is about quasi supports of measures. We prepare a simple lemma. Lemma 4.6.1. A set F is quasi closed if and only if there exists a non-negative quasi continuous function u in F with F D u1 .¹0º/ q.e. Proof. The “if” part is evident. Conversely, given a quasi closed set F , we can e being replaced take as a required function u the function v of (4.6.2) with X n E e q.e. equivalent to F in accordance with by a nearly Borel finely closed set F Theorem 4.6.1. Let us consider a positive Borel measure  on X charging no set of zero cae is said to be a quasi support of  if the next two conditions are pacity. A set F satisfied: e is quasi closed and .X n F e / D 0; F e  FL q.e. if FL is another set with property (4.6.3), then F

(4.6.3) (4.6.4)

e of  is unique up to the q.e. equivalence. Let F D suppΠThe quasi support F be the topological support of . Since any closed set is quasi closed, we have that e  F q.e., and by deleting a set of zero capacity from F e if necessary, we can F e  F. always assume that F Theorem 4.6.2. The following three conditions are equivalent to each other for any Borel measure  on X charging no set of zero capacity and for any quasi closed set F  X:

4.6

Quasi notions, fine notions and global properties

191

(i) F is a quasi support of . (ii) u D 0 -a.e. on X if and only if u D 0 q.e. on F for any quasi continuous u2F. (iii) Condition (ii) holds for any quasi continuous function u. Proof. (ii) ) (i): We set ² ³ Z N D u 2 F W je ujd D 0 ;

(4.6.5)

FF c D ¹u 2 F W e u D 0 q.e. on F º

(4.6.6)

X

and assume that N D FF c . Take a function u of Lemma 4.6.1. Then u 2 FF c and hence u 2 N , which means .X n F / D 0. Consider another quasi closed set F1 with .X n F1 / D 0 and take again a function u1 satisfying the condition of Lemma 4.6.1 for F1 . Then u1 2 N and hence u1 2 FF c , which implies that F  F1 q.e., proving that F is a quasi support of . (i) ) (iii): Suppose F is a quasi support of , then the “if” part of condition (ii) is clearly satisfied for any Borel function u. If u is quasi continuous and u D 0 -a.e., then the set FL D ¹u D 0º has the property (4.6.3) and hence F  FL q.e. and u D 0 q.e. on F . The implication (iii) ) (ii) is trivial. In view of Lemma 2.1.4, we get Corollary 4.6.1. If  D f  m for some measurable function f strictly positive m-a.e. on X , then  has the full quasi support X . Exercise 4.6.1. Suppose that F is a Borel quasi closed set,  is a positive Radon measure charging no set of zero capacity, .X n F / D 0 and the 1-order hitting distribution HF1 .x;  / is absolutely continuous with respect to  for m-a.e. x 2 X. Show that F is a quasi support of . Recall the class S of smooth measures introduced in §2.2. Each  2 S charges no set of zero capacity. Theorem 4.6.3. Any  2 S admits a quasi support. Proof. For any  2 S , the space N defined by (4.6.5) is a closed subspace of the closed sets separable Hilbert S space .E1 ; F / as 1Fn   2 S0 for some increasing e is continuously embedded into L1 .XI 1F  / Fn with .X n n Fn / D 0 and F n

192

4 Analysis by symmetric Hunt processes

for each n by Theorem 2.2.2. Choose a countable dense subcollection ¹uk º of N and let 1 X je uk j.x/ 1 F D v .¹0º/ for v.x/ D 2k : kuk kE1 kD1

Since the series defining v is E1 -convergent, v is quasi continuous and v 2 N . Hence F is quasi closed and .X n F / D 0. Accordingly we arrive at the identity N D FF c for FF c defined by (4.6.6) for this set F . By virtue of Theorem 4.6.2, we can conclude that F is a quasi support of . In §5.1, we shall identify the quasi support of  2 S with the support of the positive continuous additive functional A associated with . Recall Theorem 4.4.2 stating that the part EG of E on an open set G  X is a regular Dirichlet form on L2 .GI m/ and it is associated with the part MG of M on G which is a Hunt process. Exercise 4.6.2. Consider  2 S with the quasi support F: Let G be a open subset of X: ˇ (i) Show that G \ F is a quasi support of ˇG relative to the Dirichlet form EG : (ii) Suppose a quasi continuous function u on G vanishes -a.e. on G: Show that u D 0 q.e. on G \ F: We turn to the study of the ergodic decomposition of the state space X into M-invariant sets. The quasi notions work well in this respect. Recall that an mmeasurable set A is said to be T t -invariant if T t .1A f / D 1A T t f m-a.e. for any f 2 L2 .XI m/ and t > 0 (§1.6). Given two m-measurable sets A1 and A2 , we say that A1 is a modification of A2 if m.A1 A2 / D 0. Lemma 4.6.2. Any T t -invariant set A admits a modification which is simultaneously quasi open and quasi closed. Proof. If A is T t -invariant, then 1A  u 2 F for any u 2 F by Theorem 1.6.1. Therefore 1A possesses a quasi continuous modification, say ', on each relatively compact open set and hence on the whole space X . Since ' 2 D ' q.e., ' D 0 or 1 q.e., and there is a nearly Borel set e A with ' D 1 e A q.e. A set B  X has been called M-invariant if B is nearly Borel and Px .X t 2 B ; X t  2 B ; 8t  0/ D 1;

8x 2 B:

Obviously any M-invariant set is T t -invariant. The Hunt process M is said to be a diffusion if X t is continuous in t 2 Œ0; / Px -a.s. for any x 2 X. M admits an

4.6

Quasi notions, fine notions and global properties

193

equivalent diffusion if and only if the Dirichlet form E possesses the local property (Theorem 4.5.1). Lemma 4.6.3. Suppose that a set B is simultaneously quasi open and quasi closed. We further assume that one of the following two conditions is satisfied: B is T t -invariant;

(4.6.7)

M is a diffusion:

(4.6.8)

Then there is a properly exceptional set N such that both B n N and .X n B/ n N are M-invariant. Proof. Let B be quasi open and quasi closed. Then 1B is quasi continuous and by virtue of Theorem 4.2.2 there is a properly exceptional set N such that 1B is nearly Borel on X n N and, for any x 2 X n N ,   Px 1B .X t / is right continuous and lim 1B .X t 0 / D 1B .X t  /; 8t 2 Œ0; / D 1; t 0 "t



(4.6.9)



Px lim 1B .X t 0 / D 1B .X  /; X  2 X D Px .X  2 X/: t 0 "

(4.6.10)

Assume first condition (4.6.8). Then 1B .X t / is continuous in t 2 Œ0; / Px -a.s. for x 2 X n N . Consequently Px .X t 2 B; 8t 2 Œ0; // D 1;

8x 2 B n N;

Px .X  2 B/ D Px .X  2 X/;

8x 2 B n N

and the same equations hold with B being replaced by X n B. Therefore both B n N and .X n B/ n N are M-invariant. Next assume (4.6.7) instead of (4.6.8). In view of Lemmas 4.2.3 and 2.1.4, p t .1B u/ D 1B  p t u q.e.;

8t > 0; 8u 2 C0 .X/:

By taking un 2 C0C .X/ increasing to 1, we get p t 1B D 1B  p t 1 q.e. for each t > 0, which in conjunction with (4.6.9) and (4.6.10) implies that B n N and .X n B/ n N are M-invariant for an appropriate properly exceptional set N . Here are two immediate consequences of the preceding two lemmas. Corollary 4.6.2. The following conditions are equivalent to each other for an mmeasurable set B:

194

4 Analysis by symmetric Hunt processes

(i) B is T t -invariant, (ii) there is a nearly Borel modification B1 (resp. B2 ) of B (resp. X n B) such that both B1 and B2 are M-invariant, X D B1 C B2 C N

.disjoint sum/

and m.N / D 0. Exercise 4.6.3. Suppose ¹p t º is strongly Feller. Show that, if the state space X is connected, then M is irreducible. Corollary 4.6.3. Suppose that the Dirichlet form E has the local property. Then the following conditions are equivalent to each other for an m-measurable set B: (i) B is T t -invariant, (ii) B admits a modification which is quasi open and quasi closed simultaneously. The Dirichlet form E or the associated semigroup T t is said to be irreducible if any T t -invariant m-measurable set B is m-trivial in the sense that either B or B c is m-negligible. We may say by Corollary 4.6.3 and Lemma 2.1.7 that, under the condition of the local property of E, E is irreducible if and only if the space X is quasi-connected in the sense that, if B is simultaneously quasi open and quasi closed, then either B or B c is of zero capacity. Theorem 4.6.4 (comparison of irreducibility). Consider two regular Dirichlet forms E .i / on L2 .XI m.i / /; i D 1; 2. We suppose that E .1/ possesses the local property. We further assume that m.2/ is absolutely continuous with respect to m.1/ ;

(4.6.11)

E .2/ -quasi continuity implies E .1/ -quasi continuity:

(4.6.12)

Then the irreducibility of E .1/ implies the same property of E .2/ . Proof. By Theorem 7.2.2, we may assume that a diffusion process is associated .i / with E .1/ . We denote by T t the semigroup on L2 .XI m.i / / associated with E .i / , .2/ i D 1; 2. Let B be a T t -invariant m.2/ -measurable subset of X. Then 1 e B is .2/ .2/ e with m .B B/ e D 0 by virtue of E -quasi continuous for some Borel set B .1/ Lemma 4.6.2. By assumption (4.6.12), 1 e -quasi continuous and hence B is E .1/ .1/ e is m.1/ -trivial and T t -invariant by Corollary 4.6.3. If E is irreducible, then B .2/ consequently B is m -trivial by assumption (4.6.11) proving the irreducibility of E .2/ .

4.6

Quasi notions, fine notions and global properties .1/

.2/

is locally dominated by E1

The condition (4.6.12) is fulfilled if E1 formulated below.

195 as is

Corollary 4.6.4. Consider two Dirichlet forms E .i / on L2 .XI m.i / / possessing a common special standard core C. We suppose that E .1/ has the local property. We further assume that there are open sets Gn increasing to X and satisfying the next conditions: d m.2/ D d m.1/

with ess inf .x/ > 0; n D 1; 2; : : : ; x2Gn

E .2/ .u; u/  n E .1/ .u; u/;

8u 2 CGn

(4.6.13) (4.6.14)

where n is a positive constant and CGn is defined by (2.3.12) for each n. Then the irreducibility of E .1/ implies the same property of E .2/ . Proof. A function is quasi continuous on X if and only if it is so on each Gn . Therefore (4.6.12) follows from (4.6.13) and (4.6.14) by virtue of Theorem 4.4.3.

Example 4.6.1. Let X be a domain D of Rd and m be an everywhere dense positive Radon measure on D. We consider a Dirichlet form E on L2 .DI m/ possessing C01 .D/ as its core and admitting the expression (1.2.1) on C01 .D/. Denote by V the Lebesgue measure. We assume the following coerciveness condition: d X i;j D1

1 i j ij .K/  K jj2 V .K/ 2

for any compact set K  D and  2 Rd , where K is a positive constant. Then 1 E.u; u/  EN D.u; u/; 2

u 2 C01 .E/

for any relatively compact open subset E of D. We further assume that d m D dV

with inf .x/ > 0 K

for any compact K  D. Since the Sobolev space . 12 D; H01 .D// of Example 1.2.3 is irreducible, E is also irreducible in virtue of Corollary 4.6.4.

196

4 Analysis by symmetric Hunt processes

Returning to the previous setting that M is an m-symmetric Hunt process on X whose Dirichlet form E on L2 .XI m/ is regular, we consider a nearly Borel set B  X. We are interested in a specific ergodic decomposition of X related to the hitting probability of B; we put Y D ¹x 2 X W Px .B < 1/ > 0º:

.4:6:15/

Lemma 4.6.4. There is a properly exceptional set N such that both Y n N and .X n Y / n N are M-invariant. Proof. It suffices to prove that Px .Y < 1/ D 0;

8x 2 X n Y:

.4:6:16/

Indeed (4.6.16) implies that Y is T t -invariant by Lemma 1.6.1 and that X n Y is finely open. Since Y is also finely open, we see by Theorem 4.6.1 that Y is simultaneously quasi open and quasi closed and hence Lemma 4.6.3 is applicable. To prove (4.6.16), we introduce the finely closed set ² ³ 1 Yn D x 2 X W Px .B < 1/  : n Then, for x 2 X n Y , Px .Yn < 1/ D Px .B D 1; Yn < 1/ D Ex .PXY .B D 1/ I B D 1; Yn < 1/ n   1 Px .Yn < 1/;  1 n which means Px .Yn < 1/ D 0. We arrive at (4.6.16) by letting n ! 1. e the M-invariant set Y n N and by G the fine interior of the Let us denote by Y e set Y n B. Accordingly e n B W Px .B > 0/ D 1º: G D ¹x 2 Y

.4:6:17/

Owing to Theorem 4.4.2, we see that the part EG of the Dirichlet form E on the set G is a Dirichlet form on L2 .GI m/ and associated with the part MG of M on e n B, the set G. If we denote by R˛G the resolvent of MG , then, for any x 2 Y   Z B ^ G t R1 1.x/ D Ex e dt  1  Ex .e  B / < 1: 0

Therefore we get the next lemma by virtue of Lemma 1.6.5.

4.6

Quasi notions, fine notions and global properties

197

Lemma 4.6.5. The part EG of E on the set G is a transient Dirichlet form on L2 .GI m/. By making use of the preceding two lemmas, we can prove a useful property of the extended Dirichlet space Fe for the Dirichlet form E. We do not assume the transience of E. Nevertheless we know from Theorem 2.1.7 that any u 2 Fe admits a quasi continuous modification, which we shall denote by e u. The next lemma is a key step to extend Theorem 4.3.2 to the present general case. uj.x/ is finite m-a.e. Lemma 4.6.6. For any u 2 Fe , HB je Proof. We may assume by Corollary 1.6.3 that u 2 Fe is non-negative. Since HBe u.x/ D e u.x/ < 1 q.e. x 2 B [ B r by Theorem 4.1.3 and HBe u.x/ D 0, 8x 2 X n Y , it suffices to show that u.x/ < 1 HBe

m-a.e. on G:

.4:6:18/

Since EG is transient, it admits a reference functionR g on G: g is a strictly positive bounded m-integrable function on G such that G g  RRG gd m  1. By replacing g with g=.u _ 1/ if necessary, we may suppose that G gud m < 1. Recall that the domain of EG is given by FG D ¹v 2 F W e v D 0 q.e. on X n Gº: We let un D u ^ n. For a fixed n, we can take on account of Corollary 1.6.3 and Theorem 2.1.7 a sequence of functions vk 2 F such that 0  vk  n, ¹vk º is E-Cauchy and limk!1 e vk D e un q.e. In view of Theorem 1.6.1 and Theo.v  HB˛e v k / 2 FG and rem 4.3.1, we have that 1 e Y k Z

G

˛ gjvk  HB˛e v k jd m D E˛ .R˛G g; 1 e Y jvk  HB vk j/



1=2

Z G

gR˛G gd m

 1=2 E˛ vk  HB˛e v k ; vk  HB˛e vk  E˛ .vk ; vk /1=2 :

By letting ˛ # 0 and k ! 1, we get Z gjun  HBe un jd m  E.un ; un /1=2  E.u; u/1=2 : G

Hence

Z

Z G

g  HBe un d m 

G

g  un d m C E.u; u/1=2

198

4 Analysis by symmetric Hunt processes

and consequently Z G

Z g  HBe ud m 

G

g  ud m C E.u; u/1=2 < 1;

proving (4.6.18). e e the collection of all quasi continuous functions in Fe . We further Denote by F set Fe;X nB D ¹u 2 Fe W e u D 0 q.e. on Bº: We are now in a position to extend Theorem 4.3.2 from the transient case to the present general case in the following manner. ee Theorem 4.6.5. For any u 2 Fe and any nearly Borel set B  X , HBe u 2 F and E.HBe u; v/ D 0; 8v 2 Fe;X nB . We prove this by a method of perturbation embodied in Lemma 1.6.7. Let us fix a function g in the class K of (1.6.14) for a moment and consider the associated perturbed Dirichlet form .E g ; F / on L2 .XI m/. We also consider the canonical subprocess Mg D .X tg ;  g ; Px / of M relative to the multiplicative functional e C t where Z t Ct D g.Xs /ds: .4:6:19/ 0

According to Theorem A.2.11, Mg is a Hunt process on X with transition function p gt f .x/ D Ex .e C t f .X t //;

x 2 X:

.4:6:20/

Lemma 4.6.7. p gt is an m-symmetric transition function on X whose associated Dirichlet form on L2 .XI m/ coincides with E g . g g Proof. Denote by R˛ f the Laplace transform of p t f :

Z R˛g f .x/ D Ex

1

 e ˛t C t f .X t /dt ;

0

x 2 X:

We have then the well-known formula R˛g f  R˛ f C R˛ .g  R˛g f / D 0:

.4:6:21/

4.6

Quasi notions, fine notions and global properties

199

Rt In fact, using the identity e C t  1 D 0 e Cs g.Xs /ds and the Markov property, we have for a bounded Borel f  Z 1 g ˛t C t C t e e .e  1/f .X t /dt R˛ f .x/R˛ f .x/ D Ex Z D Ex

e 1

e

Z

˛s

Z

e

˛s

0

e

EXs

  D R˛ g  R˛g f .x/;

˛t C t

e

 f .X t /dt

s 1

e 0

1

1

g.Xs /ds

0

Z D Ex

Cs

0

Z D Ex

0

1

˛t C t ıs

Z

e

1

e

 f .X t ı s /dt g.Xs /ds

˛t C t

e







f .X t /dt g.Xs /ds

0

x 2 X:

(4.6.21) implies that, for any bounded Borel f 2 L2 .XI m/,   R˛g f 2 F ; E˛g R˛g f; v D .f; v/; 8v 2 F ;

.4:6:22/

which readily extends to any f 2 L2 .XI m/, as was to be proved. E g is a transient regular Dirichlet form and shares the quasi notion in common   with E because E1g and E1 are equivalent metrics on F . Let Feg ; E g be the extended Dirichlet space of .E g ; F /. It is a Hilbert space and admits an orthogonal decomposition g Feg D Fe;X ˚ HBg ; nB

g Fe;X D ¹u 2 Feg W e u D 0 q.e. on Bº: nB

The orthogonal projection operator P g on HBg is given by P g u.x/ D HBge u.x/ and HBge u.x/ D Ex .exp.C B /e u.X B //; because u.x/ D Ex .e u.X gB // D Ex .e u.X B /I C B < Z/ D Ex .exp.C B /e u.X B // HBge where Z is a relevant exponentially distributed random variable independent of X t (see A.2). The 0-order resolvent of the part Mg;X nB of Mg on X n B is given by   Z B g;X nB C t R f .x/ D Ex e f .X t /dt : 0

Lemma 4.6.8. (i) For any non-negative Borel w HB w.x/  HBg w.x/ D Rg;X nB .g  HB w/.x/ q.e.

.4:6:23/

200

4 Analysis by symmetric Hunt processes

(ii) If w 2 Feg and w is bounded, then HB w e 2 Feg and E.HB w e; v/ D 0; 8v 2 g . Further E.H w e ; H w e /  E.w; w/. Fe;X B B nB Proof. (i) The right-hand side of (4.6.23) equals  Z B  Z 1  C t C t Ex e.X B // dC t e HB w e.X t /dC t D Ex e 1Œ0; B  .t /EX t . w 0

Z

0



0

D Ex

1

e

C t

 1Œ0; B  .t /e w .X B . t !//dC t

Z

D Ex w e.X B /

B

e

C t

 dC t

0

e.X B /.1  e CB //: D Ex . w (ii) Since g is a reference function for .F g ; E g /, we know from Theorem 4.4.1 g that, for any bounded Borel h, Rg;X nB .h  g/ 2 Fe;X and nB E g .Rg;X nB .h  g/; v/ D .h; v/gm ;

g 8v 2 Fe;X : nB

If w 2 Feg and w is bounded, then (i) implies that HB w e 2 Feg and e; v/ D E g .HB w e; v/  .HB w e; v/gm E.HB w D .HB w e; v/gm  .HB w e; v/gm D 0;

g 8v 2 Fe;X : nB

g Since w  HB w e 2 Fe;X , we further obtain nB

e; HB w e/ C E.w  HB w e; w  HB w e /  E.HB w e; HB w e /: E.w; w/ D E.HB w Proof of Theorem 4:6:5. For arbitrarily fixed u 2 Fe and v 2 Fe;X nB , we can g find g 2 K with u; v 2 Feg owing to Lemma 1.6.7. Then v 2 Fe;X . Applying nB g C Lemma 4.6.8 to the truncated functions un D .0 _ u/ ^ n 2 Fe , we get that g HBe uC n 2 Fe and uC E.HBe n ; v/ D 0; C C uC uC kHBe n  HB e m kE  kun  um kE ! 0;

n; m ! 0:

uC uC which is finite m-a.e. owing to Lemma 4.6.6, Since HBe n converges to HB e C u 2 Fe ; kHBe uC uC kE ! 0 and E.HBe uC ; v/ D 0. we conclude that HBe n  HB e The same is true for HBe u . u is quasi continuous. We use Lemma 1.6.7 again It remains to prove that HBe g2 to find g1 with HBe uC 2 Feg1 . If we let g2 D g1 ^ g, then HBe uC n 2 Fe are

4.7 Irreducible recurrence and ergodicity

201

uC 2 L2 .XI g2  m/. Working in the transient extended E g2 -convergent to HBe g 2 uC . Of course, Dirichlet space .F ; E g2 /, we can see the quasi continuity of HBe  the same is true for HBe u . Exercise 4.6.4. Suppose that E is recurrent and that, for u 2 Fe , E.u; u/ D 0 if and only if u D constant. Let B be a nearly Borel set of positive capacity. We put HB D ¹u 2 Fe W E.u; v/ D 0; 8v 2 Fe;X nB º. Show that any u 2 Fe can be expressed uniquely as u D u 0 C u1 ;

u0 2 Fe;X nB ; u1 2 HB ;

u in this case. and e u1 D HBe

4.7

Irreducible recurrence and ergodicity

Throughout this section, we assume that X is a locally compact separable metric space, m is a positive Radon measure on X with full support and M is an m-symmetric Borel right process (a normal strong Markov process with right continuous sample paths and Borel transition function). We denote by .E; F / the Dirichlet form of M on L2 .XI m/ but we do not assume its regularity. The transition function and the resolvent of M are denoted by ¹p t I t  0º and ¹R˛ I ˛ > 0º, respectively, and the L2 -semigroup associated with E is denoted by ¹T t I t > 0º: In §1.5, each T t was extended from L2 \ L1 to a linear operator on L1 .XI m/ satisfying (1.5.2). ¹T t I t > 0º is then a contraction semigroup on L1 .XI m/: We now verify that it is strongly continuous: kT t f  f kL1 .X Im/ ! 0;

t # 0; 8f 2 L1 .XI m/:

On account of (1.5.2), it suffices to show this for any f 2 C0 .X/: Let K be the support of f; G be a relatively compact open set containing K and g be a nonnegative function which equals 1 on X n G and 0 on K: Then Z Z kT t f  f kL1 .X Im/  jp t f .x/  f .x/jm.dx/ C p t jf j.x/m.dx/: G

X nG

The first integral on the Rright-hand side tends to 0 as t # 0; while the second integral is dominated by K p t g.x/jf .x/jm.dx/ which tends to 0 as t # 0: We note that, under the present setting, the concept of an exceptional set introduced in §4.1 still makes sense for M and we use the term ‘q.e.’ to mean ‘except for an exceptional set’. Lemma 4.1.4 remains valid so that, if a finely continuous function for M is non-negative m-a.e., then so it is q.e. A nearly Borel measurable

202

4 Analysis by symmetric Hunt processes

set N  X is said to be m-inessential if m.N / D 0 and X nN is M-invariant in the sense that Px .X t 2 .X n N / [ ¹º; 8t  0/ D 1 for any x 2 X n N: Analogously to Theorem 4.1.1, any exceptional set is contained in some Borel m-inessential set.12 We also note that, if for ˛  0 an ˛-excessive function u is finite m-a.e., then so it is q.e. Indeed, the set N D ¹x 2 X W u.x/ D 1º is then m-negligible and, for x 2 X n N; Ex .e ˛ N u.X N //  u.x/ < 1 so that Px .N < 1/ D 0 and hence N is m-inessential. Theorem 4.7.1. (i) If E is irreducible, then, for any nearly Borel non-exceptional set B, .4:7:1/ Px .B < 1/ > 0; for q.e. x 2 X: (ii) If E is recurrent, then any bounded excessive function u satisfies p t u.x/ D u.x/;

8t > 0; 8x 2 X n N;

.4:7:2/

for some Borel m-inessential set N: (iii) Assume that E is irreducible recurrent. Then, for any nearly Borel nonexceptional set B, Px .B ı n < 1; 8n  0/ D 1;

for q.e. x 2 X:

.4:7:3/

Moreover any excessive function is constant q.e. Proof. (i) If we let Y D ¹x 2 X W Px .B < 1/ > 0º, then m.Y / > 0 by the assumption. On the other hand, (4.6.16) says that X n Y is M-invariant and hence ¹T t º-invariant by Lemma 1.6.1. Therefore m.X n Y / D 0 by the irreducibility assumption. Since X n Y is finely open, it must be exceptional by Lemma 4.1.4. (ii) For a strictly positive function v 2 L1 .XI m/; we have .Rˇ v; u  ˛R˛ u/ D .v; Rˇ u  ˛Rˇ R˛ u/ D .v; R˛ u  ˇR˛ Rˇ u/  .v; R˛ u/ < 1: By letting ˇ ! 0; we get from Lemma 1.6.4 (ii) that ˛R˛ u D u m-a.e. and consequently q.e. Since p t u.x/ is right continuous in t > 0; we get (4.7.2). (iii) We let f .x/ D Px .B < 1/. Then the set E D ¹x 2 X W f .x/ D 1º is T t -invariant. Indeed (4.7.2) implies that f .X t / is a Px -martingale for each x 2 X n N . For any 2 Œ0; 1/, denote by  the hitting time of the finely closed 12 Cf.

Getoor–Sharpe [1].

4.7 Irreducible recurrence and ergodicity

203

set E D ¹x 2 X W f .x/  º. Then, by the optional sampling theorem, we have for any x 2 E n N and T > 0 1 D f .x/ D Ex .f .X  ^T // D Ex .f .X  /I   T / C Ex .f .XT /I  > T /  Px .  T / C Px . > T /; which means that Px .  T / D 0. Hence Px .E c < 1/ D 0;

8x 2 E n N;

.4:7:4/

and E must be T t -invariant in view of Lemma 1.6.1. The set E n N is finely open by (4.7.4) and contains the non-exceptional set B \B r nN: Therefore using Lemma 4.1.4 and the irreducibility again, we conclude that m.X n E/ D 0 and consequently Px .B < 1/ D 1 q.e. by Lemma 4.1.5. (4.7.3) then follows from the Markov property. Finally we take an arbitrary excessive function f . If f were not constant q.e., then we can find 0 < ˛ < ˇ < 1 such that the sets E˛ D ¹x 2 X W f .x/  ˛º and Eˇ D ¹x 2 X W f .x/  ˇº are both non-exceptional. Let g.x/ D f .x/ ^ ˇ, x 2 X. Then g is a bounded excessive function and E˛ D ¹g  ˛º; Eˇ D ¹g D ˇº: By (4.7.3) and the fine continuity of g   Px lim g.X t /  ˛ < ˇ D lim g.X t / D 1 q.e. t !1

t !1

contradicting the fact that g.X t / is a bounded martingale by (4.7.2) and possesses a limit as t ! 1 Px -a.s. for q.e. x 2 X. We note that the last statement of the above theorem holds true for any excessive ˇ function of the restricted Borel right process MˇX nN for any m-inessential set N: Exercise 4.7.1. Show that, under the extra condition of the absolute continuity (4.2.9) for M, the phrase ‘for q.e. x 2 X’ in (4.7.1) and (4.7.3) can be replaced by the stronger one ‘for every x 2 X’. In the rest of this section, we assume that the Dirichlet form E of a given m-symmetric Borel right process M D .X t ; Px / is irreducible recurrent. By Lemma 1.6.5, E is then conservative and the m-symmetry of M implies that m is an invariant measure of ¹p t I t > 0º: Z p t .x; B/m.dx/ D m.B/; 8t > 0; 8B 2 B.X/; .4:7:5/ X

204

4 Analysis by symmetric Hunt processes

0 ; P / induced from which is equivalent to the stationarity of the process .; F1 m the Hunt process M D .; M; X t ; Px / in the following sense:

Pm . t1 ƒ/ D Pm .ƒ/;

0 ƒ 2 F1 :

.4:7:6/

0 is the -field defined by (A.2.1),  is the translation operator (shift) on Here F1 t R 0 : P is a -finite measure on  (§A.2) and Pm D X Px .ƒ/m.dx/; ƒ 2 F1 m 0 .; F1 / but not necessarily a finite measure. Denote by F m the Pm -completion 0 : We say that a statement holds P -a.s. if it holds for all ! 2  n ƒ for of F1 m 0 some ƒ0 2 F m with Pm .ƒ0 / D 0:

Lemma 4.7.1. Suppose u 2 L1C .XI m/ is ¹T t º-invariant, namely, T t u D u; 8t > 0: Then u is constant. Proof. Let u be an m-integrable non-negative Borel function such that p t u .D T t u/ D u; m-a.e. for every t > 0: It suffices to show that u is a constant m-a.e. Using the resolvent of M; define b u.x/ D R1 u.x/; x 2 X: Since ¹T t I t > 0º is a strongly continuous contraction semigroup on L1 .XI m/ it holds then that Z 1 Z 1 b uD e s Ts uds; p t b uD e s T t Cs uds; t > 0; 0

0

where the integrals are taken in Bochner’s sense. By the assumption, we have u Db u m-a.e., t > 0: b u D u m-a.e. p t b u are 1-excessive, they are finely continuous. Further Since both b u and e t p t b u.x/ is right continuous in t  0 for each x 2 X: Therefore there exists an ptb m-inessentialˇ set N such that p t b u.x/ D b u.x/; 8t > 0; 8x 2 X n N: This ˇ means that b u X nN is an excessive function of the Borel right process MX nN : By Theorem 4.7.1 (iii), we conclude that b u D c q.e. for some constant c and u D c m-a.e. Theorem 4.7.2. The stationary process .; F m ; Pm / is ergodic in the following sense: if ˆ 2 L1C .; Pm / is shift invariant, namely, ˆ ı  t D ˆ Pm -a.s.;

t > 0;

.4:7:7/

then ˆ is constant Pm -a.s. 0 -measurable. Define u.x/ D E .ˆ/; x 2 X: Proof. We may assume that ˆ is F1 x For any non-negative bounded v 2 B.X/; we have

0 D Em ..ˆ ı  t  ˆ/v.X0 // D Em ..p t u.X0 /  u.X0 //v.X0 //;

4.7 Irreducible recurrence and ergodicity

205

and we conclude that T t u D u; t > 0; by choosing v suitably. By the preceding lemma, u D c m-a.e. for some constant c and so u.X R t / D c Pm -a.s. for each t > 0: For a positive bounded Borel function h with X hd m D 1; it then holds Phm -a.s. that c D u.Xn / D EXn .ˆ/ D Ehm .ˆ ı n jFn0 / D Ehm .ˆjFn0 / ! ˆ;

n ! 1:

It follows from this that ˆ D c Pm -a.s. We are in a position to formulate an ergodic theorem for M under the present assumption that E is irreducible recurrent. For f 2 L1 .XI m/; we let ´R f d m=m.X/ when m.X/ < 1 cf D E .4:7:8/ 0 when m.X/ D 1: Theorem 4.7.3. Let f be a Borel measurable m-integrable function. (i) It holds Pm -a.s. that Z 1 t f .Xs /ds D cf : lim t !1 t 0

.4:7:9/

(ii) For any non-negative bounded m-integrable function h; the convergence (4.7.9) takes place in L1 .I Phm /: (iii) The convergence (4.7.9) holds Px -a.s. for q.e. x 2 X: (iv) Assume that M satisfies the absolute continuity condition (4.2.9). Let f be a Borel m-integrable function such that for any x 2 X there exists its neighbourhood U.x/ with supy2U.x/ jf .y/j < 1: Then the convergence (4.7.9) holds Px -a.s. for every x 2 X: Proof. (i), (ii) Without loss of generality, we may assume that f is non-negative Borel m-integrable function. Define Z 1 ‰.!/ D f .Xs .!//ds; T ! D 1 !; ! 2 : 0

In view of (4.7.6), T is then a Pm -measure preserving transformation on  and Z 1 hm; ps f ids D kf kL1 .X Im/ ; Em .‰/ D 0

Z ‰.T k !/ D

kC1

f .Xs .!//ds; k

k  1:

206

4 Analysis by symmetric Hunt processes

Z 1 n ˆn .!/ D f .Xs .!//ds; n 0 then, due to the Birkhoff ergodic theorem, If we let

! 2 ;

lim ˆn .!/ D ˆ.!/ Pm -a.s.

n!1

.4:7:10/

for some ˆ 2 L1 .I Pm / and furthermore ¹ˆn º is Pm -uniformly integrable:13 lim sup Em .ˆn I ˆn > r/ D 0:

r!1 n

.4:7:11/

If h satisfies the condition in (ii), then ¹ˆn º is Phm -uniformly integrable, namely, (4.7.11) holds with Ehm in place of Em ; so that the convergence (4.7.10) takes place in L1 .I Phm / because Phm is a finite measure. Obviously, ˆ satisfies the shift invariance (4.7.7) so that ˆ D c Pm -a.s. for some constant c by virtue of Theorem 4.7.2 and furthermore Z 1 t lim f .Xs /ds D c; Pm -a.s: .4:7:12/ t !1 t 0 When m.X/ < 1; (ii) holds for h D 1 and (4.7.12) is also the L1 .I Pm /convergence. Therefore Z 1 n hm; f i D lim hm; ps f ids D cm.X/; n!1 n 0 yielding (4.7.9). When m.X/ D 1; choosing an increasing sequence ¹E` º of Borel sets of finite m-measures increasing to X; we obtain by Fatou’s lemma Z 1 n hm; ps f ids  cPm .X0 2 E` / D cm.E` /; hm; f i D lim inf n!1 n 0 yielding c D 0 D cf by letting ` ! 1: (iii) Assume that f is a non-negative Borel m-integrable function. R1 f is then 1-excessive and finite m-a.e. so that, as was noted above, R1 f .x/ < 1, 8x 2 X n N , for some m-inessential Borel set N: If we let ² ³ Z 1 s 0 D ! 2  W e f .Xs /ds < 1 ; 0

then  t 0  0 and Px .0 / D 1, 8x 2 X n N .ˇ Therefore we may and we do replace  by 0 for the restricted Hunt process MˇX nN : 13 Cf.

Theorem 10.6 and its proof of O. Kallenberg [1].

4.8 Recurrence and Poincaré type inequalities

207

We denote by ƒ0 the set of ! 2 0 for which (4.7.9) is valid. Define g.x/ D Px .ƒ0 /; x 2 X n N Then g D 1 m-a.e. by (i). Clearly ƒ0 D  t1 ƒ0 ; t  0; so that p t g D g; t  0; x 2 X n N; namely, g is an excessive function of MX nN : Therefore g D 1 q.e. (iv) Denote by ƒ the set of ! 2  for which (4.7.9) is valid. By (i), Px .ƒ/ D 1 for every x 2 X n N where N is some Borel set with m.N / D 0: Let n D R 1=n ¹! 2  W 0 f .Xs .!//ds < 1º: For a fixed x 2 X; Px .n / " 1; n ! 1; 1 by the right continuity of X t and the assumption on f: Clearly 1=n ƒ \ n  ƒ: Since Px .X1=n 2 N / D 0 under the absolute continuity condition (4.2.9), we have 1 Px .ƒ/  Px .1=n ƒ \ n / D Ex .PX1=n .ƒ/I n ; X1=n 2 X n N / D Px .n /:

By letting n ! 1; we get Px .ƒ/ D 1: The above theorem remains valid for any m-symmetric general (not necessarily Borel) right process M on a Lusin space whose Dirichlet form is irreducible recurrent by the following reason: if M is a general right process on a Lusin space X which is m-symmetric, then there exists a Borel m-inessential set N such that the restricted process MX nN becomes a special Borel standard process.14 Exercise 4.7.2. Prove the following: (i) If .E; F / is irreducible, then for f 2 L2 .XI m/, lim t !1 T t f D cf in L2 .XI m/ where cf is defined by (4.7.8). (ii) Assume additionally that M satisfies the absolute continuity condition and its transition density is finite, p t .x; y/ < 1; t > 0; x; y 2 X: Then Ex .f .X t // converges to cf for any x 2 X .

4.8

Recurrence and Poincaré type inequalities

Let X be a locally compact separable metric space, m be a positive Radon measure on X with full support and M D .X t ; Px / be an m-symmetric Hunt process on X whose Dirichlet form .E; F / on L2 .XI m/ is regular. ¹p t I t > 0º and ¹R˛ I ˛ > 0º denote the transition function and the resolvent of M, respectively. Throughout this section, E is assumed to be irreducible recurrent. We also assume throughout this section that M satisfies the condition (4.2.15), that is, there exists a Borel properly exceptional set N such that p t .x;  / is absolutely continuous with respect to m for each t > 0 and x 2 X n N . Theorem 4.2.7 14 Cf.

Chen–Fukushima [1; §3.1].

208

4 Analysis by symmetric Hunt processes

provides us with a tractable condition for this: the Sobolev type inequality (4.2.16) holding for the Dirichlet form .E; F / implies the validity of (4.2.15). Indeed the Sobolev type inequality (4.2.16) holds true for a considerably large family of the Dirichlet forms E on account of Theorem 2.4.3. First of all, we make a heuristic observation that the identity Z 1Z 1 Z 1 ˛t 2 ˛R˛ f .x/ D ˛ e p t f .x/dt D ˛ e ˛s p t f .x/dsdt Z D

0

0

1

0

  Z s=˛ s ˛ e p t f .x/dt ds; s 0

t

combined with Theorem 4.7.3 leads us, under the assumption m.X/ < 1, to   1 lim ˛R˛ f  hm; f i .x/ D 0 m-a.e. ˛!0 m.X/ holding for all f 2 L1 .XI m/. In the first part of this section, under the much stronger assumption (4.8.3) stated below, we shall prove the existence of a stronger limit     1 1 R f  hm; f i D lim R˛ f  hm; f i : .4:8:1/ ˛!0 m.X/ m.X/ P n nC1 f for 0 < ˛ < 1 by the resolvent equation, Since R˛ f D 1 nD0 .1  ˛/ R1 (4.8.1) is equal to  R f 

 X   1 1 1 n hm; f i D R1 f  hm; f i m.X/ m.X/

.4:8:2/

nD1

provided that the right-hand side converges. In this case, we shall derive a Poincaré type inequality and show that Rf is a solution of the Poisson equation E.Rf; u/ D R .f; u/ holding for f 2 L2 .XI m/ with X f d m D 0. Generally we can not expect the existence of the limit (4.8.2). In the second half of this section however, we shall employ a simple method of time change and killing studied below in §6.1 and §6.2 to reduce a general case to the above special one in finding a solution Rf of the Poisson equation. We shall then utilize the transient Poincaré type inequality obtained in Theorem 2.4.2 to derive a recurrent Poincaré type inequality. As a consequence, the quotient space FPe of the extended Dirichlet space Fe by its subspace of constant functions will be shown to be a Hilbert space and Rf will be seen to be a representative of a FPe -valued P potential Gf:

4.8 Recurrence and Poincaré type inequalities

209

Exercise 4.8.1. Let .X t ; Px / be the 1-dimensional Brownian motion and r.x; y/ D jxjCjyjjxyj. For any bounded measurable function f with compact support R R such that R1 f .x/dx D 0, show that Rf .x/ D R1 r.x; y/f .y/dy 2 He1 .R1 / and it satisfies 12 D.Rf; u/ D .f; u/ for all u 2 He1 .R1 /; where He1 .R1 / is the space appearing in Exercise 1.6.2. Z

M is said to be Harris recurrent if 1 0

1B .Xs /ds D 1

Px -a.s. for any x 2 X whenever B 2 B.X/; m.B/ > 0:

ˇ Lemma 4.8.1. Under the condition (4.2.15), the restricted Hunt process MˇX nN is Harris recurrent. In particular, for each x 2 X n N and ˛ > 0; we have R˛ .x; B/ > 0 whenever B 2 B.X/; m.B/ > 0: Proof. For any non-negative bounded function g, we get by putting f D g in (4.6.21) R˛g g D R˛ .g.1  R˛g g//  R˛ .g.1  Rg g//: Since R˛g g  1, by letting ˛ to 0, we obtain from Lemma 1.6.4 (iii) that g.1  Rg g/ D 0 m-a.e. In particular, by taking g D 1B for any Borel set B such that m.B/ > 0, we have R1B 1B D R1 a.e. on B. Using the relation R1B 1B .x/ D t Ex .1  e C1 / holding for C t D 0 1B .Xs /ds, this implies that Px .C1 D 1/ D 1 for m-a.e. x 2 B. Since Ex .PX t .C1 D 1// D Px .C1 D 1/ for any t > 0, Px .C1 D 1/ is a T t -invariant function. Hence, by virtue of Lemma 4.7.1, Px .C1 D 1/ D 1 m-a.e. x 2 X and consequently for all x 2 X n N from the assumption (4.2.15). This implies the Harris recurrence of MjX nN . Since p t .x;  / is absolutely continuous with respect to m for all x 2 X n N , Lemma 4.2.4 yields that there exists a non-negative jointly measurable function r˛ .x; y/ satisfying R˛ .x; dy/ D r˛ .x; y/m.dy/, r˛ .x; y/ D r˛ .y; x/ and Z r˛ .x; z/rˇ .z; y/m.dz/ D 0 r˛ .x; y/  rˇ .x; y/ C .˛  ˇ/ X

for all x; y 2 X n N . Moreover, r˛ .  ; x/ and hence r˛ .x;  / are ˛-excessive functions for each fixed x 2 X n N . First we assume that the measure m and the density r1 .x; y/ satisfy the following additional condition: m.X/ < 1 and r1 .x; y/ > c for some c > 0 and all x; y 2 X n N : .4:8:3/ R Note that cm.X/ < X r1 .x; y/m.dy/  1. For a signed measure , denote by kk its total variation. Further put D 1  cm.X/=2 < 1 and m1 .dx/ D

210

4 Analysis by symmetric Hunt processes

m.dx/=m.X/. The Lp -norm of f 2 Lp .XI m/ will be denoted by kf kp for 1  p  1. Lemma 4.8.2. Assume that (4.8.3) holds. Then, for any n  1, sup x;x 0 2X nN

k.R1 /n .x;  /  .R1 /n .x 0 ;  /k  2 n :

.4:8:4/

Furthermore, if f 2 L2 .XI m/ then p k.R1 /n f  hm1 ; f ik2  2. /n kf  hm1 ; f ik2 :

.4:8:5/

Proof. For any x; x 0 2 X n N , put F  Fx;x 0 D ¹y W r1 .x; y/  r1 .x 0 ; y/ > 0º. Then kR1 .x;  /  R1 .x 0 ;  /k Z Z 0 D .r1 .x; y/  r1 .x ; y//m.dy/ C

X nF

F

.r1 .x 0 ; y/  r1 .x; y//m.dy/

 2  cm.X/ D 2 : This implies (4.8.4) for n D 1. To show (4.8.4) for general n  1, for any kernels P1 ; P2 such that Pi .x; X/ D Pi .x; X n N / D 1 for x 2 X n N and i D 1; 2, put Q.Pi / D

1 sup kPi .x;  /  Pi .x 0 ;  /k: 2 x;x 0 2X nN

Then it suffices to show the inequality Q.P1 P2 /  Q.P1 /Q.P2 / . Let X C and X  be the sets which support the positive and negative parts of P1 P2 .x;  /  P1 P2 .x 0 ;  /, respectively. Then kP1 P2 .x;  /  P1 P2 .x 0 ;  /k D P1 P2 .x; X C /  P1 P2 .x 0 ; X C /  P1 P2 .x; X  / C P1 P2 .x 0 ; X  / Z D P1 .x; dy/.P2 .y; X C /  P2 .y; X  // X nN

Z

 Z D

X nN

X nN

P1 .x 0 ; dy/.P2 .y; X C /  P2 .y; X  //

.P1 .x; dy/  P1 .x 0 ; dy//.P2 .y; X C /  P2 .y; X  /  a/

4.8 Recurrence and Poincaré type inequalities

211

for any constant a. In particular, for p.y/ D P2 .y; X C /  P2 .y; X  /, let a D 1 2 .supy2X nN p.y/ C infy2X nN p.y//. Then we have the estimate sup jp.y/  aj 

y2X nN

1 sup jp.y/  p.y 0 /j 2 y;y 0 2X nN

1 sup jP2 .y; X C /  P2 .y 0 ; X C / C P2 .y 0 ; X  /  P2 .y; X  /j 2 y;y 0 2X nN



 Q.P2 /: Therefore sup x;x 0 2X nN

kP1 P2 .x;  /  P1 P2 .x 0 ;  /k  Q.P2 /

sup x;x 0 2X nN

kP1 .x;  /  P1 .x 0 ;  /k

D 2Q.P1 /Q.P2 /: .n/

For the proof of (4.8.5), let r1 .x; y/ be the density of .R1 /n .x; dy/ relative to m.dy/. By using the notation f0 .x/ D f .x/  hm1 ; f i, we have from the .n/ .n/ symmetry r1 .x; y/ D r1 .y; x/ and the identity .R1 /n 1.x/ D 1; x 2 X n N; that k.R1 /n f  hm1 ; f ik22 D k.R1 /n f0 k22 ³2 Z ²Z Z .n/ .n/ 0 0 D .r1 .x; y/  r1 .x ; y//f0 .y/m.dy/m1 .dx / m.dx/ X

X

X

X

X

X

Z ²Z Z 

 

jr1 .x; y/  r1 .x 0 ; y/jm.dy/m1 .dx 0 / .n/

Z Z X

sup x;x 0 2X nN

X

 4

.n/ jr1 .x; y/

³



.n/ r1 .x 0 ; y/jf0 .y/2 m.dy/m1 .dx 0 /

m.dx/

k.R1 /n .x;  /  .R1 /n .x 0 ;  /k Z Z Z 

n

.n/

X

X

X

.r1 .x; y/ C r1 .x 0 ; y//f02 .y/m.dy/m1 .dx 0 /m.dx/ .n/

.n/

kf0 k22 ;

yielding (4.8.5). By virtue of Lemma 4.8.2, we can define a linear operator R from L2 .XI m/ to it by Rf D

1 X nD1

..R1 /n f  hm1 ; f i/ D lim .R˛ f  hm1 ; f i/; ˛!0

f 2 L2 .XI m/;

212

4 Analysis by symmetric Hunt processes

and we get from (4.8.5) the bound p 2 kRf k2  p kf  hm; f ik2 ; 1

f 2 L2 .XI m/:

R .n/ .n/ Since .R1 /n f .x/  hm1 ; f i D X .R1 .x; dy/  R1 .x 0 ; dy//f .y/m1 .dx 0 /, (4.8.4) implies, for any f 2 L1 .XI m/, k.R1 /n f  hm1 ; f ik1  2 n kf k1 ;

kRf k1 

2 kf k1 ; 1

.4:8:6/

namely, R is also a linear operator from L1 .XI m/ to it. In order to obtain a little finer L2 estimate of R, we consider any f 2 L2 .XI m/ with hm; f i D 0 and we let, for an integer k and t > 0, .ˆ2k f /.t; x/ D .R1 /k f .x/ and .ˆ2kC1 f /.t; x/ D .R1 /k p t =2 f .x/. Then ˆn Rf D Rˆn f and we get from the above Z 1 n e t .ˆn f .t;  /; Rˆn f .t;  // dt ..R1 / f; Rf / D 0

p Z 1 2 e t kˆn f .t;  /k22 dt;  p 1 0

n  1:

Since kˆ2k f .t;  /k2 D .f; .R1 /2k f / and kˆ2kC1 f .t;  /k2 D .f; .R1 /2k p t f /, we have p 2 n ..R1 / f; Rf /  p .f; .R1 /n f /; n  1: 1 By taking sum in n  1, we obtain the inequality kRf k22  any f 2 get the bound

L2 .XI m/;

kRf

k22

p 2  p .f; Rf 1 

/: For

we substitute f  hm1 ; f i in this inequality in place of f to

 p 2 p 2 2  kf hm1 ; f ik22 : .4:8:7/ p .f  hm1 ; f i; Rf /  p 1 1

Along with the estimates (4.8.6) and (4.8.7), the following is a main result under the condition (4.8.3). (4.8.9) will be called a recurrent Poincaré type inequality. Theorem 4.8.1. Assume that condition (4.8.3) is satisfied. Then, for any f 2 L2 .XI m/ such that hm; f i D 0, Rf 2 F and E.Rf; u/ D .f; u/

.4:8:8/

213

4.8 Recurrence and Poincaré type inequalities

for all u 2 F . Furthermore, for any u 2 F , ku 

hm1 ; uik22

p 2  p E.u; u/: 1

.4:8:9/

Proof. For any f 2 L2 .XI m/ such that hm; f i D 0, R.n/ f D

n X

.R1 /k f 2 F

kD1

and Rf D limn!1 R.n/ f m-a.e by taking a subsequence if necessary. Since E.R.n/ f; R.n/ f / D E1 .R.n/ f; R.n/ f /  .R.n/ f; R.n/ f / D

n X

.f; .R1 /kCm1 f  .R1 /kCm f /

k;mD1

D

n X

.f; .R1 /k f  .R1 /kCn f /;

kD1

is uniformly bounded relative to n, Rf belongs to F D Fe \ L2 .XI m/ and E.Rf; Rf / D .f; Rf /. The relation (4.8.8) follows similarly. For any u 2 F , we have from (4.8.8) ku  hm1 ; uik22 D E.R.u  hm1 ; ui/; u  hm1 ; ui/  E.R.u  hm1 ; ui/; R.u  hm1 ; ui//1=2  E.u  hm1 ; ui; u  hm1 ; ui/1=2 D .u  hm1 ; ui; R.u  hm1 ; ui//1=2  E.u  hm1 ; ui; u  hm1 ; ui/1=2 p 2 1=2 . p / ku  hm1 ; uik2 E.u; u/1=2 1 which gives (4.8.9). Now we return to the general case that, for a certain properly exceptional set N  X; p t .x;  / is absolutely continuous with respect to m for each x 2 X n N: Let r˛ .x; y/ be the density function of the resolvent R˛ .x; dy/ with respect to m.dy/ specified in the paragraph after the proof of Lemma 4.8.1. Lemma 4.8.3. (i) There exist compact sets F; K  X n N and a positive constant a > 0 such that 0 < m.F / < 1; 0 < m.K/ < 1 and r1 .x; y/ > a for all x 2 F and y 2 K.

214

4 Analysis by symmetric Hunt processes

(ii) There exists a positive constant c and a nearly Borel measurable finely closed set B  X n N such that 0 < m.B/ < 1 and r1 .x; y/ > c for all x; y 2 B. Proof. (i) Let a D ¹.x; y/ 2 .X n N /  .X n N / W r2 .x; y/ > aº for a  0: By Lemma 4.8.1 and the symmetry of r2 ;Rwe have, for any x; y 2 X nN , r2 .x; z/ > 0, r2 .z; y/ > 0 m-a.e. and r1 .x; y/  r2 .x; z/r2 .z; y/m.dz/ > 0: Consequently 0 D .X n N /  .X n N /: Take a compact set A  X n N with m.A/ > 0 and set ƒa D .A  A/ \ a : We can then find a > 0 such that m  m.ƒa / > 0: For ƒ  .X n N /  .X n N / and x 2 X n N; we denote by ƒ.x/ the x-section of ƒ: S .n/ .nC1/ º is a reLet A D i Ai be a sequence of partitions of A such that ¹Ai .n/ finement of ¹Ai º and ƒa belongs to the -field generated by direct products of elements in the sequence. For x; z 2 A; denote by i.x/; i.z/ the indices of the cells containing them, respectively, we see from the differentiation theorem that, there exists a set   A  A with m  m./ D 0 and .n/

.n/

.m  m/.ƒa \ .Ai.x/  Ai.y/ // .n/

.n/

.m  m/.Ai.x/  Ai.y/ /

! Iƒa .x; y/;

n!1

for .x; y/ 2 .A  A/ n : If we let B D ¹z 2 A W m.ƒa .z/ n .z// > 0º; then m.B/ > 0: Therefore, for z0 2 B and x0 ; y0 such that x0 ; y0 2 ƒa .z0 / n .z0 /; we can choose n with .n/

.n/

.n/

.n/

.n/

.n/

.n/

.n/

.m  m/.ƒa \ .Ai.x0 /  Ai.z0 / // > .3=4/.m  m/.Ai.x0 /  Ai.z0 / /; .m  m/.ƒa \ .Ai.y0 /  Ai.z0 / // > .3=4/.m  m/.Ai.y0 /  Ai.z0 / /: The first inequality in the above and the Fubini theorem yields m.F / > 0 for the set .n/ .n/ .n/ F D ¹x 2 Ai.x0 / W m.Ai.z0 / \ ƒa .x// > .3=4/m.Ai.z0 / /º: Similarly the second inequality yields m.K/ > 0 for the set .n/

.n/

.n/

K D ¹y 2 Ai.y0 / W m.Ai.z0 / \ ƒa .y// > .3=4/m.Ai.z0 / /º: It holds for x 2 F and y 2 K that .n/

m.ƒa .x/ \ ƒa .y//  m.ƒa .x/ \ ƒa .y/ \ Ai.z0 / / .n/

.n/

.n/

 m.Ai.z0 / \ ƒa .x// C m.Ai.z0 / \ ƒa .y//  m.Ai.z0 / / 1 .n/ > m.Ai.z0 / / > 0 2

4.8 Recurrence and Poincaré type inequalities

215

and consequently Z r1 .x; y/ 

ƒa .x/\ƒa .y/

r2 .x; z/r2 .z; y/m.dz/ 

a2 .n/ m.Ai.z0 / /: 2

F and K can be taken to be compact due to the inner regularity of the measure m: (ii) By the irreducibility of M and Theorem 4.7.1 (i), we have for the set F; K of (i) Ex .e  F / > 0 and Ex .e  K / > 0 for q.e. x 2 X. Hence there exists a constant b > 0 such that the set B D ¹x 2 X nN W Ex .e  F /  b; Ex .e  K /  bº is of positive m-measure. As the functions involved are 1-excessive, B is nearly Borel and finely closed. Since r1 .  ; x/ and r1 .x;  / are 1-excessive, we have for any x; y 2 B r1 .x; y/  Ex .e  F r1 .X F ; y//  Ex .e  F Ey .e  K r1 .z; X K //jzDXF / > a Ex .e  F /Ey .e  K /  ab 2 : It suffices to put c D ab 2 . By replacing B by its intersection with a certain relatively compact open set, m.B/ can be also assumed to be finite. Let us fix a nearly Borel finely closed subset B of X n N such that 0 < m.B/ < 1 and 0 < c < r1 .x; y/; x; y 2 B; according to Lemma 4.8.3 (ii). To proceed further, we apply the results in §6.1 and §6.2 on the killing and time change of M by means of the simple positive continuous additive functional Rt C t D 0 1B .Xs /ds; t  0; with Revuz measure mB : L D .X t ; Px / be the time changed process of MjX nN where  t is the right Let M L is a right process with state space being the support continuous inverse of C t : M e e e is a quasi support of mB and B is quasi closed. B of C t : B  B q.e. because B  e Denote by B . B/ the topological support of B: According to Theorem 6.2.1 and L FL / L is mB -symmetric and recurrent, and its Dirichlet form .E; Theorem 6.2.3, M 2 2  on L .BI mB / D L .B I mB / is regular admitting the expression (6.2.4) with e in place of ; Y; Y e: In particular, mB ; B  ; B L E.'; '/ D E.H '; H '/;

' 2 FL D Fe jB \ L2 .BI mB /;

.4:8:10/

L where H '.x/ D Ex .'.X /I  e B < 1/; x 2 X: Let ¹pL t I t  0º; ¹Rp I p > 0º e B L respectively. As M L is recurrent, be the transition function and the resolvent of M, it is conservative and RL 1 1B D 1B : In view of (6.2.5), we have the identity H.RLp '/.x/ D RpC .1B  '/.x/;

q.e. x 2 X; ' 2 BC .B/:

.4:8:11/

216

4 Analysis by symmetric Hunt processes

Here 1B ' denotes the zero extension of ' from B to X and  Z 1 pC pC ˛t pC t R˛ f .x/ D Ex e f .X t /dt ; x 2 X; RpC D R0 ; 0

the resolvent of the subprocess MpC of M by the multiplicative functional e pC t , which is m-symmetric with the Dirichlet form on L2 .XI m/ given by E pmB .u; v/ D E.u; v/ C p.u; v/mB ;

u; v 2 F pmB D F \ L2 .mB / .D F /

in view of Theorem 6.1.1. Due to the irreducibility assumption on M; we can see that this Dirichlet form is transient in view of the proof of Lemma 6.2.6 (i). If a Borel function f satisfies .jf j; RC jf j/ < 1; then by Theorem 1.5.4, RC f 2 FemB ;

8v 2 FemB ; .4:8:12/ mB m m B B where Fe denotes the extended Dirichlet space of .E ; F / which equals Fe \ L2 .BI mB / by virtue of Lemma 6.2.5. Since R1C .x;  /  R1 .x;  /, R1C .x;  / and hence RC .x;  / is absolutely continuous relative to m for x 2 X n N . Therefore, RL 1 .x;  / is absolutely continuous e \ .X n N /. As in Lemma 4.2.4, we can take a density relative to mB for any x 2 B e function rL1 .x; y/; x; y 2 B; to be symmetric and 1-excessive in each variable relL Since R1 .x; A \ B/ e  RL 1 .x; A \ B/ e for any x 2 B e and any Borel set ative to M: e A, it follows that c  rL1 .x; y/ for any x; y 2 B \ .X n N /. Hence the condition L (4.8.3) is fulfilled by M: Therefore by putting D 1  cm.B/=2, we can conclude from Lemma 4.8.2, (4.8.6) and (4.8.7) that, if we define the operator RL by L D R'

f  v 2 L1 .XI m/;

1  X .RL 1 /n '  nD1

E mB .RC f; v/ D .f; v/;

 1 hmB ; 'i ; m.B/

' 2 L2 .BI mB /;

then RL is a linear operator making the spaces L1 .BI mB / and L2 .BI mB / invariant and satisfying the bounds L 1  2 k'k1 ; ' 2 L1 .BI mB /; kR'k 1  p  2 1 2 L L kR'kmB  ' hmB ; 'i; R' p 1 m.B/ mB  p 2 2 1 hmB ; 'ik2mB ;  k'  p 1 m.B/

(4.8.13)

' 2 L2 .BI mB /: (4.8.14)

4.8 Recurrence and Poincaré type inequalities

217

Moreover, by Theorem 4.8.1, we have for any ' 2 L2 .BI mB / with hmB ; 'i D 0; L 2 FL ; R'

L R'; L E. / D .'; /mB ;

2 FL :

.4:8:15/

We next prove that mB RC D m:

.4:8:16/

In fact we have from (4.4.9) and the symmetry of Rˇ and RC 0 D .g; Rˇ 1B  RC 1B C ˇRˇ RC 1B  Rˇ .1B RC 1B // D .Rˇ g  RC g C ˇRC Rˇ g  Rˇ g; 1B / D hmB RC ; g  ˇRˇ gi: Denote by h the density of mB RC with respect to m: h is then ¹T t º-invariant by the above and a constant by Lemma 4.7.1. Since mB RC .B/ D m.B/, h D 1: Define now a recurrent potential Rf of a function f on X by L B RC f C RC 1B RC f C RC f  Rf D RC 1B R1

2 hm; f i: m.B/

D.R/ D ¹f 2 L1 .XI m/ \ L1 .XI m/ W .jf j; RˇC jf j/ < 1; RC jf jˇB 2 L1 .BI mB /º:

(4.8.17)

(4.8.18)

Take f 2 D.R/: Then Rf 2 FemB because of (4.8.12), (4.8.13) and (4.8.17). L D 0 for ' 2 Further hmB ; Rf i D 0 on account of (4.8.16) and hmB ; R'i 2 L .BI mB /: Theorem 4.8.2. (i) If f 2 D.R/ and hm; f i D 0; then Rf 2 FemB and E.Rf; u/ D .f; u/

.4:8:19/

for all u 2 Fe . (ii) There exists a strictly positive bounded m-integrable function g with kRC gk1 < 1; g  1B such that a recurrent Poincaré type inequality ˇ2 p Z ˇ ˇ ˇ ˇu.x/  1 hmB ; uiˇ g.x/m.dx/  4.3 p / kRC gk1 E.u; u/ ˇ ˇ m.B/ 1 X .4:8:20/ is valid for all u 2 Fe . Proof. We first note that it is sufficient to prove (i) and (ii) holding for all u 2 FemB : To see this, assume that (4.8.20) is true for any u 2 FemB : Then it holds for any u 2 F : Take any u 2 Fe and its approximating sequence ¹un º  F : Then

218

4 Analysis by symmetric Hunt processes

(4.8.20) holding for un implies that un 

1 hmB ; un i m.B/

is L2 .g  m/-convergent

1 hmB ; un i ! k, to some v 2 L2 .g  m/ as n ! 1: Since un ! u m-a.e., m.B/ n ! 1; m-a.e. for some constant k by taking a subsequence if necessary, so that u D v C k 2 L2 .g  m/ and u 2 L2 .mB / because g  1B : This means that actually Fe D FemB : (i) For any f 2 D.R/ with hm; f i D 0; we have ' D RC f jB 2 L1 .BI mB / and hmB ; 'i D 0: It follows from (4.8.11) and (4.8.17) that

L C RL 1 ' D R'; L .Rf  RC f /jB D RL 1 R'

L Rf  RC f D H R':

For any u 2 FemB ; ujB 2 FL in view of (4.8.10). Therefore (4.8.10), (4.8.11), (4.8.12) and (4.8.15) yield that L E.Rf; u/ D E..Rf  RC f /jB ; ujB / C E.RC f; u/ D .'; u/mB C E.RC f; u/ D E mB .RC f; u/ D .f; u/: (ii) Take a strictly positive function g 2 L1 .XI m/ \ L1 .XI m/ such that kRC gk1 < 1. To show the existence of such function, let h be a bounded strictly positive function of L1 .XI m/ such that RC h < 1. SFor an increasing sequence of closed sets ¹Cn º such that m.Cn / < 1 and n Cn D X, let An D ¹x 2 Cn W RC h.x/  nº. Using the hitting probability HACn of An relative to the subprocess MC , we have RC .1An h/ D HACn RC .1An h/  n. Hence P 1 n C it is enough to put g D 1 nD1 n.m.An /_1/ 2 1An h. Since R 1B  1; we may assume that g  1B by replacing g with g C 1B if necessary. Let f be a Borel function such that jf j  kg for some k > 0 and hm; f i D 0. Then f 2 D.R/ and Rf 2 FemB : Since E mB is a transient Dirichlet form, we get by Theorem 2.4.2 and (4.8.19) Z .Rf /2 g d m  4kRC gk1 E mB .Rf; Rf / X

2 D 4kRC gk1 Œ.f; Rf / C kRf kL 2 .m / : B

L C ' for ' D RC f jB ; we obtain from (4.8.14) Since Rf jB D R' 2 2 L 2 L kRf kL 2 .m / D kR'kL2 .m / C 2.R'; '/mB C k'kL2 .m / B B B



2 2 L '/mB C k'k2 2  p .R'; p .Rf; RC f /mB : L .m / B 1 1

As E.Rf; RC f / D .f; RC f / D E mB .RC f; RC f /  0; the last expression in the above is dominated by 12p E mB .Rf; RC f / D 12p .f; Rf /: Therefore we

4.8 Recurrence and Poincaré type inequalities

have

219

p 4.3  / .Rf / .x/g.x/m.dx/  kRC gk1 E.Rf; Rf /; p 1 X

Z

2

proving (4.8.20) for u D Rf as hmB ; Rf i D 0: mB We next let Fe;0 D ¹u 2 FemB W hmB ; ui D 0º and prove (4.8.20) for mB any u 2 Fe;0 . It suffices to show that the family R D ¹Rf W jf j  kg mB mB : Since Fe;0 is an E mB for some k > 0 , hm; f i D 0º is E mB -dense in Fe;0 mB must closed subspace of FemB , it is enough to show that a function u 2 Fe;0 m vanish under the assumption E B .Rf; u/ D 0 for all Rf 2 R: b D f  1 hm; f i1B for any measurable Under this assumption, we put f m.B/ b 2 R so that, by (4.8.19), function f satisfying jf j  kg. Then Rf b; u/ D .f b; u/ C .Rf b; u/m D .f b; u/ C .R' L C '; 1B u/mB ; 0 D E mB .Rf B ˇ bˇ : Since hmB ; 'i D 0; u 2 L2 .BI mB / and hmB ; ui D 0; it where ' D RC f B holds that L D R'

1 X

.RL 1 /n ';

L B u/ D R.1

nD1

1 X

.RL 1 /n .1B u/:

.4:8:21/

nD1

By the symmetry of RL 1 on L2 .BI mB / and the symmetry of RC on L2 .XI m/; we then get b; u/ C .RC f b; 1B R.1 L B u/ C 1B u/ D .f; u C RC .1B R.1 L B u/ C 1B u//: 0 D .f L B u/ C 1B u/ D 0: But then (4.8.11) and (4.8.21) imply Therefore u C RC .1B R.1 L B u/ C 1B u D 1B .u C RC .1B R.1 L B u/ C 1B u// D 0; 1B R.1 and we arrive at the desired conclusion that u D 0: mB We have shown that (4.8.20) holds for any u 2 Fe;0 : But it then holds for any u 2 FemB as we may substitute v D u  Let K.g/ D . the following

hmB ;ui 1 .2 m.B/ B

p 4.3  / p kgkL1 .X Im/ kRC gk1 /1=2 . 1 

L1 -estimate

mB Fe;0 / in (4.8.20).

Then, by virtue of (4.8.20),

holds.

Corollary 4.8.1. It holds for all u 2 Fe that ˇ Z ˇ ˇ ˇ ˇu.x/  1 hmB ; uiˇg.x/m.dx/  K.g/E.u; u/1=2 : ˇ ˇ m.B/ X

.4:8:22/

220

4 Analysis by symmetric Hunt processes

(4.8.20) implies for u 2 Fe that E.u; u/ D 0 if and only if u is constant. We can further deduce the following from (4.8.20) in the same way as Lemma 1.5.5 for the transient case. Corollary 4.8.2. The quotient space FPe of Fe by the subspace of constant functions is a Hilbert space with inner product E: Any E-Cauchy sequence ¹un º  Fe admits u 2 Fe and constants cn such that un is E-convergent to u and un C cn is L2 .XI g  m/-convergent to u as n ! 1: Consider the space L0 D ¹f 2 L1 .XI m/ W hm; f i D 0; .jf j; RC jf j/ < 1º: For each f 2 L0 ; ˆf .u/ D .u; f /;

u 2 Fe .D FemB /;

defines a linear functional on the quotient space FPe on account of (4.8.12). Let L D ¹f 2 L0 W ˆf is boundedº: A function f 2 L0 belongs to L if and P 2 FPe uniquely such that only if there exists Gf P u/ D .f; u/; E.Gf;

u 2 FPe :

Theorem 4.8.2 (i) states that, if f 2 L0 is bounded on X and RC jf j is bounded P with on B; then f 2 L and Rf defined by (4.8.17) is a representative of Gf P hmB ; Rf i D 0: The proof of Theorem 4.8.2 (ii) implies that G.L/ is an E-dense P P subspace of Fe : We may call Gf a recurrent potential of f 2 L:

Chapter 5

Stochastic analysis by additive functionals

We assume throughout this chapter that we are given a symmetric Hunt process whose Dirichlet form is regular. It is shown in §5.1 that all equivalence classes of PCAF’s of the Hunt process are in one-to-one correspondence with all smooth measures introduced in §2.2. Such a correspondence is useful in that it enables us to reduce the calculus of functionals to the calculus of measures. After §5.2, we are concerned with additive functionals which are not necessarŒu ily non-negative. In particular, the AF A t D e u.X t /  e u.X0 / generated by the function u in the Dirichlet space admits a unique decomposition AŒu D M Œu C N Œu ; where M Œu is a martingale AF of finite energy and N Œu is a continuous additive Œu functional of zero energy (§5.2). N t is not of bounded variation in t unless u is a potential of a signed smooth measure in a certain sense (§5.4 and §5.5). Thus the decomposition in §5.2 may be regarded as a generalization (towards the Dirichlet space) of the Doob–Meyer decomposition of supermartingales and Itô’s formula on semi-martingales. The Beurling–Deny formula of §3.2 results from a further decomposition of M Œu (§5.3). Several methods of computing and studying M Œu ; N Œu and their extended versions are provided in §5.2, §5.4 and §5.5. The LeJan formula of §3.2 is translated into a transformation rule involving stochastic integrals (§5.6). In §5.7, the Lyons–Zheng decomposition of AŒu into a sum of forward and backward martingale AF’s will be presented, which will then be applied to the study of the conservativeness and tightness for symmetric diffusions. AF’s in the strict sense are also investigated under the absolute continuity condition for the transition function at the ends of §5.1, §5.2 and §5.5. Most sections contain examples concerning the Brownian motion and the diffusions associated with second order differential operators of divergence form.

222

5.1

5 Stochastic analysis by additive functionals

Positive continuous additive functionals and smooth measures

Throughout Chapter 5, let .X; m/ be as in (1.1.7) and M D .; M; X t ; Px / be an m-symmetric Hunt process on X whose Dirichlet form .E; F / on L2 .XI m/ is regular.1 Potential theories relevant to .E; F / and M have been developed in Chapter 2 and Chapter 4. We now utilize these theories and study the structures of some important classes of additive functionals of the Hunt process M.2 From now on in the main text, we shall call an additive functional of M in the sense of Appendix A.2 and A.3 an additive functional in the strict sense in order to distinguish it from the somewhat relaxed notion we are going to introduce. Let us call an extended real valued function A t .!/; t  0; ! 2 , an additive functional (AF in abbreviation) if A t .!/ is an additive functional in the strict sense but with respect to the restricted Hunt process MjX nN ; N being a properly exceptional set which depends on A in general. More precisely, in order for A t .!/ to be an AF the following conditions must hold: (A.1) A t .  / is F t -measurable, ¹F t º being the minimum completed admissible filtration (see §A.2); (A.2) there exist a set ƒ 2 F1 and an exceptional set N  X such that Px .ƒ/ D 1, 8x 2 X n N ,  t ƒ  ƒ, 8t > 0, and moreover, for each ! 2 ƒ, A:.!/ is right continuous and has the left limit on Œ0; .!//, A0 .!/ D 0, jA t .!/j < 1, 8t < .!/, A t .!/ D A .!/ .!/, 8t  .!/ and A t Cs .!/ D As .!/ C A t .s !/, 8t; s  0. The sets ƒ and N referred to in the preceding paragraph are called a defining set and an exceptional set of the AF A t .!/ respectively. .1/ Two AF’s A.1/ and A.2/ are said to be equivalent if for each t > 0 Px .A t D .2/ A t / D 1 q.e. x 2 X . By virtue of Theorem 4.1.1, we can then find a common defining set ƒ and a common properly exceptional set N of A.1/ and A.2/ such .1/ .2/ that A t .!/ D A t .!/; 8t > 0; 8! 2 ƒ. An AF A t .!/ is said to be finite (resp. continuous) if jA t .!/j < 1; 8t 2 Œ0; 1/ (resp. A t .!/ is continuous in t 2 Œ0; 1// for any ! in a defining set. We also call A t .!/ cadlag if A t .!/ is right continuous and possesses the left limit on Œ0; 1/ for any ! in a defining set. A Œ0; 1-valued continuous AF is called a positive continuous AF (PCAF in abbreviation). The set of all PCAF’s is 1 As

for a reduction of the case where X is a Lusin space to the present setting, see §7.3 and Theorem A.4.3 of the Appendix A.4. 2 The convention made in the last chapter is still in force in this chapter: any numerical function f on X is extended to X by setting f ./ D 0.

5.1

223

Positive continuous additive functionals and smooth measures

C denoted by AC c : The aim of this section is to characterize Ac by the class S of smooth measures associated with .E; F / (see §2.2). To this end we first construct a PCAF for any  2 S0 – the positive Radon measure of finite energy integral. After establishing the relation of AC c and S , it is not difficult to give an analytical characterization of the family of all PCAF’s in the strict sense, as we shall do at the end of this section under the absolute continuity assumption of the transition function.

Lemma 5.1.1. (i) For any u 2 F ;  2 S0 ; 0 < T < 1 and " > 0, P



p  eT p u.X t /j > "  E1 ./ E1 .u; u/: sup je " 0t T

.5:1:1/

.0/

(ii) Suppose that E is transient. For any u 2 Fe ;  2 S0 and " > 0 P



p  1p sup je E./ E.u; u/: u.X t /j > "  " 0t <1

.5:1:2/

Proof. (i) Taking a quasi continuous Borel version e u, we let E RD ¹x 2 X W je u.x/j > "º. Then the left-hand side of (5.1.1) is dominated by e T X p.x/.dx/ with p.x/ D Ex .e  E /. By virtue of Theorem 2.1.5 and Theorem 4.2.5, we have Z p p p.x/.dx/ D E1 .p; U1 /  E1 ./ Cap.E/ X



p 1p E1 ./ E1 .u; u/: "

(ii) can be proved analogously to the above by making use of the 0-order version of Theorem 2.1.5 together with Theorem 4.3.3. Lemma 5.1.2. (i) Let ¹un º be a sequence of quasi continuous functions in F . If ¹un º is an E1 -Cauchy sequence, then there exists a subsequence ¹unk º satisfying the condition that for q.e. x 2 X Px .unk .X t / converges uniformly in t on each compact interval of Œ0; 1// D 1: (ii) Suppose that E is transient. Let ¹un º be an E-Cauchy sequence of quasi continuous functions in Fe . There exists then a subsequence ¹unk º satisfying Px .unk .X t / converges uniformly in t 2 Œ0; 1// D 1

q.e. x 2 X:

224

5 Stochastic analysis by additive functionals

p Proof. (i) Take ¹nk º such that E1 .unkC1  unk ; unkC1  unk /  22k . By p virtue of Lemma 5.1.1 (i), P .ƒk /  e T 2k E1 ./ with ƒk D ¹sup0t T junk .X t /  unkC1 .X t /j > 2k º. By the Borel–Cantelli lemma, the P -measure of the set ƒ D limk ƒk vanishes for every  2 S0 . Hence, Px .ƒ/ D 0 q.e. by virtue of Theorem 2.2.3. (ii) can be proved similarly to the above by using Lemma 5.1.1 (ii) and the 0-order version of Theorem 2.2.3. Henceforth we use the notations h; ui D hu; i D  and a function u.

R X

u.x/.dx/ for a measure

C Theorem R 1 t5.1.1. For any  2 S0 , there exists a finite A 2 Ac such that Ex . 0 e dA t / is a quasi continuous version of U1 .

Proof. Take a non-negative finite Borel and quasi continuous version u of U1  such that ´ nRnC1 u.x/ " u.x/; n ! 1; 8x 2 X n N; .5:1:3/ u.x/ D 0; 8x 2 N; where N is some properly exceptional set and ¹R˛ ; ˛ > 0º is the resolvent of the process M. We let ´ n .u.x/  nRnC1 u.x// ; x 2 X n N gn .x/ D 0; x 2 N: Then it is easy to see that gn  m !  vaguely, R1 gn .x/ " u.x/; 8x 2 X n N , and, moreover, R1 gn is E1 -convergent to u. We define the approximating 1-order PCAF e An of M by e An .t; !/ D

Z

t

e s gn .Xs .!// ds:

0

Then, for any  2 S00 E ..e An .1/  e Al .1//2 /  2M

p

E1 ./kR1 gn  R1 gl kE1 3

.5:1:4/

with M D kU2 k1 . In fact, by setting gn;l D gn  gl ; n > l, the left-hand side 3 kvk E1

D

p

E1 .v; v/

5.1

Positive continuous additive functionals and smooth measures

of (5.1.4) is equal to Z 1 Z e s gn;l .Xs /ds 2E 0

D 2E

1

e u gn;l .Xu /du

s

Z

1

e 2s gn;l .Xs /R1 gn;l .Xs /ds

225





0

D 2h; R2 .gn;l R1 gn;l /i D 2.U2 ; gn;l R1 gn;l /  2.U2 ; gn R1 gn;l /  2M .gn ; R1 gn  R1 gl / D 2M E1 .R1 gn ; R1 gn  R1 gl /: Applying the Schwarz inequality and noting E1 .R1 gn ; R1 gn / D .gn ; R1 gn /  .gn ; u/ D h; R1 gn i  h; ui D E1 ./; we obtain (5.1.4). An .1/jF t / D e An .t/ C e t EX t .e An .1// D e An .t / C e t R1 gn .X t /, Since E .e we see that Mn .t/ D e An .t/ C e t R1 gn .X t /; 0  t  1; .5:1:5/ is a martingale with respect to .F t ; P /;  2 S00 . By Doob’s inequality4 P



 1 An .1/  e Al .1//2 / sup jMn .t/  Ml .t/j > "  2 E ..e " 0t 1

which, combined with (5.1.4), means that there exists a subsequence ¹nk º such that P .Mnk .t / converges uniformly on Œ0; 1// D 1;

8 2 S00 :

.5:1:6/

In view of (5.1.5), (5.1.6), Theorem 2.2.3 and Lemma 5.1.2, we conclude that, by e , where selecting a new subsequence if necessary, Px .ƒ/ D 1; x 2 X n N eWe ƒ D ¹! 2  Ank .1; !/ < 1; e Ank .t; !/ converges uniformly in t on each finite interval of Œ0; 1/º5

.5:1:7/

e is some properly exceptional set containing N . and N Let us set e A.t; !/ D limnk !1 Re Ank .t; !/ for ! 2 ƒ and e A.t; !/ D 0 for t s e ! … ƒ. Furthermore, let A.t; !/ D 0 e d A.s; !/: A is then a PCAF with ƒ and e being its defining set and exceptional set respectively. N 4 Cf.

N. Ikeda and S. Watanabe [2]. e. the set defined by (A.2.22) for e S DX nN

5e  is

226

5 Stochastic analysis by additive functionals

In order to complete the proof, it suffices to show E .e A.1// D h; ui, 8 2 An .1/ is L2 .P /-convergent, S00 , by virtue of Theorem 2.2.3. Since Mn .1/ D e so is the martingale Mn .t/. Hence, E .e A.t // C e t h; p t ui D lim E .Mn .t // D lim E .e An .1// n!1

n!1

D lim h; R1 gn i D h; ui: n!1

By letting pt tend to infinity and noting the bound h; p t ui D E1 .U1 ; p t u/  p E1 ./ E1 .u; u/, we get the desired equality. Lemma 5.1.3. Consider  and A of the preceding Then forany ˛ > 0 R theorem. 1 and bounded non-negative Borel function f; Ex 0 e ˛t f .X t /dA t is a quasi continuous version of U˛ .f  /. Proof. It is sufficient to consider the case where ˛R D 1 and f D 1G ; G being 1 an open with .@G/ D 0. Let .x/ D Ex . 0 e t 1G .X t /dA t /; .x/ D R 1 set Ex . 0 e t 1X nG .X t /dA t /; then both  and are 1-excessive and  C D U1 . By Theorem 2.2.1 and Lemma 2.3.2,  and are quasi continuous versions of 1-potentials of some measures  and  2 S0 respectively. We know from Lemma 4.3.1 and the equalities  D HG1 ; D HX1 nG that suppŒ  G and suppŒ  X n G. Since  D  C , we have  D 1G  .

Theorem 5.1.2. For  2 S0 ; A 2 AC c of Theorem 5:1:1 is unique (up to equivalence). are associated with  2 S0 in the manner Proof. Suppose that A.1/ ; A.2/ 2 AC R 1 tc .1/ R1 .2/ of Theorem 5.1.1. Then Ex . 0 e dA t / D Ex . 0 e t dA t / D u.x/; x 2 X n N , for some properly exceptional set N . In the same way as in the proof of Theorem 5.1.1, Z 1  Z 1 .i / .j / e t dA t e t dA t vij .x/ D Ex 0

D

X

Z

Ex

kDi;j

0

0

1

e 2t u.X t /dA t

.k/

 ;

x 2 X n N; i; j D 1; 2. Let un D u ^ n. Then, by Lemma 5.1.3, Z 1  X .k/ E e 2t un .X t /dA t h; vij i D lim n!1

kDi;j

0

D lim 2h; U2 .un /i D 2hU2   u; i < 1 n!1

5.1

Positive continuous additive functionals and smooth measures

227

for any  2 S00 . Hence, ² Z

1

e

E

t

0

.1/ dA t 

Z

1

e

t

0

.2/ dA t

³2  D h; v11 2v12 Cv22 i D 0;  2 S00 ;

from which it easily follows that A.1/ is equivalent to A.2/ . Hereafter we use the notations Z Z t f .Xs /dAs ; UA˛ f .x/ D Ex .fA/ t D 0

1

e ˛t f .X t /dA t



0

C for A 2 AC c and f 2 B .

Lemma 5.1.4. Let  2 S0 and A 2 AC c be related as in Theorem 5:1:1. Then for any nearly Borel set B  X  Z B  Z .˛C /t ˛Ehm e f .X t /dA t " h.x/f .x/.dx/; ˛ " 1; X nB r

0

where h is any -excessive function .  0/; f 2 B C and B r is the set of regular points of B, i.e., B r D ¹x 2 X W Px .B D 0/ D 1º. Proof. It suffices to give the proof when h is a bounded -excessive function belonging to L2 .XI m/ and f 2 BbC . By Lemma 5.1.3, UA˛ f is a quasi continuous version of U˛ .f/. By virtue of Theorem 4.3.1, Z B  ˛t e f .X t /dA t D ˛.h; UA˛ f  HB˛ UA˛ f / ˛Ehm 0

D ˛E˛ .R˛ h; UA˛ f  HB˛ UA˛ f / D ˛E˛ .R˛ h  HB˛ R˛ h; UA˛ f / Z X nB R˛X nB h.x/f .x/.dx/ D ˛E˛ .R˛ h; U˛ .f// D ˛ X

which increases to

R X nB r

h.x/f .x/.dx/ as ˛ " 1. Here

R˛X nB h.x/

Z D Ex

B

e

˛t

 h.X t /dt :

0

C Lemma 5.1.5. Let A 2 AC c and f 2 Bb .X/.

(i)

ˇ

ˇ

UA˛ f  UA f C .˛  ˇ/R˛ UA f D 0;

˛; ˇ > 0:

228

5 Stochastic analysis by additive functionals

(ii) We put Z R˛A f .x/

D Ex

1

e

˛t A t

e

 f .X t /dt ;

0

x 2 X n N;

where N is a properly exceptional set for A. Then R˛A f is quasi continuous and R˛A f  R˛ f C UA˛ R˛A f D 0:

.5:1:8/

Proof. (i) is easy. (ii) was shown already in (4.6.21) in the special case that A t D Rt 0 g.Xs /ds.R The present case can be proved in the same way by using the identity t e A t  1 D 0 e As dAs and noting the next easily verifiable fact: R˛ f  R˛A f is ˛-excessive and consequently R˛A f is, together with R˛ f , finely continuous on X n N . In particular R˛A f is quasi continuous by virtue of Theorem 4.6.1. Before formulating the main assertions of this section, we need one more lemma. Recall the definition of the smooth measure given in §2.2:  2 S if  is a positive Borel measure charging no set of zero capacity and admits an associated generalized nest. Thus the class S contains all positive Radon measures charging no set of zero capacity. Lemma 5.1.6. Let ¹Fn º be an increasing sequence of closed sets. ¹Fn º is a generalized nest if and only if   Px lim X nFn <  D 0 q.e. x 2 X: .5:1:9/ n!1

Proof. Condition (2.2.17) for ¹Fn º to be a generalized nest is equivalent to   .5:1:10/ Px lim GnFn D 1 D 1 q.e. n!1

for any relatively compact open set G on account of Theorem 4.2.1. (5.1.9) implies (5.1.10) because of the quasi-left-continuity of the Hunt process M. To get (5.1.9) from (5.1.10), it suffices to choose a sequence ¹Gl º of relatively compact open sets with Gl  GlC1 ; Gl " X, and observe the inequality X nFn  Gl nFn ^ X nGl . Theorem 5.1.3. For  2 S and A 2 AC c , the following conditions are equivalent to each other: (i) For any -excessive function h (  0) and f 2 B C , 1 lim Ehm ..fA/ t / D hf  ; hi: t #0 t

.5:1:11/

5.1

Positive continuous additive functionals and smooth measures

229

(ii) For any -excessive function h (  0) and f 2 B C , ˛C

˛.h; UA

f / " hf  ; hi;

(iii) For any t > 0; f; h 2 B C , Ehm ..fA/ t / D

Z 0

˛ " 1:

.5:1:12/

t

hf  ; ps hids:

.5:1:13/

(iv) For any ˛ > 0; f; h 2 B C , .h; UA˛ f / D hf  ; R˛ hi:

.5:1:14/

When  2 S0 , each of the above four conditions is also equivalent to each of the following three conditions: (v) UA1 1 is a quasi continuous version of U1 . (vi) For any h 2 B C \ F and f 2 BbC , 1 hi: lim Ehm ..fA/ t / D hf  ; e t #0 t

.5:1:15/

(vii) For any h 2 B C \ F and f 2 BbC , hi: lim ˛.h; UA˛ f / D hf  ; e

˛!1

.5:1:16/

A PCAF A related to a given  2 S by .5:1:11/ is unique up to the equivalence. Conversely, given A 2 AC c , a smooth measure  satisfying .5:1:11/ is unique. 1 ˛t Proof. Let  2 S and A 2 AC Ehm ..fA/ t / increases as t dec . Note that t e creases if h and f are as in (i). Therefore it suffices to prove the equivalence of (i)– (iv) when the right-hand sides are finite, for instance, when h 2 BbC , f D g  1Fn , g 2 BbC , ¹Fn º being a generalized nest associated with . Then, the uniqueness of the Laplace transform yields the implication (iv))(iii). (iii))(i))(ii) are obvious. (ii))(iv) is a consequence of Lemma 5.1.5 (i). Suppose next that  2 S0 and A 2 AC c . The implications (iii))(vi) and (iv))(vii) are clear. Lemma 5.1.3 means the equivalence (iv),(v). Let us derive the implication (vi))(v). Suppose that (vi) is satisfied and put c t .x/ D Ex .A t /. Then Z t Z Z 1 t 1 t h; ps hids D lim .ps h; cu /ds D lim .h; cuCs  cs /ds u#0 u 0 u#0 u 0 0 Z Z 1 t Cu 1 u .h; cs /ds  lim .h; cs /ds D .h; c t / D lim u#0 u t u#0 u 0

230

5 Stochastic analysis by additive functionals

for any -excessive function h in F . By taking the Laplace transform we get (iv) for h of this type and f D 1, which is enough to obtain (v). In the same way, we can get the implication (vii))(v). Suppose that A 2 AC c is related to a given  2 S by the condition (i). Take a generalized nest ¹Fn º such that 1Fn   2 S0 ; n D 1; 2; : : :, in accordance with Theorem 2.2.4. Then 1Fn  A and 1Fn   are related by (i) and consequently by (v). Hence 1Fn  A is unique by virtue of Theorem 5.1.2. Therefore A is uniquely determined by  on account of Lemma 5.1.6. The last statement in the theorem is trivial. We say that A 2 AC c and  2 S are in the Revuz correspondence if they satisfy the relation (5.1.11). In this case,  is called the Revuz measure of A and sometimes denoted by A . Theorem 5.1.4. The family of all equivalence classes of AC c and the family S are in one to one correspondence under the Revuz correspondence. The proof of Theorem 5.1.4 is accomplished by the following two lemmas and the uniqueness statements in Theorem 5.1.3. Lemma 5.1.7. Any A 2 AC c admits its Revuz measure  2 S . Proof. For a given A 2 AC c with a properly exceptional set N  X , we put .x/ D R1A f .x/; x 2 X n N , where f is a bounded Borel function in L2 .XI m/ such that f .x/ > 0; 8x 2 X. Then, .x/ > 0; 8x 2 X n N . Since  is quasi continuous by Lemma 5.1.5, there exists a sequence ¹En º of increasing closed sets T such that Cap.X n En / # 0; n ! 1; N  1 .X n En / and jEn is continuous nD1 for each n. Let us put ² ³ 1 Fn D x 2 En W .x/  .5:1:17/ n and prove that ¹Fn º satisfies condition (5.1.9). To this end, set Bn D ¹x 2 X n N W .x/  n1 º, n D Bn and  D limn!1 n . Since  is finely continuous on X n N , we have for x 2 X n N ,  Z 1 1 t A t Ex e f .X t /e dt D Ex .e  n e An .X n //  : n

n By letting n tend to infinity, we can see Px . < / D 0; x 2 X n N , in view of the strict positivity of f . Hence ¹Fn º satisfies (5.1.9) because of the inclusion X n Fn  .X n En / [ Bn and Theorem 4.2.1.

5.1

Positive continuous additive functionals and smooth measures

231

Now let An D 1Fn  A. Then UA1n 1 is 1-excessive (hence quasi continuous by Theorem 4.6.1) and satisfies the inequality UA1n 1  nUA1 '  nR1 f .2 F / by Lemma 5.1.5. Therefore we can use Theorem 2.2.1 and Lemma 2.3.2 to conclude that there exists a unique n 2 S0 such that UA1n 1 is a quasi continuous version of the potential U1 n . But then n D 1Fn  l ;

n
because UA1n 1 D UA1l 1Fn is a version of U1 1Fn  l by Lemma 5.1.3. S We can now define a measure  by 1Fn   D 1Fn  n ; n D 1; 2; : : : ; .X n n Fn / D 0.  is smooth in view of Theorem 2.2.4 and Lemma 5.1.6. It remains to prove that the measure  constructed above is the Revuz measure of A. In view of Theorem 5.1.3, it suffices to derive the relation (5.1.12). But the same theorem implies that, for any f 2 B C and -excessive function h; ˛C

˛C

hf  ; hi D lim lim ˛.h; UAn f / D lim lim ˛.h; UA n!1 ˛!1

˛!1 n!1 ˛C

By (5.1.9), we get hf  ; hi D lim˛!1 ˛.h; UA

.1Fn  f //:

f / as was to be proved.

Lemma 5.1.8. Any  2 S admits an A 2 AC c whose Revuz measure is . Proof. By Theorem 2.2.4, there exists a generalized nest ¹Fn º satisfying (2.2.18) and 1Fn   2 S0 for each n. By Theorem 5.1.1 and Theorem 5.1.3, there exists A.n/ 2 AC c with Revuz measure 1Fn   uniquely up to the equivalence. But then A.n/ 1Fn  A.l/ ;

n < l;

because 1Fn   D 1Fn 1Fl   is the Revuz measure of 1Fn  A.l/ . Choose a properly exceptional set N and a defining set ƒ which are common to .n/ .nC1/ / .!/; 8t > 0; n D all A.n/ such that, for t  any ! 2 ƒ; A t .!/ D .1Fn  A 1; 2; : : : ; and .!/ D limn!1 X nFn .!/  .!/. For ! 2 ƒ let ´ .n/ A t .!/ D A t .!/;

X nFn1 .!/  t < X nFn .!/;

A t .!/ D A .!/ .!/;

t  .!/: .n/

Obviously A 2 AC c . Since A t D A t Z

XnFn

e

˛Ehm 0

.˛C /t

n D 1; 2; : : :

for t < X nFn , we see from Lemma 5.1.4,

 f .X t /dA t

Z "

.0/

Fn

h.x/f .x/.dx/;

˛ " 1;

232

5 Stochastic analysis by additive functionals

S S .0/ .0/ Fn being the fine interior. Note that the set n Fn n n Fn is exceptional because of (5.1.9). By letting n tend to infinity, we get ˛C

lim ˛.h; UA

˛!1

f / D hf  ; hi;

proving that  is the Revuz measure of A. The proof of Theorem 5.1.4 is now completed. Here we state two lemmas on A 2 AC c which will play important roles in the subsequent sections. The first one concerns an estimate of A, while the second one does its behaviors on a finely open subset. Lemma 5.1.9. For any A 2 AC c ;  2 S00 and t > 0, E .A t /  .1 C t/ kU1 k1  A .X/ . 1/:

.5:1:18/

Proof. (i) Assume first that the Revuz measure  D A of A is of finite energy integral:  2 S0 . By setting c t .x/ D Ex .A t /; x 2 X, we claim that ct 2 F

and

E.c t ; v/ D h;e v  p te v i;

8v 2 F :

.5:1:19/

By Theorem 5.1.3, the potential UA1 1.x/ of A is a quasi continuous version of the potential U1  of . Hence, h; c t i  e t h; UA1 1i D e t E1 ./ < 1. On the other hand, .c t ; c t  ps c t / D .c t ; cs  p t cs / D .c t  p t c t ; cs / which is, by Theorem 5.1.3, equal to Z s Z s h; pu .c t  p t c t /idu D h; 2c t Cu  cu  c2t Cu idu: 0

0

Therefore, we get 1 lim .c t ; c t  ps c t / D h; 2c t  c2t i D h; c t  p t c t i < 1; s#0 s proving that c t 2 F and E.c t ; c t / D h; c t  p t c t i. Similarly, we have the relation E.c t ; v/ D h;e v  p te v i; 8v 2 F . For  2 S0 and  2 S00 , (5.1.19) gives us

e

e

E .A t / D h; c t i D E1 .U1 ; c t / D h; U1   p t .U1 /i C .c t ; U1 /   Z c t .x/m.dx/ ;  kU1 k1 .X/ C X

5.1

Positive continuous additive functionals and smooth measures

233

which proves the inequality (5.1.18) by noting that Z 1 1 c t .x/m.dx/ D lim Em .A t / D .X/: sup t #0 t t >0 t X When  D A is a general smooth measure, we may consider the increasing sequence ¹Fn º of closed sets of Theorem 2.2.4.   Since 1Fn  A has the Revuz measure 1Fn   2 S0 , we have E .1Fn  A/ t  .1 C t / kU1 k1  .Fn /. By letting n tend to infinity, we arrive at (5.1.18). C For a nearly Borel finely open set G, A 2 AC c and f 2 B , we use the notations Z 1 G pG f .x/ D E .f .X / W t <  /; R f .x/ D e ˛t p G x t X nG t ˛ t f .x/dt; 0 Z XnG  Z 1  G;˛ ˛t ˛t UA f .x/ D Ex e f .X t /dA t D Ex e f .X t /dA t ^ XnG : 0

0

Lemma 5.1.10. Let A 2 AC c and G be a nearly Borel finely open set. C (i) If h is -excessive (  0) on G with respect to ¹p G t º and f 2 B , then G;˛C

˛.h; UA

f /m " hf 1G  A ; hi;

˛ " 1:

(ii) For any t > 0; f; h 2 B C Z t  Z t Ehm f .Xs /dAs^ XnG D hf 1G  A ; psG hids: 0

0

(iii) Suppose that Px .A t D 0; 8t < X nG / D 1

.5:1:20/

for m-a.e. x 2 X. Then A .G/ D 0 and .5:1:20/ holds for q.e. x 2 X. Proof. (i) When A 2 S0 , this has been proved in Lemma 5.1.4 and indeed the proof of Lemma 5.1.4 works for any -excessive function h with respect to ¹p G t º rather than ¹p t º. When A is a general smooth measure, we choose closed sets ¹Fn º as in the proof of the previous lemma. We may then let n ! 1 in the relation (i) holding for 1Fn  A and 1Fn  A . (ii) We have G;ˇ

UAG;˛ f  UA

G;ˇ

f C .˛  ˇ/R˛G UA

f D 0;

˛; ˇ > 0;

an equation analogous to Lemma 5.1.5 (i). We then see the equivalence of the statements (i) and (ii) exactly in the same manner as in the proof of Theorem 5.1.3. (iii) (5.1.20) holding m-a.e. x 2 X implies 1G  A D 0 by (ii) and 1G  A D 0 by Theorem 5.1.4.

234

5 Stochastic analysis by additive functionals

Let us study the relationship between the support of PCAF and that of the Revuz measure. For A 2 AC c with a properly exceptional set N , we let e D ¹x 2 X n N W Px .R D 0/ D 1º Y

.5:1:21/

e is called a support of A. We consider where R.!/ D inf¹t > 0 W A t .!/ > 0º. Y R eD the function 'A .x/ D Ex .e /. 'A is easily seen to be 1-excessive. Since Y e is nearly Borel, finely closed with respect to MX nN ¹x 2 X n N W 'A .x/ D 1º, Y e is actually a and quasi closed in virtue of Theorem 4.6.1. We shall prove that Y quasi support (in the sense of §4.6) of the Revuz measure of A. e. For this, we need a lemma stating that A t .!/ increases only when X t .!/ 2 Y e Denote by  the first hitting time of Y . Lemma 5.1.11. For q.e. x 2 X, Px .R D / D 1;

(5.1.22)

Px .A t D .1 e Y  A/ t ; 8t > 0/ D 1:

(5.1.23)

e Px -a.s. by Lemma A.2.7 and accordingly Proof. For x 2 X n N; X 2 Y Px . < R/ D P . < R; R ı  > 0/ D Ex .PX .R > 0/I  < R/ D 0: We next show Px .R < / D 0 q.e. x 2 X:

.5:1:24/

e and t > 0 In fact we have for any x 2 .X n N / n Y Px .R < / D Px .ARCt > 0; R < / D Ex .PXR .A t > 0/; R < / e/;  Ex .PXR .R < t/; XR 2 .X n N / n Y er , it holds which tends to zero as t # 0. Since (5.1.24) is trivially true for x 2 Y r e e except for x 2 N [ .Y n Y / and hence for q.e. x by Theorem 4.1.3. The proof of (5.1.22) is complete. We next introduce the random time sets I.!/ D ¹t W A t C" .!/  A t .!/ > 0; 8" > 0º; eº; Z.!/ D ¹t W X t .!/ 2 Y and show that Px .I  Z/ D 1 q.e. x 2 X:

.5:1:25/

5.1

Positive continuous additive functionals and smooth measures

235

Indeed, by virtue of the right continuity of 'A .X t /, we have  [ Px .I 6 Z/  Px ¹t 2 I; 'A .X t / < 1º t

X



Px .Aq  Ar > 0; r C  ı r  q/;

r
P QC denoting non-negative rationals. The last probability is dominated by r2QC   Ex PXr .R < / which vanishes for q.e. x 2 X in view of (5.1.22). Since I.!/ differs from the set of the increase times of A t .!/ at most by a countable set and A t .!/ is continuous, we can now get from (5.1.25) that Z dAs .!/ A t .!/ D Z D

Œ0;t \I.!/

Œ0;t \I.!/

1e Y .Xs .!//dAs .!/ D .1 e Y  A/ t .!/

Px -a.s. for q.e. x 2 X . Theorem 5.1.5. The support .5:1:21/ of A 2 AC c is a quasi support of the Revuz measure of A. e of A is quasi closed. The Revuz meaProof. We already noticed that the support Y e sure A does not charge the set X n Y on account of Theorem 5.1.3 and (5.1.23). Therefore it suffices to verify the condition (iii) of Theorem 4.6.2: if v is quasi e. continuous and vanishes A -a.e. on X, then v D 0 q.e. on Y Put vn D jvj ^ n. Theorem 5.1.3 then implies that the AF vn  A vanishes identically. If we set  t .!/ D inf¹s > 0 W As .!/ > t º; then Z

1

e

ˇEx 0

ˇ t

 vn .X t / dt

Z D ˇEx

1

e 0

ˇA t

 vn .X t /dA t

D 0 q.e. x 2 X:

Since vn is q.e. finely continuous by Theorem 4.2.2, we get after letting ˇ ! 1 e and hence v D 0 q.e. on Y e. that vn D 0 q.e. on Y Finally, we turn to the study of PCAF’s in the strict sense. An AF in the strict sense is by definition an AF admitting a defining set ƒ with Px .ƒ/ D 1; 8x 2 X,

236

5 Stochastic analysis by additive functionals .1/

namely, an AF without exceptional set N . Two such functionals A t regarded to be equivalent if .1/

Px .A t

.2/

D A t / D 1;

8t > 0; 8x 2 X:

.2/

and A t

are

.5:1:26/

We can then find a common defining set ƒ with Px .ƒ/ D 1; 8x 2 X, such that .1/ .2/ A t .!/ D A t .!/; 8t  0; 8! 2 ƒ. We denote by AC c;1 the family of all PCAF’s in the strict sense. In the rest of this section, we assume that M satisfies the absolute continuity condition (4.2.9). Recall a consequence of this assumption stated at the end of §4.2: the ˛-potential U˛  R of  2 S0 admits a quasi continuous and ˛-excessive modification R˛ .x/ D X r˛ .x; y/.dy/ where r˛ .x; y/ is the resolvent density appearing in Lemma 4.2.4. In characterizing the family AC c;1 analytically, the subfamily S00 of S0 defined by (2.2.10) plays a primary role.  2 S00 if and only if  is a positive Borel measure such that .X/ < 1 and R1 .x/ is uniformly bounded in x 2 X . We first prove the following as a direct consequence of Theorem 5.1.1. Theorem 5.1.6. For any  2 S00 , there exists A 2 AC c;1 uniquely such that Z Ex

1

e t dA t



0

D R1 .x/;

8x 2 X:

.5:1:27/

Proof. Since  2 S0 , Theorem 5.1.1 provides us with a finite A 2 AC c , with a defining set ƒ and an exceptional set N such that (5.1.27) holds for m-a.e. x 2 X. Since both sides are 1-excessive with respect to MjX nN , (5.1.27) holds for any x 2 X n N . We shall construct a desired e A 2 AC c;1 by modifying A. We set M D supx2X R1 .x/ .< 1/. We may assume that P .ƒ/ D 1 and P .! 2  W A t .!/ D 0; 8t > 0/ D 1. Let A"tT .!/ D A t " ." !/, t > " > 0, ! 2 , and, for a sequence "n # 0, ƒ0 D n "1 ƒ. Then Px .ƒ0 / D 1; 8x 2 X , because n Px ."1 ƒ/ n

Z D Ex .PX"n .ƒ// D

X

p"n .x; dy/Py .ƒ/ C Px .  "n /P .ƒ/

which is equal to 1 by the assumption (4.2.9). If we set Ant .!/ D A"t n .!/; ! 2 , then for each ! 2 ƒ0 Ant .!/  Am t .!/ D A"m "n ."n !/;

m < n;

5.1

Positive continuous additive functionals and smooth measures

237

which means that Ant .!/ is increasing as n ! 1 and the convergence is locally uniformly in t 2 .0; 1/ if the limit is finite. On the other hand, from Z 1 Z 1 t s t e dAs  e e s dAs At  e 0

t

holding on ƒ, we have Ex .A t /  e t R1 .x/  p t R1 .x/;

x 2 X n N;

and consequently Ex .A"t / D Ex .EX" .A t " //  e t " p" R1 .x/  p t R1 .x/;

8x 2 X:

Put, for ! 2  and t > 0, ´ limn!1 Ant .!/ if the limit exists e A t .!/ D 0 otherwise: e A t is then an adapted process and further A t /  lim Ex .Ant /  e t R1 .x/  p t R1 .x/  e t M; Ex .e n!1

A0C / D lim Ex .e At /  0 Ex .e t #0

for every x 2 X. Therefore we can conclude that e A 2 AC c;1 with defining set e D ¹! 2 ƒ0 W e A t .!/ < 1 for any t > 0; e A0C .!/ D 0º: ƒ R R t s n t e we see that e dAs D e t Ant C "n e s Ans ds increases, as For ! 2 ƒ, " n Rt n ! 1, to 0 e s d e As . Hence, for any x 2 X, Z 1  Z 1  s e s n Ex e d As D lim Ex e dAs n!1

0

"n

 Z "n D lim Ex e EX"n n!1

1

e

s

 dAs

0

D lim p"n R1 .x/ D R1 .x/; n!1

getting (5.1.27) for e A. To prove the uniqueness, take A.1/ ; A.2/ 2 AC c;1 satisfying (5.1.27). We already know by Theorem 5.1.2 that (5.1.26) holds for q.e. x 2 X. Hence, for any x 2 X, .1/

Px .A t

.2/

 A.1/ "n D A t

It now suffices to let n ! 1.

.1/

.2/

 A.2/ "n / D Ex .PXs .A t "n D A t "n // D 1:

238

5 Stochastic analysis by additive functionals

A positive Borel measure  on X is said to be smooth in the strict sense if there exists a sequence ¹En º of Borel sets increasing to X such that 1En   2 S00 for each n and   Px lim X nEn   D 1; 8x 2 X: .5:1:28/ n!1

The totality of the smooth measures in the strict sense is denoted by S1 . Theorem 5.1.7. (i) The equivalence class of AC c;1 is in one to one correspondence with the class S1 under the Revuz correspondence .5:1:11/. (ii) S1  S . (iii)  2 S1 if and only if there exists a sequence ¹En º of Borel finely open sets increasing to X such that 1En   2 S00 for each n. Proof. (i) The uniqueness of the correspondence follows from the following observation and Theorem 5.1.6: A 2 AC c;1 and  2 S00 are under the Revuz correspondence if and only if the equation (5.1.27) holds. In fact, by virtue of Theorem 5.1.3, the Revuz correspondence is equivalent to (5.1.27) holding for m-a.e. x 2 X, and consequently for every x 2 X because both sides of (5.1.27) are 1-excessive on X. The construction of A 2 AC c;1 for a given  2 S1 can be done analogously to the proof of Lemma 5.1.8 with the help of Theorem 5.1.6. The construction of  2 S1 for a given A 2 AC c;1 is also similar to the proof of Lemma 5.1.7, but this time we may set ² ³ 1 En D x 2 Gn W .x/ > .5:1:29/ n where Gn are relatively compact open sets increasing to X. Note that En is then Borel and finely open because  D R1A f is a difference of 1-excessive functions by Lemma 5.1.5 and hence Borel measurable in view of assumption (4.2.9) and Theorem A.2.5. C (ii) This follows from (i) and the inclusion AC c;1  Ac . (iii) If ¹En º satisfies the stated condition, then (5.1.28) is valid on account of the quasi-left-continuity of M. If  2 S1 , then we can take by (i) the associated A 2 AC c;1 , which in turn enables us to redefine ¹En º by (5.1.29). Example 5.1.1. Let M D .; M; X t ; Px / be the Brownian motion on Rd . The associated Dirichlet form E on L2 .Rd / is given by . 12 D; H 1 .Rd // (see Example 4.2.1). The absolutely continuous measure .dx/ D f .x/dx;

f  0;

5.1

Positive continuous additive functionals and smooth measures

239

is smooth with respect to the form E when f is locally integrable. In fact,  is then a positive Radon measure charging no set of zero capacity since  vanishes on any set of Lebesgue measure zero. The PCAF of M with Revuz measure  is Z t f .Xs .!//ds: .5:1:30/ A t .!/ D 0

If f is locally bounded, then R1 .1Br f /.x/  supx2Br f .x/ < 1 and consequently  2 S1 . (5.1.30) then gives an associated PCAF in the strict sense. Similar assertions hold even when  diverges around a set of zero capacity. Suppose d  2. Then, for example, f .x/ D jxjˇ ;

x 2 Rd ;

gives a smooth measure for any real ˇ. Indeed, Fn D ¹x 2 Rd W 1=n  jxj  nº serves as a generalized nest because the one point set ¹0º is of zero capacity (see Example 2.2.2 and Example 2.2.3). In view of (2.2.27), we see that  2 S1 if d  3 and ˇ > 2. The associated PCAF A t is still given by (5.1.30). A t satisfies the finiteness condition  Z t ˇ Px jXs j ds < 1 D 1; t > 0; .5:1:31/ 0

whenever x ¤ 0, since ¹0º is polar (see Example 4.2.1). Because of the law of the iterated logarithm, however, (5.1.31) is violated for x D 0 when ˇ  2. In this case, A t is not a PCAF in the strict sense. But it is always a PCAF in our sense because we can ignore the polar set ¹0º as an exceptional set for A t . When d D 1, the Brownian motion on R1 admits no non-empty exceptional set (Example 4.5.1). Hence, the present definition of AF reduces to the strict one. By virtue of the remark at the end of Example 2.1.2, we easily see that a measure  on R1 is smooth if and only if  is a positive Radon measure. In particular, the ı-measure ı¹yº concentrated at one-point y admits a PCAF 2L.t; y; !/ according to Theorem 5.1.4. L.t; y/ is called the Brownian local time. It is known that there exists a set ƒ   with Px .ƒ/ D 1; 8x 2 R1 , such that for each ! 2 ƒ, L.t; y; !/ is continuous in two variables t and y.6 Notice that the formula (5.1.13) serves as a characterization of the correspondence of  2 S and A 2 AC c . By integrating the corresponding formula for  D ı¹yº and A t D 2L.t; y/ by 1E dy, we get Z t Z 1E .Xs /ds D 2 L.t; y/dy; E  R1 : .5:1:32/ 0

6 Cf.

N. Ikeda and S. Watanabe [2].

E

240

5 Stochastic analysis by additive functionals

Furthermore, we get the expression A t .!/ D 2

Z L.t; y; !/.dy/

.5:1:33/

R1

for the PCAF A with a Revuz measure being a positive Radon measure  on R1 . The quasi support of  coincides with the topological support suppŒ on account of Theorem 4.6.2. Hence the support of A equals suppŒ by Theorem 5.1.5. Example 5.1.2. We continue to consider the d -dimensional Brownian motion. Let d  3. We exhibit a measure  2 S00 whose topological support and quasi support differ by a set of positive capacity. Let Fn be the set of points in Rd defined by Fn D ¹z D .x1 ; : : : ; xd / W xk D 2n .1 C 2l/; l 2 N; k D 1; 2; : : : ; d º: S S Put Dn D z2Fn Brn .z/ for rn D 2n.d C1/=.d 2/ . Then D D 1 nD1 Dn is an d open set dense in R . Consider a smooth measure  2 S00 such that .Rd / < 1, .Rd n D/ D 0, .Brn .z// > 0, 8z 2 Fn ; n  1. Note that such measure exists. Indeed, by enumerating the points of Fn as Fn D ¹zn;k ºk1 , it suffices to set 1 X

D

2nk 1Brn .zn;k /  m

.5:1:34/

n;kD1

where m denotes the Lebesgue measure. Since .Rd / is finite and the Newtonian .0/ potential of  is bounded,  2 S00  S00 (see Example 2.2.1). Z t X nk 2 1Brn .zn;k / .Xs /ds At D n;k

0

e) is the associated PCAF in the strict sense of the Brownian motion. Let Y (resp.Y be the topological support (resp. quasi support) of . Then, by virtue of Theoe D ¹z 2 Rd W 'A .z/ D 1º where 'A .z/ D Ez .e R / and R.!/ rem 5.1.5 Y D inf¹t > 0 W A t .!/ > 0º. e/ > 0, which implies that Obviously Y D Rd . We shall show that m.Rd n Y d e e Cap.Y n Y / > 0. It suffices to prove that R n Y D ¹z W 'A .z/ < 1º contains at e, then least one point, for, if there exists a point z0 2 Rd n Y e t p t 'A .z0 / " 'A .z0 / < 1;

t # 0:

e/ > 0, because otherwise This implies that m.Rd n Y e t p t 'A .z0 / D e t p t 1.z0 / ! 1 as t ! 0 by the absolute continuity of p t .z0 ;  / relative to m.

241

5.2 Decomposition of additive functionals of finite energy

Since A increases only when the Brownian motion stays in D r , to show that e, it suffices to prove Pz0 .D D 0/ D 0. We shall prove this for z0 2 Rd n Y z0 D 0 by using Wiener’s test:7 P0 .D D 0/ D 0 if and only if

1 X

2k.d 2/ Cap.Bk / < 1;

kD1

where Bk D D \ .B2k .0/ n B2k1 .0//. Since the function ur .x/ D .jxj2d = r 2d / ^ 1 belongs to H 1 .Rd / and ur .x/  1 on Br .0/ for any r > 0, Cap.0/ .Br .0// 

1 D.ur ; ur / D C.d /r d 2 2

for some constant C.d / depending on d . Noting that the number of points of z 2 Fn for which Brn .z/ intersects with B2k .0/ does not exceed .2nk C 2/d , we have Cap .Dn \ B2k .0//  .2nk C 2/d Cap .Brn .0//  C.d /.2nk C 2/d 2n.d C1/  C 0 .d /2kd 2n e by for some constant C 0 .d /. We can see now that 0 … Y 1 X

2k.d 2/ Cap .D \ B2k .0// 

kD1

1 X

2k.d 2/ Cap .Dn \ B2k .0//

k;nD1

 C 0 .d /

1 X

22k < 1:

kD1

5.2

Decomposition of additive functionals of finite energy

From now on till the end of Chapter 5, we shall deal with those AF’s of the process M which are not necessarily non-negative. For any AF A t of M, we set e.A/ D lim t #0

1 Em .A2t / 2t

.5:2:1/

whenever the limit exists. e.A/ is called the energy of A. First of all, we shall exhibit three important classes of AF’s of finite energy. 7 Cf.

K. Itô and H. P. McKean [1].

242

5 Stochastic analysis by additive functionals

(I) AF’s generated by functions Suppose that a function u on X possesses a version e u .e u D u m-a.e.) such that e u is finely continuous q.e. and finite q.e. Then Œu

At

De u.X t /  e u.X0 /;

t >0

.5:2:2/

defines a finite AF in our sense and, indeed, a unique equivalence class independent of the choice of the version e u by virtue of Lemma 4.1.5. u 2 Fe a quasi AŒu is well-defined for any u 2 Fe because we may take as e Œu continuous version of u (Theorem 2.1.7 and Theorem 4.2.2). Moreover, A t is of finite energy and Z 1 Œu res e.A / D E .u; u/ C .5:2:3/ e u.x/2 k.dx/  E.u; u/; u 2 Fe ; 2 according to the formula (4.5.22). Here E res is the resurrected Dirichlet form defined by (4.5.20). E res equals E if and only if there is no killing inside X. If u 2 F or u 2 Fe in transient case, AŒu is cadlag in view of Lemma 4.2.2 and Corollary 4.3.1. (II) Martingale AF’s of finite energy Consider the family M D ¹M W M is a finite cadlag AF such that for each t > 0 Ex .M t2 / < 1 and Ex .M t / D 0 q.e. x 2 Xº:

.5:2:4/

Since Em .M t2 / is subadditive in t, e.M / is well-defined and 1 Em .M t2 / . 1/ t >0 2t

e.M / D sup

.5:2:5/

for any M 2 M. We set MV D ¹M 2 M W e.M / < 1º:

.5:2:6/

M 2 M if and only if M is a square integrable perfect AF in the strict sense with mean zero of the Hunt process MjX nN , N being some properly exceptional set (depending on M in general). In other words, M belongs to the space Mad with respect to the process MjX nN in the sense of Appendix A.3. Hence we can conclude from the following Theorem A.3.18 that M admits its sharp bracket hM i

5.2 Decomposition of additive functionals of finite energy

243

as a PCAF in the strict sense of MjX nN . Thus each M 2 M admits a PCAF hM i 2 AC c such that Ex .hM i t / D Ex .M t2 / q.e. x 2 X; t > 0:

.5:2:7/

Such hM i is unique for M up to the equivalence in the sense of §5.1. We call M 2 M a martingale additive functional (MAF in abbreviation). hM i is called its quadratic variation or the sharp bracket associated with M . Denote by hM i the Revuz measure of hM i according to Theorem 5.1.3. hM i is called the energy measure of the MAF M . From (5.2.1), (5.2.7) and (5.1.11), we see that the energy of an MAF is just a half of the total mass of its energy measure: 1 e.M / D hM i .X/; 2

M 2 M:

.5:2:8/

For M; L 2 M let hM; Li t D

1 ¹hM C Li t  hM i t  hLi t º : 2

.5:2:9/

Then hM; Li t is a CAF of bounded variation on each finite interval of t and satisfies Ex .M t L t / D Ex .hM; Li t /; 8t > 0; q.e. x 2 X: .5:2:10/ Furthermore the signed measure hM;Li defined by 1 hM;Li D ¹hM CLi  hM i  hLi º 2

.5:2:11/

is related to hM; Li by the relation (5.1.11). Since hM;N i is bilinear, symmetric with respect to M; L 2 M and hM;M i D hM i  0 for all M 2 M, we get for all non-negative f 2 L2 .XI hM i C hLi / sZ ˇZ ˇ sZ ˇ ˇ ˇ f dhM;Li ˇˇ  f dhM i  f dhLi ; ˇ X X X sZ ˇsZ ˇ sZ ˇ ˇ ˇ f dhM i  f dhLi ˇˇ  f dhM Li : (5.2.12) ˇ X

X

X

(III) CAF’s of zero energy Nc D ¹N W N is a finite continuous AF; e.N / D 0; Ex .jN t j/ < 1 q.e. for each t > 0º:

.5:2:13/

244

5 Stochastic analysis by additive functionals

The quadratic variation of N 2 Nc vanishes in the following sense: ŒnT X

.N.kC1/=n  Nk=n /2 ! 0;

n ! 1; in L1 .Pm /

.5:2:14/

kD1

because the expectation of the left-hand side equals ŒnT X

2 2 Em .EXk=n .N1=n //  nT  Em .N1=n / ! 0;

n ! 1:

kD1

An example of N 2 Nc is given by Z Nt D

t

f .Xs /ds

.5:2:15/

0

for any nearly Borel function f 2 L2 .XI m/. Then Z

t

Ex 0

 jf .Xs /jds  e t R1 jf j.x/

is finite q.e. by Theorem 4.2.3 and N is a CAF. Furthermore, Z Em .N t2 /

D 2Em

Z

t

f .Xs / 0

Z t Z D 2Em

0

f .Xv /dvds s



t s

f .Xs /pv f .Xs /dvds 0 0 t s

Z tZ D2



t

0

.ps 1; f  pv f /dvds:

Hence 1 1 Em .N t2 / D 2t t

Z 0

Z

t

.p t s 1; f  Ss f /ds;

Ss f .x/ D

Since the right-hand side of (5.2.16) is dominated by (5.2.15) is of zero energy, i.e., N 2 Nc .

s

pv f .x/dv: .5:2:16/ 0 1 t

Rt 0

sds  .f; f /, N of

5.2 Decomposition of additive functionals of finite energy

245

e

Another example is the PCAF A with Revuz measure  belonging to S0 . Then Ex .A t /  e t U1 .x/ < 1 q.e. and Z t  2 Em .A t / D 2Em .A t  As /dAs 0

D 2 lim Em

n X

n!1

 .A t  Ak t =n /.Ak t =n  A.k1/t =n /

kD1

D 2 lim Em

n X

n!1

 .EXkt=n .A t k t =n //.Ak t =n  A.k1/t =n /

kD1

 2 lim Em

n X

n!1

kD1

Z D 2Em

 .EXkt=n .A t //.Ak t =n  A.k1/t =n / 

t 0

 2t h; E:.A t /i

EXs .A t /dAs

by (5.1.13), and further h; E:.A t /i  e t h; UA1 1  e t p t UA1 1i ! 0;

t ! 0:

We are particularly interested in the sum of the classes (II) and (III): A D MV ˚ Nc

.5:2:17/

namely, A consists of AF’s A such that At D Mt C Nt ;

V N 2 Nc : M 2 M;

.5:2:18/

Clearly A is a linear space of AF’s of finite energy. Moreover, the expression (5.2.18) of A 2 A is unique because MV \ Nc D ¹0º where 0 denotes the additive functional identically zero. In fact, if A 2 MV is of zero energy, then hAi vanishes by (5.2.8) and so does hAi. Hence, A D 0 by (5.2.7). We define the mutual energy of A; B 2 A by e.A; B/ D lim t #0

1 Em .A t B t /: 2t

.5:2:19/

We know by Schwarz inequality that e.A; B/ D 0 when either A or B is in Nc . Therefore, e.A/ D e.M /

if

A D M C N;

V N 2 Nc : M 2 M;

.5:2:20/

246

5 Stochastic analysis by additive functionals

The main purpose of this section is to show that any AF AŒu .u 2 Fe / in the first class (I) actually belongs to A, i.e., AŒu is expressed as a sum of elements of MV and Nc uniquely. To this end we first prove that the space MV is a Hilbert space with inner product (5.2.19). Theorem 5.2.1. MV is a real Hilbert space with inner product e. More specifiV there exists a unique M 2 MV and cally, for any e-Cauchy sequence M .n/ 2 M, .n/ a subsequence nk such that limn!1 e.M  M / D 0 and for q.e. x 2 X,   .n / Px lim M t k D M t uniformly on any finite interval of t D 1: k!1

Proof. By virtue of Lemma 5.1.9 and identities (5.2.7), (5.2.8), we have E .M t2 /  2.1 C t/kU1 k1 e.M /;

 2 S00 :

.5:2:21/

.M t ; P / t 0 is then a square integrable martingale and Doob’s inequality implies P



 2 sup jMs j > ˛  2 .1 C T /kU1 k1 e.M /: ˛ 0sT

.5:2:22/

Assume that M .n/ 2 MV constitutes an e-Cauchy sequence and select a subsequence nk ! 1 such that e.M .nkC1 /  M .nk / / < 1=23k . By (5.2.22),   1 2.1 C T /kU1 k1 .n / P : sup jMs kC1  Ms.nk / j > k  2 2k 0sT Applying the Borel–Cantelli lemma to this inequality, we see P .ƒc0 / D 0; .n /

 2 S00

.5:2:23/

where ƒ0 D ¹! 2  W Ms k converges uniformly in s on each finite intervalº. In view of Theorem 2.2.3, (5.2.23) implies T Px .ƒ0 / D 1 q.e. x 2 X . Denote by .n / k and put ƒ1 D k k . Furthermore, we set Ms .!/ D k a defining set of M .n / limk!1 Ms k .!/; s  0, for ! 2 ƒ0 \ ƒ1 . Ms is then an AF with defining set .n/ ƒ0 \ ƒ1 . Since M t is L2 .P / convergent to M t by virtue of (5.2.21), we see E .M t2 / < 1 and E .M t / D 0 for any  2 S00 . It is easy to conclude from this with the help of Theorem 2.2.3 that Ex .M t2 / < 1 and Ex .M t / D 0 q.e. x 2 X , that is, M 2 M. For any " > 0, choose N such that e.M .n/  M .m/ / < ", .n/ n; m > N . We have .1=2t/Em ..M t  M t /2 /  " by Fatou’s lemma and (5.2.5). Then e.M .n/  M /  " by (5.2.5) again. Hence, M 2 MV and M .n/ is e-convergent to M .

5.2 Decomposition of additive functionals of finite energy

247

Theorem 5.2.2. For any u 2 Fe , the AF AŒu can be expressed uniquely as AŒu D M Œu C N Œu ;

V M Œu 2 M;

N Œu 2 Nc :

.5:2:24/

Moreover, it holds that e.M

Œu

1 / D E .u; u/ C 2 res

Z e u.x/2 k.dx/  E.u; u/:

.5:2:25/

Proof. We already know the uniqueness of the decomposition (5.2.24) and it suffices to show the existence of such M Œu and N Œu . We start with the special case where u is in the range of the resolvent: u D R1 f , f being a Borel function in L2 .XI m/. In this case (5.2.24) is reduced to the usual semimartingale decomposition, namely, we may set ´ Rt Œu N t D 0 .u.Xs /  f .Xs // ds .5:2:26/ Œu Œu M t D u.X t /  u.X0 /  N t ; t  0: As we have seen, N Œu of (5.2.26) belongs to Nc . Since u is an element of F , Œu A t D u.X t /u.X0 / is of finite energy and M Œu of (5.2.26) satisfies the relation V (5.2.25) in view of (5.2.3). We easily see that M Œu 2 M and hence M Œu 2 M. Next, take any Borel function u 2 F and define the approximating function un by un D nRnC1 u D R1 fn ; fn D n.u  nRnC1 u/: .5:2:27/ By the uniqueness of the decomposition (5.2.24) for un ’s, we have then M Œun   M Œum  D M Œun um  and Z 1 Œun  Œum  res M / D E .un  um ; un  um / C .un  um /2 d k: .5:2:28/ e.M 2 R Since un 2 F and E1 -convergent to u and E.v; v/ dominates E res .v; v/C12 v 2 d k V for any v 2 F , we conclude that M Œun  is an e-Cauchy sequence in the space M. By virtue of Theorem 5.2.1, the formula ´ V e/ M Œu D limn!1 M Œun  in .M; .5:2:29/ N Œu D AŒu  M Œu ; u 2 F makes sense and M

Œu

V 2 M;

e.M

Œu

1 / D E .u; u/ C 2 res

Z e u.x/2 k.dx/;

u2F:

.5:2:30/

248

5 Stochastic analysis by additive functionals

It only remains to show that N Œu of (5.2.29) belongs to the space Nc . Note that a subsequence nk exists and Px .N Œunk  converges to N Œu uniformly on any finite interval of t / D 1 q.e. x 2 X; (5.2.31) because the same statements for AŒun  and M Œun  hold on account of Lemma 5.1.2 and Theorem 5.2.1 respectively. From this and (5.2.26), we know that N Œu is a CAF. On the other hand, we have from (5.2.29) that Œu

Nt

Œuun 

D At

Œu

 .M t

Œun 

 Mt

Œun 

/ C Nt

and, consequently lim t #0

1 Œu Em ..N t /2 /  3e.AŒuun  / C 3e.M Œu  M Œun  /: 2t

By virtue of (5.2.3), (5.2.28) and (5.2.29), the right-hand side is dominated by 6E.u  un ; u  un /, which can be made arbitrarily small with large n. Therefore, e.N Œu / D 0 and N Œu 2 Nc . Finally suppose that u 2 Fe . Let ¹un º  F be an approximating E-Cauchy sequence such that un ! u m-a.e. In view of (5.2.28) and Theorem 5.2.1, Œu  limn!1 M t n converges uniformly on any finite interval Px -a.s. Therefore, similarly to the previous case, it suffices to prove the uniform convergence of ¹AŒun  º on any finite interval. If M is transient, this follows from Lemma 5.1.2. In the general case, by virtue of Lemma 1.6.7, there exists a bounded strictly positive function g 2 K such that u 2 Feg . Then the approximating sequence ¹un º can be taken in such a way that ¹un º is contained in F g and Cauchy relative to E g . Owing to Lemma 4.6.7, we can take the canonical subprocess .X tg ;  g ; Px / of M by the multiplicative Rt functional exp. 0 g.Xs /ds/ as a Hunt process associated with .E g ; F g /. Since .E g ; F g / is transient, Lemma 5.1.2 implies that Px .ƒgT / D 0 q.e., where ® ƒgT D ! 2  W lim

¯ sup je un .Xsg /  e u.Xsg /j > 0 :

n!1 0sT

Therefore, for ® ƒT D ! 2  W lim

¯ sup je un .Xs /  e u.Xs /j > 0 ;

n!1 0sT

5.2 Decomposition of additive functionals of finite energy

249

RT

Px .ƒT /  e kgk1 T Ex .e  0 g.Xs /ds I ƒT /   Z T g kgk1 T Px ƒT I g.Xs /ds < Z D 0 q.e. x 2 X; De 0

as was to be proved for general u 2 Fe .8 Corollary 5.2.1. The following linearity and continuity hold: (i) M ŒauCbv D aM Œu C bM Œv ; N ŒauCbv D aN Œu C bN Œv ; a; b 2 R1 , u; v 2 Fe . (ii) un , u 2 F (2 Fe in transient case), E1 .un  u; un  u/ ! 0; n ! 1 Œunk 

) there exists a subsequence ¹unk º such that limnk !1 M t Œunk 

limnk !1 N t q.e. x 2 X .

Œu

D Nt

Œu

D Mt ;

uniformly on each finite interval of t , Px -a.s. for

Proof. The relation (i) follows from the uniqueness of the decomposition (5.2.24). The equality (5.2.25) means that the linear transform ˆ W u 7! M Œu

.5:2:32/

V e/. Combinis continuous from the Dirichlet space .Fe ; E/ into the space .M; ing this with Theorem 5.2.1 and Lemma 5.1.2, we conclude the continuity statement (ii). (5.2.24) also implies the following embedding statement telling us that the structure of the Dirichlet space may be studied entirely within the framework of the space MV of martingales of finite energy. Corollary 5.2.2. Suppose that the process M has no killing inside X . Then the transformation ˆ of .5:2:32/ is an isometry and the extended Dirichlet space V e/. .Fe ; E/ can be identified by ˆ with a closed subspace of .M; Having established decomposition (5.2.24), we are now in a position to study some basic properties of AFs M Œu and N Œu in (5.2.24). In this section we shall give basic formulae of computing M Œu and N Œu . First of all, we identify the energy measure of the martingale part M Œu with the energy measure hui of the function u in the sense of §3.2. Denote by Fb and Fe;b the family of all 8Z

is an exponentially distributed random variable independent of X t (see A.2).

250

5 Stochastic analysis by additive functionals

m-essentially bounded functions in F and Fe , respectively. We note that Fe;b is an algebra (Corollary 1.6.3) and further that 1 .u  p t u; v/ ! E.u; v/; t

t # 0; u 2 Fe ; v 2 F :

.5:2:33/

Indeed u  p t u 2 L2 .XI m/ even when u 2 Fe by virtue of Lemma 1.5.4, and hence the left-hand side of (5.2.33) is finite and equal to the polarized expression of the left-hand side of the formula (4.5.7). Therefore we get (5.2.33) from (4.5.7). Keeping this in mind, let us proceed to the proof of Theorem 5.2.3. hM Œu i equals hui for u 2 Fe . In particular, for u 2 Fe;b , Z e.x/ Œu .dx/ D 2E.u  f; u/  E.u2 ; f /; f 2 Fb : .5:2:34/ f hM i X

Proof. We first show (5.2.34) for u 2 Fe;b . By (5.1.11) and (5.2.7), Z e.x/ Œu .dx/ D lim 1 Ef m ..M Œu /2 / f t hM i t #0 t X

.5:2:35/

Œu

when f is a bounded -excessive function .  0/ in F . Since M t D e u.X t /  Œu Œu 2 Œu 2 e u.X0 /  N t and Ef m ..N t / /  kf k1 Em ..N t / /, we can see that the right-hand side of (5.2.35) equals ³ ² 2 1 1 2 2 2 lim Ef m ..e u.X t /  e u.X0 // / D lim .uf; u  p t u/  .f; u  p t .u // t t #0 t t #0 t which is equal to the right-hand side of (5.2.34) because u; u2 2 Fe ; f 2 F ; uf 2 Fe \ L2 D F and (5.2.33) applies. e. Suppose E is Next take any f 2 Fb . (5.2.34) is now valid for fn D nRn f transient, then fn and ufn are E-weakly convergent to f and uf respectively on account of Corollary 1.5.1. Since fn converges boundedly to f q.e. as well, we get (5.2.34) for f from that for fn . In the general case, we choose g 2 K so that u 2 Feg . Since g 2 L1 .XI m/, the same E-weak convergences as above take place on Feg and we can get (5.2.34) in the same way. (5.2.34) along with (3.2.14) leads us to the identity hM Œu i D hui for u 2 Fb . This identity readily extends to u 2 Fe by noting the inequality (5.2.12) and the bound ˇZ ˇ ˇ ˇ ˇ f dhM Œu i ˇˇ  2kf k1 E.u; u/ ˇ X

which follows from (5.2.8) and (5.2.25).

5.2 Decomposition of additive functionals of finite energy

251

We next state a characterization of N Œu . Theorem 5.2.4. The following conditions are equivalent for an AF A and for a function u 2 Fe : (i) A D N Œu , (ii) A 2 Nc and, for each t > 0, Ex .A t / D p t u.x/  u.x/; (iii) A 2 Nc and

m-a.e. x 2 X;

1 lim Evm .A t / D E.u; v/; t #0 t

8v 2 F :

.5:2:36/

.5:2:37/

(iv) A 2 Nc and .5:2:37/ holds for v D R1 f for any bounded Borel f 2 L1 .XI m/. Proof. The implication (i) ) (ii) is clear. (ii) ) (iii) follows from the equation (5.2.33). (iii) ) (iv) is trivial. Assume that an AF A satisfies the condition (iv). Let us put c t .x/ D Ex .A t /, then we see from A 2 Nc and the relation c t Cs .x/ D c t .x/ C p t cs .x/ q.e. for each t; s > 0;

.5:2:38/

that c t is in L2 .XI m/ and L2 -right continuous in t > 0. By (5.2.37), (5.2.38), (1.5.5) and Lemma 1.5.4, we have that for any v of the stated type, 1 1 .v  pT v; c t / D lim .ST v  ps ST v; c t / D lim .ST v  p t ST v; cs / s#0 s s#0 s D  E.ST v  p t ST v; u/ D  E.ST v; u  p t u/ D  .v  pT v; u  p t u/;

t > 0; T > 0:

Therefore, l t D .v; c t C u  p t u/ satisfies the equation l t D l t CT  lT of the linearity. Since lim t #0 .1=t/l t D 0 by (5.2.37) and (5.2.33), l t vanishes and we get (5.2.36). Finally, we assume condition (ii) for A. In order to derive (i), we set B t D Œu Œu u.X t /  e u.X0 /  M t  A t ; q t .x/ D Ex .B t2 / and claim that Nt  At D e q t .x/ D 0 q.e. for each t > 0:

.5:2:39/

Clearly B t is a CAF of zero energy R and Ex .B t / D 0 for m-a.e. x 2 X on account of the assumption (ii). But then X q t .x/m.dx/ D 0 because the left-hand side

252

5 Stochastic analysis by additive functionals

is subadditive in t. We thus have q t .x/ D 0 m-a.e., from which we can conclude ps q t D 0 q.e. for each s > 0. By using Fatou’s lemma, q t .x/ D Ex



 lim .B t C1=n  B1=n /2  lim p1=n q t .x/ D 0 q.e.;

n!1

n!1

getting (5.2.39). So far in this section, we have treated AF’s of various types but always admitting exceptional sets. Under the extra condition of the absolute continuity of the transition function, we may get a decomposition like (5.2.24) involving only AF’s in the strict sense as will be suggested by the next theorem. We call M t .!/ a MAF in the strict sense if it is a finite cadlag AF in the strict sense and .5:2:40/ Ex .M t / D 0; Ex .M t2 / < 1; 8x 2 X: According to Theorem A.3.18, there exists then a unique hM i 2 AC c;1 such that Ex .hM i t / D Ex .M t2 /;

8x 2 X:

We call hM i the sharp bracket of M in the strict sense. Theorem 5.2.5. Assume that M satisfies the absolute continuity condition .4:2:9/. Suppose that a function u satisfies the following conditions: (i) u 2 F ; u is bounded and finely continuous. (ii) hui 2 S00 . (iii) 9 D  .1/   .2/ with  .1/ ;  .2/ 2 S00 and, for a dense subset L of F , E.u; v/ D h;e v i;

8v 2 L:

Then Œu

u.X t /  u.X0 / D M t

Œu

C Nt

Px -a.s.; 8x 2 X:

.5:2:41/

Œu

Here M t is an MAF in the strict sense whose sharp bracket in the strict sense has the Revuz measure hui . Further N Œu D A.1/ C A.2/

Px -a.s.; 8x 2 X;

.5:2:42/

where A.1/ (resp. A.2/ ) is a PCAF in the strict sense with Revuz measure  .1/ (resp.  .2/ ).

5.2 Decomposition of additive functionals of finite energy

253

Proof. The existence of the PCAF A.i / in the strict sense with Revuz measure  .i / has been assured by Theorem 5.1.7, i D 1; 2. By virtue of Theorem 5.1.3 and condition (iii), A D A.1/ C A.2/ satisfies the equation (5.2.37) for any v 2 F and consequently A is a strict version of N Œu according to Theorem 5.2.4. We define N Œu by (5.2.42) and set Œu

Mt

Œu

D u.X t /  u.X0 /  N t ;

t > 0:

Œu V In particular Ex .M Œu / D 0 Then M t is an AF in the strict sense and M Œu 2 M. t .i / q.e. By noting that Ex .A t / < 1; 8x 2 X; i D 1; 2, we get Œu

Œu

Ex .M t C" /  Ex .M"Œu / D Ex .EX" .M t // Z Œu D p" .x; dy/Ey .M t / D 0; X

8x 2 X;

and by letting " # 0, we are led to Œu

Ex .M t / D 0;

8x 2 X:

Denote by B the PCAF in the strict sense with Revuz measure hui (Theorem 5.1.7). Then by Theorem 5.2.3 Œu

Ex ..M t /2 / D Ex .B t /

q.e.

Therefore as in the above Œu

Ex ..M t C"  M"Œu /2 / D Ex .B t C"  B" /;

8x 2 X

.5:2:43/

and by Fatou’s lemma Œu

Ex ..M t /2 /  Ex .B t /;

8x 2 X:

.5:2:44/

The right-hand side is dominated by e t R1 hui .x/ < 1. We have proved that M Œu is an MAF in the strict sense. Since the sharp bracket in the strict sense of M Œu has hui as its Revuz measure by Theorem 5.2.3, it coincides with B by virtue of Theorem 5.1.7, and consequently the equality takes place in (5.2.44). But we derive the equality more directly for the sake of later use. Œu Denoting M t by M t , we then have for any x 2 X q ˇ q ˇq ˇ 2 ˇ 2 ˇ Ex ..M t C"  M" / /  Ex .M t /ˇ  Ex ..M t C"  M t  M" /2 / q q q 2 2 D Ex ..M" ı  t  M" / /  Ex .EX t .M" // C Ex .M"2 / p p  Ex .EX t .B" // C Ex .B" /;

254

5 Stochastic analysis by additive functionals

which tends to 0 as " # 0. Hence, by letting " # 0 in (5.2.43), we can replace the inequality in (5.2.44) by the equality as was desired. Example 5.2.1. Consider the case where X is an Euclidean domain D and M is an m-symmetric diffusion process on D whose Dirichlet form E on L2 .DI m/ possesses C01 .D/ as its core. We assume for simplicity that M admits no killing inside D. From Theorem 3.2.3 and Theorem 4.5.3, we know then that E should have the expression E.u; v/ D

d Z X i;j D1 D

@u.x/ @v.x/ ij .dx/; @xi @xj

u; v 2 C01 .D/;

with some measures ij satisfying (1.2.3). Now the right-hand side of (5.2.34) equals 2

d Z X i;j D1 D

@u.x/ @u.x/ f .x/ij .dx/ for u; f 2 C01 .D/; @xi @xj

and thus (5.2.34) implies hM Œu i .dx/ D 2

d X @u.x/ @u.x/ ij .dx/; @xi @xj

u 2 C01 .D/;

.5:2:45/

i;j D1

which equals the expression (3.2.30) of hui of course. Suppose that every ij is absolutely continuous with respect to the Lebesgue P measure. Take Borel density functions aij .x/ such that dij D1 aij .x/i j  0, 8x 2 D, 8 2 Rd . In view of Theorem 5.1.3, the identity (5.2.45) then implies hM

Œu

Z t X d  @u @u it D 2 aij .Xs /ds; @xi @xj 0

u 2 C01 .D/:

.5:2:46/

i;j D1

Furthermore, if aij satisfies 0 jj2 

d X

aij .x/i j  ƒ0 jj2 ;

8x 2 D; 8 2 Rd ;

.5:2:47/

i;j D1

for some positive constants 0 and ƒ0 and if m.dx/ D dx, then (5.2.46) holds for any u 2 Fb namely, for any bounded function in H01 .D/, because so does (5.2.45).

5.2 Decomposition of additive functionals of finite energy

255

Example 5.2.2 (Skorohod representation of the reflecting Brownian motion). Let M D .X t ; Px / be the reflecting Brownian motion on a Lipschitz domain D  Rd considered in Example 4.5.3. Denote by m the Lebesgue measure on D. M is an m-symmetric conservative diffusion on D satisfying the absolute continuity condition (4.2.9) and possessing the regular Dirichlet form .E; F / D . 12 D; H 1 .D// on L2 .D/. C01 .D/ is actually a core of E. The surface measure  on the boundary @D can be defined as follows. Consider a coordinate neighbourhood D \ U D ¹.x 0 ; xd / W xd > F .x 0 /º \ U with a Lipschitz function F . Since F is differentiable a.e. with bounded rF , we may let for a Borel B  @D \ U Z .1 C jrF .x 0 /j2 /1=2 dx 0 ; B D ¹x 0 W .x 0 ; F .x 0 // 2 Bº: .B/ D B

-a.e. point x 2 @D then admits a unit inward normal vector n.x/ D .n1 .x/; : : : ; nd .x// and an associated divergence theorem holds. Consider the coordinate functions 'i .x/ D xi ; 1  i  d , and let '.x/ D .'1 .x/; : : : ; 'd .x// ; x 2 D. Then 'i 2 F and d X @'i @'j  m D ıij m; @xk @xk kD1 Z Z 1 1 @v E.'i ; v/ D dx D  e v .x/ni .x/.dx/; v 2 C01 .D/; 2 D @xi 2 @D

h'i ;'j i D

(5.2.48) (5.2.49)

due to the divergence theorem. Let r˛ .x; y/; ˛ > 0; x; y 2 D, be the R resolvent density of M in accordance with Lemma 4.2.4. Then R1 m.x/ D D r1 .x; y/m.dy/ D 1, x 2 D, and so m 2 S00 . Using a known bound9 r1 .x; y/  C jx  yj.d 2/ R ; d > 2 ; r1 .x; y/  1 C.log jxyj _ 1/; d D 2, we also have that R1 .x/ D @D r1 .x; y/.dy/ is bounded and hence  2 S00 . Therefore, in virtue of Theorem 5.1.7, there are unique PCAF’s in the strict sense with Revuz measure m and  respectively. The former coincides with the constant AF t and the R t latter denoted by L t is called the local time of M on the boundary. Obviously 0 ni .Xs /dLs is then a CAF in the strict sense with (signed) Revuz measure ni  . Theorem 5.2.5 now applies in getting Œ'

'.X t / D '.X0 / C M t 9 Cf.

R. F. Bass and P. Hsu [2].

Œ'

C Nt

Px -a.s.; 8x 2 D

256

5 Stochastic analysis by additive functionals Œ'

where M t is a continuous MAF in the strict sense with hM Œ'i  ; M Œ'j  i t D ıij  t Rt Œ'  and N t i D 12 0 ni .Xs /dLs . Using a martingale characterization of Brownian motion10 , we arrive at the representation Z 1 t X t D X0 C B t C n.Xs /dLs ; Px -a.s.; 8x 2 D; .5:2:50/ 2 0 where B t denotes a d -dimensional Brownian motion with respect to the filtration of M. Notice that the third term in the right-hand side represents a singular inward drift at the boundary which forces X t to stay inside D against the isotropic nature of the Brownian motion B t . We call (5.2.50) a Skorohod representation of the reflecting Brownian motion X t . Theorem 5.2.2, Theorem 5.2.3 and Theorem 5.2.5 will be extended in §5.5 when E is local.

5.3

Martingale additive functionals and Beurling–Deny formulae

In §3.2, we derived the Beurling–Deny formulae in an analytical way. In this section we give a probabilistic approach to them by making use of the decomposition Œu

u.X0 / D M t e u.X t /  e

Œu

C Nt ;

V N Œu 2 Nc ; u 2 F ; M Œu 2 M;

in Theorem 5.2.2 combined with the general theory of martingale additive functionals summarized in Appendix A.3. The latter will enable us to give a further decomposition of the martingale part M Œu . Let us first make the following observation. As was stated in the beginning of 5:1, a functional B of M is an AF in the present sense if and only if B is an AF in the strict sense of the Hunt process MjX nN , N being a properly exceptional set depending on B in general. Therefore Theorems A.3.17, A.3.18, A.3.19 of Appendix A.3 still work in producing AF’s in the present sense. We shall take this for granted in the sequel. In particular, Theorem A.3.18 and the paragraph following it provide us, for a given M 2 M, with its square bracket Œ M  and sharp bracket hM i as PAF’s. Actually hM i is a PCAF. We have made this observation for hM i in the last section already. Theorem A.3.18 further implies that M 2 M is uniquely decomposed as M D Mc C Md; 10 Cf.

N. Ikeda and S. Watanabe [2].

M c 2 Mc ; M d 2 Md ;

5.3

Martingale additive functionals and Beurling–Deny formulae

257

where Mc D ¹M 2 M W Px .M t is continuous in t / D 1; q.e. xº;

(5.3.1)

Md D ¹M 2 M W hM; Li D 0; Px -a.e. for q.e. x for any L 2 Mc º: MAF’s M c and M d are called the continuous part and the purely discontinuous part of the MAF M respectively. Let us denote by M Œu;c and M Œu;d the continuous part and purely discontinuous part of the MAF M Œu for u 2 F . We further let M t D M t  M t  , t > 0, Œu M 2 M. Since N t is continuous and e u./ D 0, we have that Œu

M 1¹ t º D e u.X  /1¹ t º : By virtue of the quasi-left continuity of the Hunt process, we see that there exists a CAF .e u.X  /1¹ t º /p such that u.X  /1¹ t º  .e u.X  /1¹ t º /p M Œu;k D e is a purely discontinuous MAF. In fact, we may take as .e u.X  /1¹ t º /p the dual predictable projection of e u.X  /1¹ t º in accordance with Lemma A.3.1 and Theorem A.3.17. Put M Œu;j D M Œu;d  M Œu;k : Then the MAF M Œu is decomposed as M Œu D M Œu;c C M Œu;j C M Œu;k :

(5.3.2)

Note that hM i is the dual predictable projection Œ M p of Œ M  and Œ M  t D P hM c ; M c i t C 0<st .Ms /2 for M 2 M by (A.3.16) and (A.3.17). Accordingly P Œu;j Œu;k Œu;j Œ M Œu;j ; M Œu;k  t D 0<st Ms  Ms D 0 because M D 0 by the definition of M Œu;j . Hence hM Œu;j ; M Œu;k i D 0 and the decomposition (5.3.2) is orthogonal with respect to the sharp bracket h  ;  i. As the result, we have the identity hM Œu i D hM Œu;c i C hM Œu;j i C hM Œu;k i ;

(5.3.3)

where hM i denotes the Revuz measure of a PCAF hM i for M 2 M. Using the mutual energy of MAF’s, we introduce three symmetric forms by EO .c/ .u; v/ D e.M Œu;c ; M Œv;c /; EO .k/ .u; v/ D 2e.M Œu;k ; M Œv;k /:

EO .j / .u; v/ D e.M Œu;j ; M Œv;j /;

(5.3.4)

258

5 Stochastic analysis by additive functionals

Then, by taking the half of the total mass of each measure appearing in (5.3.3) and making polarization, we get from (5.2.25) that Z 1 1 res e ue v d k D EO .c/ .u; v/ C EO .j / .u; v/ C EO .k/ .u; v/; u; v 2 F : E .u; v/ C 2 X 2 (5.3.5) Let .N.x; dy/; H / be a Lévy system of the Hunt process M (Theorem A.3.21). Let  be the Revuz measure of the PCAF H and put 1 JO .dx; dy/ D N.x; dy/.dx/; 2 Using the identity (A.3.25), we have

O k.dx/ D N.x; /.dx/:

p

hM Œu;j i t D Œ M Œu;j  t  X p D .MsŒu /2 1¹t < º 0<st

D

 X

.e u.Xs /  e u.Xs //2 1¹Xs 2X º

(5.3.6)

(5.3.7)

p

0<st

Z tZ D

0

X

.e u.Xs /  e u.y//2 N.Xs ; dy/dHs

and Œu

hM Œu;k i t D ..M /2 1¹ t º /p

(5.3.8)

D .e u.X  /2 1¹ t º /p p  X D 1¹º .Xs /e u.Xs /2 1¹Xs ¤Xs º Z D

0<st

0

t

e u.Xs /2 N.Xs ; /dHs :

Therefore

Z hM Œu;j i .dx/ D 2

X

.e u.x/  e u.y//2 JO .dx; dy/;

O hM Œu;k i .dx/ D e u.x/2 k.dx/:

(5.3.9) (5.3.10)

Lemma 5.3.1. If u 2 F is constant on a nearly Borel finely open set G, then hM Œu;c i .G/ D 0 and Œu;c

Mt

D 0;

8t  G ; Px -a.s. for q.e. x 2 X;

where G D inf¹t > 0 W X t … Gº (D X nG ^ ).

(5.3.11)

5.3

Martingale additive functionals and Beurling–Deny formulae

259

P .n/ .n/ u.X k t /  e u.X k1 t //2 . Then B t is equal to zero on Proof. Let B t D nkD1 .e n n t < G . On the other hand, because of (5.2.10), (5.2.14) and Theorem A.3.10, X .n/ lim B t D hM Œu;c i t C .e u.Xs /  e u.Xs //2 Pm -a.s.; n!1

0<st

by taking a subsequence if necessary. Hence, we have hM Œu;c i t D 0;

t  G ; Pm -a.s.;

and we attain the assertions of the lemma by virtue of Lemma 5.1.10 (iii). Lemma 5.3.2. For any u; v 2 F \ C0C .X/ such that suppŒu \ suppŒv D ;, Z 1 u.x/v.x/JO .dx; dy/ D  E.u; v/: (5.3.12) 2 X X nd Proof. Consider a relatively compact open set G satisfying suppŒu  G  G  R c .suppŒv/ . We put Nf .x/ D X f .y/N.x; dy/. Since Z  Z 1 v  N jf jd C jf j  N vd ; e.M Œjf j;j ; M Œv;j / D  2 X X f 2 F \ C0 .G/; we have

Z X

1

1

jf j  N vd  2e.M Œjf j;j / 2  e.M Œv;j / 2  C

p E.f; f /;

and hence 1G  N v   is of finite energy integral with respect to EG . On the other hand, we have  X  HX˛nG v.x/ D Ex e ˛s .v.Xs /  v.Xs // 0<sG

Z D Ex

0

Z D Ex

G

Z

X G

e ˛s .v.y/  v.Xs //N.Xs ; dy/dHs

e ˛s N v.Xs /dHs





0

in view of the property of the Lévy system. Thus HX˛nG v.x/ is equal to U˛G .N v/ on G, which means that HX˛nG v D U˛G .N v  / C v. Since E˛ .HX˛nG v; u/ D 0 by virtue of Theorem 4.4.1, Z u.x/v.y/JO .dx; dy/ D E˛ .U˛G .N v  /; u/ D E.u; v/: 2 X X nd

260

5 Stochastic analysis by additive functionals

Theorem 5.3.1. Let J and k be the jumping measure and the killing measure appearing in Theorem 3:2:1. Then, it holds that J D JO ;

O k D k:

(5.3.13)

Further, E can be decomposed as E.u; v/ D EO .c/ .u; v/ C EO .j / .u; v/ C EO .k/ .u; v/; where EO .j / .u; v/ D EO .k/ .u; v/ D

ZZ Z X

X X nd

u; v 2 F ;

(5.3.14)

.e u.x/  e u.y//.e v .x/  e v .y//JO .dx; dy/;

O e u.x/e v .x/k.dx/:

EO .c/ has the strong local property .4:5:14/. Proof. By (3.2.4) and Lemma 5.3.2, JO is identical with J . The strong local property of EO .c/ follows from Lemma 5.3.1. Therefore, using the relation (5.3.5), we can conclude by the same argument as in Theorem 3.2.1 (I) that kO is identical with k. This theorem says that the decomposition (5.3.14) is nothing but the Beurling– Deny formula (4.5.13), and moreover the jumping measure J and the killing measure k admit specific expressions (5.3.6), which enable us to derive some properties of J and k stated in Lemma 4.5.2 and Lemma 4.5.5 by using the property of the Lévy system. For example, since for f 2 B C  Z t  Z t Ex f .Xs /N.Xs ; /dHs D Ex f .Xs /d.1¹X ¤; sº / ; 0

0

(5.3.15) we can regard the killing measure k as the Revuz measure of the PCAF .1¹X ¤; t º /p . Therefore Theorem 5.1.3 leads us to the identity (4.5.6). Lemma 5.3.3. The local part chui of the energy measure hui defined in 3:2

coincides with the energy measure of M Œu;c : chui D hM Œu;c i ;

u2F:

In particular chui vanishes on a nearly Borel finely open set G whenever u 2 F is constant on G.

5.4 Continuous additive functionals of zero energy

261

Proof. By Lemma 4.5.4 and the definitions of hui and chui , Z

hui D chui C 2

X

.e u.  /  e u.y//2 J.  ; dy/ C e u2 k;

for u 2 Fb ;

(5.3.16)

and thus chui D hM Œu;c i for u 2 Fb because of Theorem 5.2.3, (5.3.3), (5.3.9), (5.3.10) and (5.3.13). The identity extends to u 2 F owing to the inequality preceding to Lemma 3.2.3 and inequality (5.2.12). The formula (5.3.16) also extends to u 2 F . The last sentence rephrases Lemma 5.3.2.

5.4

Continuous additive functionals of zero energy

We saw in §5.2 that any AF in the class Nc is of zero quadratic variation in a certain sense. Nevertheless, Nc contains many CAF’s which are not of bounded variation. In this section we concentrate our attention on the second term N Œu 2 Nc in the decomposition (5.2.24) of Theorem 5.2.2 and give some answers to the next basic questions: (I) Where is the support of N Œu located? Œu

(II) When is N t of bounded variation in t ? An AF A is said to be of bounded variation if A t .!/ is of bounded variation in t on each compact subinterval of Œ0; .!// for every fixed ! in a defining set of A. A CAF A is of bounded variation if and only if A can be expressed as a difference of two PCAF’s: .1/

.2/

A t .!/ D A t .!/  A t .!/;

t < .!/; A.1/ ; A.2/ 2 AC c :

.5:4:1/

If an AF A is continuous and of bounded variation, then its total variation ¹Aº t Rt .1/ .D 0 jdAs j/ is a PCAF 11 and an expression as above is provided by A t D ¹Aº t , .2/ A t D ¹Aº t  A t for instance. For the sake of definiteness, we say that each measure  in S0 is of finite 1-order energy integral. In the transient case, the same role is played by a measure  of finite 0-order energy integral ( 2 S00 in notation) which is defined as a positive Radon measure satisfying Z p jv.x/j.dx/  C E.v; v/; 8v 2 F \ C0 .X/: X

Let us say that a signed Radon measure  on X is of finite 1-order (resp. 0-order) energy integral if so is the total variation jj. This happens if and only if  can be 11 Cf.

P. A. Meyer [2].

262

5 Stochastic analysis by additive functionals

expressed as  D  .1/   .2/ ;  .1/ and  .2/ being positive Radon measures of the same type. We can then define the ˛-potential of  by U˛  D U˛  .1/  U˛  .2/ (resp. 0-potential of  by U D U .1/  U .2/ ) which does not depend on the v of any v 2 F choice of  .1/ and  .2/ . Moreover, the quasi continuous version e (resp. v 2 Fe ) is integrable with respect to the total variation jj and v i; E˛ .U˛ ; v/ D h;e

v 2F;

.5:4:2/

v 2 Fe :

.5:4:2/0

E.U; v/ D h;e v i;

Given a signed Radon measure  of finite 1-order (resp. 0-order in transient case) energy integral, we express it as  D  .1/   .2/ in the above way and set A D A.1/  A.2/ ;

.5:4:3/

where A.1/ and A.2/ are finite PCAF’s corresponding to  .1/ and  .2/ respectively according to Theorem 5.1.1. A of (5.4.3) is a finite CAF of bounded variation on Œ0; 1/ and does not depend on the choice of  .1/ and  .2/ . Lemma 5.4.1. For any signed Radon measure  as above, Z t ŒU˛  D˛ U˛ .Xs /ds  A t ; t  0; Nt 0

e

.5:4:4/

and in the transient case, ŒU

Nt

D A t ;

.5:4:5/

where A is the CAF of bounded variation associated with  by .5:4:3/. Proof. In view of the linearity (Corollary 5.2.1 to Theorem 5.2.2), we may assume that  is non-negative. Take  2 S0 and denote by N t the right-hand side of (5.4.4). We have seen in §5.2 (III) that N 2 Nc . Moreover, by virtue of Theorem 5.1.3 (vi), we get 1 v i D E.U˛ ; v/; Evm .N t / D ˛.U˛ ; v/  h;e t !0 t lim

v 2F;

which proves (5.4.4) by Theorem 5.2.4. The proof of (5.4.5) is similar. In order to formulate an answer to the question (I) raised in the beginning of this section, recall the definition of the ˛-spectrum ˛ .u/ of u 2 F given in §2.3. In particular, a set .u/ D 0 .u/ is called the (0-)spectrum of u 2 Fe if .u/ is the complement of the largest open set G such that E.u; v/ vanishes for any v 2 F \ C0 .X/ with suppŒv  G.

263

5.4 Continuous additive functionals of zero energy Œu

Theorem 5.4.1. (i) For any u 2 Fe , the CAF N t vanishes on the complement of the spectrum F D .u/ of u in the following sense: Œu

Px .N t

D 0; 8t < F / D 1 q:e: x 2 X:

(ii) For any u 2 F and ˛ > 0, let F˛ D ˛ .u/. Then   Z t Œu Px N t D ˛ e u.Xs /ds; 8t < F˛ D 1 0

q:e: x 2 X:

.5:4:6/

.5:4:7/

Proof. First of all, consider a measure  2 S0 and an AF A 2 AC c which are related to each other by Theorem 5.1.1. Since 1E   and 1E  A are related to each other for any Borel set E, we can say that the support of  is just the complement of the largest open set G on which A vanishes in the sense that Px ..1G  A/ t D 0; 8t > 0/ D 1;

q.e. x 2 X:

.5:4:8/

We first prove assertion (i) for u 2 F . Since 1 .u  G1 u/  F , we can invoke the theorem of spectral synthesis (Theorem 2.3.2), to find a sequence ¹n º of signed Radon measures of finite (1-)energy integrals such that suppŒn   F and the potentials un D U1 n are E1 -convergent to u  G1 u. By subtracting a subsequence if necessary, we then conclude from Corollary 5.2.1 that Œun 

lim N t

n!1

Œu

D Nt

ŒG1 u

 Nt

.5:4:9/ .n/

uniformly on each finite interval of t, Px -a.s. for q.e. x 2 X . Denote by A t the CAF of bounded variation associated with n in the manner of (5.4.3). Lemma Rt Œu  .n/ un .Xs /ds  A t , while Lemma 5.1.2 implies that 5.4.1 then implies N t n D 0 e Rt Rt Rt ŒG u un .Xs /ds converges to 0 e u.Xs /ds  0 G1 u.Xs /ds D N t 1 in the same 0e sense as above. Therefore, we get from (5.4.9) that

e

.n/

lim A t

n!1

Œu

D N t ;

.5:4:10/

the convergence being in the same sense as (5.4.9). Since A.n/ vanishes on the complement of F in the sense of (5.4.8), we are led from (5.4.10) to (5.4.6). The assertion (ii) can be proved in a similar way. Therefore it remains to prove assertion (i) for u 2 Fe . If .E; F / is transient, the 0-order version of the above argument works again in proving it. We shall prove (i) for a general (not necessarily transient) Dirichlet space Fe . We first take u 2 Fe;b . In the same way as in the proof of Theorem 5.2.2, we consider g 2 K such as u 2 Feg and the canonical subprocess Mg D .X tg ;  g ; Px /

264

5 Stochastic analysis by additive functionals

Rt of M by the multiplicative functional C t D exp. 0 g.Xs /ds/. Since Mg is associated with .E g ; F g /, the 0-order resolvent Rg of Mg satisfies Rg .u  g/ 2 Feg and E g .u  Rg .u  g/; v/ D E.u; v/; 8v 2 F g ; in virtue of Theorem 1.5.4. This means that F D .u/ D g .u  Rg .u  g// where g denotes the 0-spectrum relative to the transient Dirichlet form E g . Therefore, by what has been proved already, we get g;ŒuRg .ug/

Px .N t

D 0; 8t < Fg / D 1 q.e. x 2 X;

.5:4:11/

g;Œw

denotes the second term in the

g;Œw

C Nt

g g where F D inf¹t > 0 W X t 2 F º and N t decomposition

w.X tg /  w.X0g / D M t

g;Œw

obtained by applying Theorem 5.2.2 to w 2 Feg and Mg . We now make use of the fact   Z t Z t Œw g;Œw C w  g.Xs /ds; g.Xs /ds < Z Px N t D N t 0 0 .5:4:12/  Z t D Px

g.Xs /ds < Z 0

q.e. x 2 X;

where Z is an exponentially distributed random variable independent of ¹X t ; t  0º (see A.2). The proof of (5.4.12) will be given in Lemma 6.1.4. From (5.4.11) and (5.4.12), we have that Œu

Px .N t

¤ 0 for some t < T ^ F /

 e kgk1 T Ex .C t I N t ¤ 0 for some t < T ^ F /   Z t g;ŒuRg .ug/ g kgk1 T De Px N t ¤ 0 for some t < T ^ F ; g.Xs /ds < Z Œu

0

D 0: Finally take u 2 Fe . Owing to Theorem 2.3.3 and Theorem 4.6.5, we have E.HF u  u; HF u  u/ D 0 for F D .u/:

5.4 Continuous additive functionals of zero energy

265

If we put un D .n/ _ .u ^ n/; n D 1; 2; : : : ; then E.HF un  u; HF un  u/ D E.HF un  HF u; HF un  HF u/  E.un  u; un  u/ ! 0;

n ! 1: ŒH u 

Since .HF un /  F , (5.4.6) applied to HF un 2 Fe;b shows that N t F n D 0, 8t < F a.s. Now it suffices to apply Corollary 5.2.1 again to get (5.4.6) for u 2 Fe . The support of an AF A is defined by SuppŒA D ¹x 2 X n N W Px .R D 0/ D 1º;

.5:4:13/

where R.!/ D inf¹t > 0 W A t .!/ ¤ 0º and N is a properly exceptional set of A. Theorem 5.4.1 means SuppŒN Œu   .u/;

8u 2 F :

.5:4:14/

The assertions of the next lemma for B closed are trivial consequences of Theorem 5.4.1. Lemma 5.4.2. Let B be an open set or a nearly Borel finely closed set. (i) For any u 2 Fe ŒHBe u

Px .N t

D 0; 8t < B / D 1 q.e. x 2 X:

(ii) For any u 2 F and ˛ > 0   Z t ˛ ŒHB e u ˛ D˛ HB e u.Xs /ds; 8t < B D 1 q.e. x 2 X: Px N t 0

Proof. We first suppose that B is an open set. Then this follows from Theorem 5.4.1 and Lemma 2.3.3. We only give the proof of (i) in transient case. Then, by the 0-order version of Lemma 2.3.3 and Theorem 4.3.2, there exist un 2 Fe such that their spectra Fn D .un / are contained in B and ¹un º is E-convergent to HBe u. By Œun  Œu  vanishes on Œ0; Fn / and consequently on Œ0; B /. But N t n Theorem 5.4.1, N t ŒH e u converges to N t B uniformly on each finite time interval by Corollary 5.2.1. Suppose next that B is a nearly Borel finely closed set. We again give only the proof of (i) for bounded u in transient case. The proof is similar to the proof of Theorem 5.4.1 if we can find a sequence of measures n 2 S0 vanishing on X n B such that kUn  HBe ukE ! 0; n ! 1. Take a sequence un D Gfn 2

266

5 Stochastic analysis by additive functionals

Fe \ L1 .XI m/ which is E-convergent to u. Then HBe un converges to HBe u with respect to E-norm by virtue of Theorem 4.3.2. Therefore it suffices to prove for Gf 2 Fe \ L1 .XI m/ that the 0-order sweeping out B of the measure f  m vanishes on X n B. By virtue of Lemma 4.6.1, there exists a non-negative quasi continuous function w 2 F such that B D w 1 .¹0º/ q.e. It then follows that B .X n B/ D 0 because w 2 FX nB and Z X

e

wdB D E.UB ; w/ D E.HB Gf ; w/ D 0:

In formulating an answer to the question (II), we need the following notion.  is called a smooth signed measure on X if there exists a generalized nest ¹Fk º such that 1Fk  is a finite signed Borel measure charging S no set of zero capacity for each k and further  charges no Borel subset of X n 1 kD1 Fk . Such a generalized nest is said to be associated with the smooth signed measure .  is a smooth signed measure with an associated generalized nest ¹Fk º if and only if  D  .1/   .2/ for some smooth measures  .1/ and  .2/ possessing ¹Fk º as their common associated generalized nest. In fact  .1/ and  .2/ can be obtained from  by applying the Jordan decomposition on each Fk . For a closed set F  X, we put FF D ¹u 2 F W e u D 0 q.e. on X n F º; u D 0 q.e. on X n F º: Fb;F D ¹u 2 Fb W e

.5:4:15/

Theorem 5.4.2. The following two conditions are equivalent to each other for u 2 Fe : (i) N Œu is a CAF of bounded variation. (ii) There exists a smooth signed measure  such that E.u; v/ D h;e v i;

8v 2

1 [

Fb;Fk ;

.5:4:16/

kD1

for a generalized nest ¹Fk º associated with . In this case, N Œu is expressible as .5:4:1/ with A.i / being a PCAF with Revuz measure  .i/ , i D 1; 2, where  .1/ ;  .2/ satisfy  D  .1/   .2/ . For the proof we prepare two lemmas.

5.4 Continuous additive functionals of zero energy

267

Lemma 5.4.3. For any N 2 Nc and compact set K  X 1 lim Evm .N t I t > K / D 0 t #0 t where K D inf ¹t > 0 W X t … Kº and v is any function expressible as Z 1 K e ˛t p K pK v.x/ D R˛ f .x/ D t f .x/dt; t f .x/ D Ex .f .X t /I t < K / 0

for a bounded Borel function f . Proof. We have ŒEvm .N t I t  K /2  kvk1 Em .N t2 /Pjvjm .t  K / which is o.t 2 /, because N is of zero energy and Z 1 1 K K Pjvjm .t  K /  .R˛ jf j; 1  p t 1/ ! .jf j  ˛R˛K jf j/d m; t t t #0 e K e being the fine interior of K. K Lemma 5.4.4. Let  D  .1/   .2/ be a difference of finite Borel measures on X charging no set of zero capacity. If u 2 Fe satisfies E.u; v/ D h;e v i;

8v 2 Fb;F

.5:4:17/

for some closed set F , then Px .N Œu D A.1/ C A.2/ on Œ0; F // D 1;

q.e. x 2 X;

.5:4:18/

where A.i / is a PCAF with Revuz measure  .i / , i D 1; 2. Proof. Under the assumption,  D  .1/  .2/ for some smooth measures  .i / with  .i / .X/ < 1; i D 1; 2. First suppose that u 2 F and u satisfies equation (5.4.17). We can choose a generalized nest ¹Fk º such that 1Fk  .i / 2 S0 ; i D 1; 2, for each b k D Fk \ F and b k by virtue of Theorem 2.2.4. We put F k D 1b  . C u  m/. Fk Then (5.4.17) implies k ;e v i; E1 .u; v/ D hb Owing to Theorem 4.3.1, we have

8v 2 Fb;b : F k

A

u; v/ D E1 .U1b k  H 1 c .U1b k /; v/; E1 .u  H 1 c e b b Fk Fk

268

5 Stochastic analysis by additive functionals

which can be readily extended to v 2 Fb . Hence Fk

A

k C H 1 c e u  H 1 c .U1b k /: u D U1b b b Fk Fk

.5:4:19/

We can conclude from (5.4.19) combined with Corollary 5.2.1 (i), Lemma 5.4.1 and Lemma 5.4.2 that (5.4.18) holds with F being replaced by b . By letting Fk k ! 1, we arrive at (5.4.18) on account of Lemma 5.1.6. Suppose next that E is transient, u 2 Fe and u satisfies the equation (5.4.17). By using the 0-order version of Theorem 2.2.4, we can find a generalized nest .0/ ¹Fk º such that 1Fk   .i / 2 S0 ; i D 1; 2, for each k (see Exercise 2.2.4). If we b k D Fk \ F and b  k D 1b  , then in the same way as above we get from put F Fk (5.4.17) and Theorem 4.3.2 that

e

E.u  Hb e u; v/ D E.Ub  k  Hb .Ub  k /; v/; Fc Fc k

k

8v 2 Fb;b : F k

For any bounded v 2 Fe;b , we can find a sequence of uniformly bounded wn 2 F Fk E-convergent to v as n ! 1. Then vn D wn  Hb w f 2 Fb;b is E-convergent Fc n F k

k

to v  Hb e v D v. Therefore the above identity holds for any bounded v 2 Fe;b Fc F

k

k

and eventually for any v 2 Fe;b . Thus we get F k

e

e u  Hb .Ub  k /; u D Ub  k C Hb Fc Fc k

k

which leads us to (5.4.18) by the same reasoning as before. Finally we consider the general case that u belongs to a general (not necessarily transient) extended Dirichlet space Fe and satisfies the equation (5.4.17). We use g Lemma 1.6.7 to find g 2 K with u 2 Fe . If we put  D  C ug  m, then  is still a finite signed Borel measure charging no set of zero capacity and moreover E g .u; v/ D h;e v i;

8v 2 Fb;F :

Since E g is transient and F g D F , we conclude from what was just proved that   Z  g g;Œu .1/ .2/ D A C A  u  g.Xs /ds on Œ0; F / D 1 q.e. x 2 X; Px N 0

.5:4:20/ with respect to the corresponding canonical subprocess Mg D .X tg ;  g ; Px / employed already in the proof of Theorem 5.4.1. We then get (5.4.18) from (5.4.20) just as in the proof of Theorem 5.4.1.

5.4 Continuous additive functionals of zero energy Œu

Proof of Theorem 5:4:2. Suppose N t Œu

Nt

.1/

D A t

.2/

C At

269

is of bounded variation on Œ0; /, then Px -a.s. on ¹t < º q.e. x;

.1/ and  .2/ the Revuz measures of A.1/ for some A.1/ ; A.2/ 2 AC c . Denote by  .2/ and A respectively. Let ¹Fk º be a generalized compact nest commonly associated with  .1/ and  .2/ . Then ¹Fk º is also associated with the smooth signed measure  D  .1/   .2/ . We may assume that IFk   .i / 2 S0 ; i D 1; 2, for every k. Fix k and denote the notions of Lemma 5.4.3 relevant to Fk by k and R˛k f . Since 1Fk  A.i / 2 Nc , i D 1; 2, we get from Lemma 5.4.3 and Theorem 5.1.3 that

1 1 Œu Œu lim Evm .N t / D lim Evm .N t I t < k / t #0 t t #0 t 1 D lim Evm ..1Fk  A.1/ / t C .1Fk  A.2/ / t I t < k / t #0 t 1 D lim Evm ..1Fk  A.1/ / t C .1Fk  A.2/ / t / D h; vi t #0 t for v D R˛k f with bounded Borel f . Hence (5.4.16) holds for this type of function v in view of Theorem 5.2.4. For k a general v 2 Fb;Fk , vn D nRnC1 v are of the above type, uniformly bounded, convergent to v and E1 -convergent to v as well. Thus we attain (5.4.16). The converse implication is immediate from Lemma 5.4.4 and Lemma 5.1.6. The following corollary extends Lemma 5.4.1. Corollary 5.4.1. Suppose u 2 F satisfies E.u; v/ D h; vi;

8v 2 C;

.5:4:21/

where C is a special standard core of E and  D 1  2 a difference of positive Radon measures charging no set of zero capacity. Then N Œu D A.1/ C A.2/ where A.i / is a PCAF with Revuz measure i ; i D 1; 2. S In fact, for relatively compact open sets ¹Gk º with G k  GkC1 , 1 kD1 Gk D X, any v 2 Fb;GN k  FGkC1 admits a uniformly bounded sequence vn 2 CGkC1 , n D 1; 2; : : : ; converging to e v q.e. and in E-metric as well by virtue of Theorem 4.4.3. Hence (5.4.21) implies (5.4.16) for ¹Fk º D ¹G k º. In connection with Theorem 5.4.2, we like to know when a finite signed measure charges no set of zero capacity.

270

5 Stochastic analysis by additive functionals

Theorem 5.4.3. Assume that the resolvent R˛ of M satisfies the following condition: .5:4:22/ R˛ .Bb .X//  Cb .X/; ˛ > 0: Suppose that  is a finite signed measure on X and, for some u 2 Fe , E.u; v/ D h; vi;

8v 2 F \ Cb .X/:

.5:4:23/

Then  charges no set of zero capacity. Proof. (5.4.23) holds for any bounded 1-excessive function v in F because it holds for R˛ v on account of the assumption (5.4.22) and further ˛R˛C1 v is increasing to v and E-convergent to v as ˛ ! 1. Let K be a compact set of zero capacity. Since (5.4.22) implies that R˛ .x;  / is absolutely continuous with respect to m and consequently K is polar on account of Theorem 4.1.2; Px .K < 1/ D 0; 8x 2 X. T Choose a sequence of relatively compact open sets Gn such that GnC1 G n and 1 nD1 Gn D K. Then eGn .x/ D  

P G K n / decreases as n ! 1 to Ex .e Ex .e / D 1K .x/ by virtue of (A.2.9) and further E1 .eGn ; eGn / ! 0; n ! 1. Since (5.4.23) is valid for v D eGn , we get .K/ D 0 by letting n ! 1. Corollary 5.4.2. Assume that X is compact and that condition .5:4:22/ is fulfilled. If u 2 Fe satisfies jE.u; v/j  C kvk1 ;

8v 2 F \ Cb .X/;

.5:4:24/

for some constant C > 0, then N Œu is of bounded variation. Indeed (5.4.24) implies by the Riesz representation theorem that u satisfies the equation (5.4.23) for some finite signed measure . We can then get the desired conclusion from Theorem 5.4.2 and Theorem 5.4.3. The first two theorems of this section will be extended in the next section when E is local.

5.5

Extensions to additive functionals locally of finite energy

The decomposition AŒu D M Œu C N Œu of Theorem 5.2.2 can be extended to u in a broad class of functions including Floc under the hypothesis .E; F / possesses the strong local property:

.5:5:1/

5.5

Extensions to additive functionals locally of finite energy

271

From now on till the end of this section, we assume (5.5.1). According to Theorem 4.5.3, we may then assume that M is a diffusion process on X with no killing inside in the sense that (4.5.19) holds for every x 2 X. The killing measure k disappears in the Beurling–Deny formula for E. We can now utilize the next lemma. u1 D e u2 q.e. on a nearly Borel finely Lemma 5.5.1. (i) For any u1 ; u2 2 F , if e open set G, then Œu1 

Mt

Œu2 

D Mt

8t < G ; Px -a.s. for q.e. x 2 X;

;

.5:5:2/

where G D inf¹t > 0 W X t … Gº the first leaving time from G. (ii) M D Mc : any MAF is continuous. Proof. Let u D u1  u2 . Under (5.5.1), M Œu D M Œu;c in view of (5.3.2) Œu and Theorem 5.3.1. Lemma 5.3.1 then implies that M t D 0; t < G . Since M ŒR˛ g D M ŒR˛ g;c ; 8˛ > 0; 8g 2 C0 .X/, the purely discontinuous part of M 2 M vanishes on account of Theorem A.3.20. We notice that, by (4.5.19) holding for all x 2 X, Px .t ^ G < / D 1;

8t > 0; 8x 2 X;

.5:5:3/

provided that G is relatively compact. At this stage, it is convenient to introduce the notion of local AF’s. We call A t .!/ a local additive functional (local AF in abbreviation) if A t .!/ satisfies all requirements for an AF stated in the beginning of §5.1 except that the additivity A t Cs .!/ D As .!/ C A t .s !/ for ! 2 ƒ is required only for non-negative t; s with t C s < .!/. Therefore a local AF is a synonym for an AF in the case that M is conservative; Px . D 1/ D 1, 8x 2 X. For two local AF’s A.1/ ; A.2/ and a stopping time   , we write .1/

At

.2/

D At ;

t < ;

if Px -probability of this event equals Px .t < / for each t  0 and q.e. x 2 X. .1/ .2/ There is then a common defining set ƒ such that A t .!/ D A t .!/ for any ! 2 ƒ and any t < .!/. If  can be taken to be  in the above, then we say that A.1/ and A.2/ are equivalent and write A.1/ D A.2/ : Let us also introduce the family „ of sequences of finely open sets defined by „ D ¹¹Gn º W Gn is nearly Borel S finely open for all n, Gn  GnC1 ; 1 nD1 Gn D X q.e.º:

.5:5:4/

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5 Stochastic analysis by additive functionals

We say that a function u is locally in F in the broad sense (u 2 FPloc in notation) if there exists a sequence ¹Gn º 2 „ and a sequence ¹un º  F such that u D un m-a.e. on Gn . Denote by FPb;loc the family of functions u 2 FPloc for which each function un of the sequence appearing in the definition of u can be chosen bounded. FPloc (resp. FPb;loc ) is obviously broader than the space Floc (resp. Fb;loc ) introduced at the end of §3.2. Lemma 5.5.2. (i) Any function u 2 FPloc has a quasi continuous modification e u. (ii) For ¹Gn º 2 „, Px



 lim Gn D  D 1 q:e: x 2 X:

n!1

(iii) For an AF A and a nearly Borel finely open set G, A.t Cs/^G D As^G C A t ^G ı s^G Px -a.s.;

8x 2 X n N;

where N is any properly exceptional set containing .X n G/ n .X n G/r and an exceptional set for A. Proof. (i) Let ¹Gn º 2 „ and ¹un º  F be associated with u 2 FPloc . Take quasi continuous modifications e un of un . Since ¹Gn º are quasi open by Theorem un jGn q.e., n D 1; 2; : : : ; in view of 4.6.1, u has a version e u such that e uj G n D e Lemma 2.1.5. e u is easily seen to be quasiScontinuous. (ii) We may assume that N D X n 1 nD1 Gn is properly exceptional. Suppose that Px . < / > 0 for some x 2 X n N where  denotes limn!1 Gn . Since XGn 2 .X n Gn / [  by Lemma A.2.7, we can conclude from the quasileft-continuity of M that Px . < ; X 2 N / > 0 contradicting the proper exceptionality of N . (iii) Since .X n G/ n .X n G/r is exceptional by Theorem 4.1.3, we have from Lemma A.2.7 that XG 2 .X n G/r [  Px -a.s., 8x 2 X n N if N is chosen as in the statement. Hence G ı G D 0 and the stated additivity of A is valid. The third statement of the above lemma will be used in the proof of Theorem 5.5.5 and Theorem 5.5.6. For any u 2 FPloc , its quasi continuous version e u is q.e. finite, q.e. finely continuous and satisfies the properties stated in Theorem 4.2.2. Hence Œu

At

De u.X t /  e u.X0 /;

t  0;

u./ D 0). defines an AF AŒu in the sense of §5.1 (we set e

5.5

Extensions to additive functionals locally of finite energy

273

By an MAF locally of finite energy, we mean a local AF M t admitting a sequence ¹Gn º 2 „ and a sequence of MAF’s ¹M .n/ º  MV such that, for each n, .n/

Mt D Mt

t < G n :

The family of MAF’s locally of finite energy is denoted by MV loc . By a CAF locally of zero energy, we mean a local AF N t admitting a sequence ¹Gn º 2 „ and a sequence ¹N .n/ º  Nc such that, for each n, .n/

Nt D Nt ;

t < Gn :

We denote by Nc;loc the family of all CAF’s locally of zero energy. The next theorem extends Theorem 5.2.2. Theorem 5.5.1. For any u 2 FPloc the AF AŒu can be decomposed as AŒu D M Œu C N Œu ;

ı

M Œu 2 Mloc ; N Œu 2 Nc;loc :

.5:5:5/

Such a decomposition is unique up to the equivalence of local AF’s. Proof. Lemma 5.5.1 readily yields the decomposition (5.5.5). Indeed, let ¹Gn º 2 „ and ¹un º  F be associated sequences of u. Let AŒun  D M Œun  CN Œun  be the Œu  Œu  corresponding decomposition given by Theorem 5.2.2. Since M t n D M t nC1 for t < Gn by Lemma 5.5.1, M Œu D limn!1 M Œun  ; t <  I M Œu D 0; t  , Œu Œu Œu is well-defined and belongs to MV loc . It then suffices to let N t D A t  M t : Before proving the uniqueness of the decomposition, we make the following observation. For a real valued process A let v kt .A/

D

Œk t  X

.A.lC1/=k  Al=k /2 ;

k D 1; 2; : : :

lD1

which approximates a quadratic variation of A. We further let A t D A t ^ for a V then M is a square integrable Pm -martingale and stopping time . If M 2 M, lim v kt .M / D hM i t ^

k!1

in L1 .Pm /

in view of Lemma 5.5.1 (ii) and Theorem A.3.10. On the other hand, for N 2 Nc , v kt .N / is L1 .Pm /-convergent to 0 by virtue of (5.2.14) and hence v kt .N / admits a subsequence converging to 0 Pm -a.s.

274

5 Stochastic analysis by additive functionals

We now assume that M t CN t D 0, M t 2 MV loc ; N t 2 Nc;loc . We can choose an associated sequence ¹Gn º 2 „ in common for M and N . We can assume that each Gn is relatively compact and we have by noticing (5.5.3) and Lemma 5.5.1 (ii) .n/

.n/

M t ^n C N t ^n D 0;

8t > 0; n D Gn ; n D 1; 2; : : : :

The preceding observation implies Em .hM .n/ i t ^n / D 0; 8t > 0; and consequently hM .n/ i t D 0; t < n , in view of Lemma 5.1.10 (iii). Hence M D 0 proving the uniqueness of the decomposition. Although Theorem 5.5.1 is formulated for the class FPloc , we shall be mainly concerned with the class Floc . For u 2 Floc , we can take as an associated sequence in „ any relatively compact open sets Gn with G n  GnC1 ;

1 [

n D 1; 2; : : : ;

Gn D X:

.5:5:6/

nD1

If u; v 2 Floc and u D v on a relatively compact open set G, then Œu

Mt

Œv

D Mt ;

Œu

Nt

Œv

D Nt ;

t < G :

.5:5:7/

The next theorem is immediate from Theorem 5.2.3. Note that, for M 2 MV loc , its quadratic variation hM i t , t  0, is well-defined as a PCAF. Theorem 5.5.2. For u 2 Floc , hM Œu i coincides with hui introduced at the end of §3.2. If u 2 Fb;loc , then the equation .5:2:34/ holds for any f 2 Fb with compact support. Two theorems concerning the support and bounded variation property of N Œu demonstrated in the preceding section can be readily extended to u 2 Floc as well. Note that the spectrum .u/; ˛ .u/ are well-defined for u 2 Floc in an obvious manner. Theorem 5.5.3. Theorem 5:4:1 remains valid for u 2 Floc if we replace F and F˛ in .5:4:6/ and .5:4:7/ by F ^  and F˛ ^  respectively. Proof. Consider relatively compact open sets ¹Gn º satisfying (5.5.6). For u 2 Floc , choose un 2 F with u D un on Gn for each n. If F D .u/, then E.un ; v/ D E.u; v/ D 0; 8v 2 F \ C0 .X/ with suppŒv  .X n F / \ Gn , and accordingly .un /  F [.X nGn /. Theorem 5.4.1 (i) yields (5.4.6) with u and F Œu Œu  being replaced by un and F [.X nGn / respectively. Since N t D N t n ; t < Gn , and Gn " , we let n ! 1 to arrive at (5.4.6) for u 2 Floc with F being replaced by F ^ . In the same way, we get the present version of (5.4.7).

5.5

275

Extensions to additive functionals locally of finite energy

Theorem 5.5.4. Theorem 5:4:2 remains valid for u 2 Floc if we take as ¹Fk º in the statement (ii) a generalized compact nest associated with . Proof. The implication (ii))(i) is clear from Lemma 5.4.4. To prove (i))(ii), we assume that N Œu for u 2 Floc is of bounded variation and consider an associated generalized nest ¹Fk º as in the proof of Theorem 5.4.2. Fix k and denote the notions of Lemma 5.4.3 relevant to K D Fk by k and R˛k f . Choose w 2 F such that w D u in a neighbourhood of Fk . Then, from Lemma 5.4.3, Theorem 5.2.4 and (5.5.7), we get for any v D R˛k f with bounded f 1 1 Œu Œw lim Evm .N t I t < k / D lim Evm .N t I t < k / t #0 t t #0 t 1 Œw D lim Evm .N t / D E.w; v/ D E.u; v/: t #0 t The rest is the same as the proof of Theorem 5.4.2. Corollary 5.5.1. Corollary 5:4:1 remains valid for u 2 Floc . We can now give an extension of Theorem 5.2.5. Theorem 5.5.5. Assume the absolute continuity condition .4:2:9/. Suppose that a function u satisfies the following conditions: (i) u 2 Fb;loc , u is finely continuous on X. (ii) 1G  hui 2 S00 for any relatively compact open set G. (iii) 9 D  .1/   .2/ with 1G   .1/ ; 1G   .2/ 2 S00 for any relatively compact open set G and E.u; v/ D h; vi; 8v 2 C; for some special standard core C of E. Let A.1/ ; A.2/ and B be PCAF’s in the strict sense with Revuz measures  .1/ ; .2/  and hui respectively in accordance with Theorem 5:1:7. Then Œu Œu u.X t /  u.X0 / D M t C N t ; Px -a.s. 8x 2 X: .5:5:8/ Here N Œu D A.1/ C A.2/ ;

Px -a.s. 8x 2 X;

.5:5:9/

M Œu

is a local AF in the strict sense such that, for any relatively compact and open set G, Œu

Ex .M t ^G / D 0; Œu

8x 2 G;

Ex ..M t ^G /2 / D Ex .B t ^G /;

(5.5.10) 8x 2 G:

(5.5.11)

276

5 Stochastic analysis by additive functionals

Proof. We first note that any exceptional set is polar by the assumption (4.2.9) and Theorem 4.1.2. As in the proof of Theorem 5.2.5, we define N Œu by (5.5.9) and then M Œu by (5.5.8). M Œu is a local AF in the strict sense and belongs to MV loc in view of Theorem 5.5.1 and Corollary 5.5.1. By thinking of v 2 F such that u D v on G, we see from (5.5.7), Theorem 5.5.2 and the optional sampling theorem that (5.5.10) and (5.5.11) hold for q.e. x 2 X and hence for any x outside a nearly Borel polar set N . On the other hand, we see from Lemma 5.5.2 (iii) and (5.5.3) that any local AF A in the strict sense satisfies A.sCt /^G D As^G C A t ^G ı s^G ;

Px -a.s. 8x 2 G:

Therefore, for any x 2 G and " > 0, Œu

Œu

Œu

Ex .M.t C"/^G  M"^G / D Ex .EX"^ G .M t ^G /I X"^G … N / D 0: Since u is bounded on G, Px -a.e. sample paths are continuous by Theorem 4.5.4 .i/ and Ex .A t ^G / is finite, i D 1; 2, we can let " # 0 in the above to obtain (5.5.10). In the same way, we have that, for x 2 G, Œu

Œu

Ex ..M.t C"/^G  M"^G /2 / D Ex .B.t C"/^G  B"^G /: By Fatou’s lemma we get Œu

Ex ..M t ^G /2 /  Ex .B t ^G / and we can proceed on the same line as in the last part of the proof of Theorem 5.2.5 to attain the identity (5.5.11). A function u 2 Floc is said to be E-harmonic on X if .u/ D ;, namely, E.u; v/ D 0;

8v 2 C

for some special standard core C of E. Corollary 5.5.2. Assume the absolute continuity condition .4:2:9/. Suppose that a function u satisfies the following conditions: (i) u 2 Fb;loc , u is E-harmonic and finely continuous on X . (ii) 1G  hui 2 S00 for any relatively compact open set G.

5.5

Then

Extensions to additive functionals locally of finite energy

Œu

D u.X t /  u.X0 /;

At

277

t < ;

is a local AF in the strict sense such that, for any relatively compact open set G, Œu

Ex .A t ^G / D 0;

8x 2 G;

Œu

Ex ..A t ^G /2 / D Ex .B t ^G /;

(5.5.12) 8x 2 G;

(5.5.13)

where B denotes a PCAF in the strict sense with Revuz measure hui . Example 5.5.1. Just as in Example 5.1.1, we consider the Brownian motion M D .X t ; Px / on Rd . Since M is a conservative diffusion, limn!1 Gn D 1 Px a.s., 8x 2 Rd , for any relatively compact open sets Gn satisfying (5.5.6) for X D Rd . Let 'i .x/ D xi ; i D 1; 2; : : : ; d , be coordinate functions. Then 1 .Rd /, D-harmonic and with co-energy measures ıij  dx. Hence ¹'i º are in Hloc Corollary 5.5.2 applies and we conclude that the processes B ti D 'i .X t /  'i .X0 /;

t > 0; 1  i  d;

are CAF’s of M in the strict sense which are local martingales with co-variation hB i ; B j i t D ıij  t Px -a.s. 8x 2 Rd : Of course, this can be seen directly using the transition function (1.4.25) and indeed ¹B ti º are continuous MAF’s in the strict sense with co-variation ıij  t : j Ex .B ti / D 0; Ex .B ti B t / D ıij  t; 8x 2 Rd . Œu 1 We next examine how the CAF N t of Theorem 5.5.1 for u 2 Hloc .Rd / looks like in the present case. If the distribution derivative u belongs to L2loc .Rd /, then Œu Nt

1 D 2

Z

t

u.Xs /ds;

.5:5:14/

0

because .u/.x/dx is then a smooth signed measure on Rd and Theorem 5.5.4 applies. In general, the following expression is valid: Z 1 t Œu 1 N t D lim u"n .Xs /ds; u 2 Hloc .Rd /; .5:5:15/ "n #0 2 0 where u" D "  u is the convolution with the mollifier " (see Exercise 1.2.1), and ¹"n º is a sequence decreasing to zero and depending only on the function u.

278

5 Stochastic analysis by additive functionals

The convergence in (5.5.15) is uniform on each finite interval of t Px -a.s. for q.e. 1 .Rd / and take, for any relatively compact open x 2 Rd . To see this, let u 2 Hloc 1 d set G, a function v 2 H .R / such that u D v on a neighbourhood of G. Then u" D v" on G for small ". Since v" converges to v in the space H 1 .Rd / as " # 0, Corollary 5.2.1 and the identity (5.5.14) for v" lead us to (5.5.15) for v, from which we get the identity (5.5.15) for u and t < G . Now (5.5.15) is clear. 1 For any u 2 Hloc .Rd /, the AF N Œu vanishes on the complement of the support of the distribution derivative u on account of Theorem 5.5.3. N Œu is of bounded variation if and only if 1=2u is represented by a signed smooth measure in the sense of (5.4.16). This condition is certainly satisfied if 1=2u is a signed Radon measure with total variation of finite energy integral. 1 For instance, the signed measure f .y/dy on R3 with f .y/ D 1B1 .y/jyjˇ sin jyj , ˇ >  52 , has the absolutely convergent energy integral based on the Newtonian 1 1 kernel v.x/ D 4 on account of (2.2.27). Consequently, the potential u.x/ D jxj R 1 3 R3 v.x  y/f .y/dy is in H .R / and, by virtue of Lemma 5.4.1, Œu Nt

Z D

t

f .Xs /ds: 0

The right side is a CAF of bounded variation of the 3-dimensional Brownian motion with a possible exceptional set being the origin 0 (see Example 5.1.1). In the case of one-dimensional Brownian motion, a non-empty exceptional set is absent and the smooth measure reduces to the positive Radon measure (Example 5.1.1). This considerably simplifies the situation. 1 Thus, in view of Theorem 5.5.4, N Œu for u 2 Hloc .R1 / is of bounded variation 00 if and only if the distribution 1=2u is a signed Radon measure or equivalently Œu u0 is of bounded variation on each finite interval. When this is the case, N t is expressed as the local time integral: Z Œu Nt D L.t; y/du0 .y/ Px -a.s. x 2 R1 : .5:5:16/ R1

In fact, from (5.1.32) we see that Z Z Z t 00 00 u" .Xs /ds D 2 L.t; y/u" .y/dy D 2 0

R1

R1

L" .t; y/du0 .y/

which combined with (5.5.15) leads us to (5.5.16). Here L" .t;  /RD "  L.t;  /. As an example, consider a function g 2 C0 .R1 / satisfying R1 g.x/dx D 0 Rx and set u.x/ D 1 g.y/dy. Then N Œu is of bounded variation if and only if the function g is of bounded variation.

5.5

Extensions to additive functionals locally of finite energy

279

In the next example we exhibit a CAF locally of zero energy but not of bounded variation. Example 5.5.2. Let M D .X t ; Px / be the 1-dimensional Brownian motion. Then the associated Dirichlet space on L2 .R1 / is . 12 D; H 1 .R1 //. Put u.x/ D x log jxj  x. Since u0 .x/ D log jxj is square integrable on any finite interval, 1 .R1 /. However u0 is not of bounded variation in the neighbouru belongs to Hloc Œu hood of the origin and hence N t is not of bounded variation in t as was noticed in the preceding example. We shall give an explicit expression of N Œu for this function. We note that the distribution derivative u00 equals Cauchy’s principal value v.p. x1 , because Z 1 1 log jxj  v 0 .x/dx E.u; v/ D 2 1  Z "  Z 1 1 0 0 D lim log jxj  v .x/dx C log jxj  v .x/dx "!0 2 1 "  Z "  Z 1 1 v.x/ v.x/ dx C dx ; D  lim "!0 2 x 1 x " for all v 2 C01 .R1 /. To give an explicit expression of N Œu , put 8 ˆ x log jxj  x  " C 12 " log "; ˆ ˆ ˆ < log " x 2 ; u" .x/ D log 2" " 2 ˆ ˆ 2" x ; ˆ ˆ :x log x  x C "  1 " log "; 2

x < " "  x < 0 0x<" "  x:

Then, for any v 2 C01 .R1 /, Z 1 1 1 .1.1;"/ .x/ C 1.";1/ .x//v.x/dx E.u" ; v/ D  2 1 x Z log " 1  .1.0;"/ .x/  1.";0/ .x//v.x/dx; 2" 1 which together with (5.5.14) implies that Z 1 t 1 Œu"  Nt D .1.1;"/ .Xs / C 1.";1/ .Xs //ds 2 0 Xs Z log " t .1.0;"/ .Xs /  1.";0/ .Xs //ds: C 2" 0

(5.5.17)

280

5 Stochastic analysis by additive functionals

To estimate the second term of the right-hand side of the above equality, put 8 1 ˆ <x C "  2 " log "; x < " w" .x/ D u" .x/; "  x < " ˆ : 1 x  " C 2 " log "; "  x: Then Z log " 1 .1.0;"/ .x/  1.";0/ .x//v.x/dx 2" 1 1  .1 C log "/.v."/  v."//; 2

E.w" ; v/ D 

and consequently we get from (5.5.14) and (5.5.16) that Œw  Nt "

Z log " t D .1.0;"/ .Xs /  1.";0/ .Xs //ds 2" 0 1 C .1 C log "/.L.t; "/  L.t; "//; 2

where L.t; "/ is the local time at ". Let ' be the coordinate function: '.x/ D x; 8x 2 R1 . For any finite interval Gk D .k; k/, take a function vk 2 C01 .R1 / such that vk D 1 on Gk . Clearly ku" vk  uvk kE1 ! 0, kw" vk  'vk kE1 ! 0 as " ! 0. Hence we get from Œu



Œu

Corollary 5.2.1 that there exists a sequence "n # 0 such that N t ^"nk ! N t ^k and Œw



Œ'

N t ^"kn ! N t ^k uniformly on each finite t -interval for any k, where k D Gk . Note that the choice of the sequence ¹"n º depends only on the E1 -norms of u" vk and w" vk ; k  1. We have seen at the beginning of the precedŒ' ing example that '.X t / is a local martingale and N t ^k D 0. Now by using the estimate that ˇ ˇ p ˇ ˇ limbaDı#0 ˇ.L.t ^ k ; b/  L.t ^ k ; a// = ı log.1=ı/ˇ r  2 max L.t ^ k ; x/ x2R1

a.s.,12 we can see that as " ! 0 .1 C log "/.L.t ^ k ; a C "/  L.t ^ k ; a  "// ! 0; 12 Cf.

K. Itô–H. P. McKean [1; §2.8].

5.5

Extensions to additive functionals locally of finite energy

281

uniformly in a Px -a.s. Therefore, by letting n ! 0 and then k ! 1 in (5.5.17) with t ^ k ; "n in place of t; "; we arrive at the expression Œu Nt

Z

t

D lim

.1.1;"n / .Xs / C 1."n ;1/ .Xs //

"n !0 0

1 ds: Xs

.5:5:18/

Rt Let us denote the right-hand side of (5.5.18) by 0 .v.p. X1s /ds: Since the E1 -norm is invariant under the shifts of the variable, by considering R t u.  a/ instead of u, we can use the same sequence ¹"n º to define the CAF 0 .v.p. Xs1a /ds of locally of zero energy by Z

 1 ds Xs  a Z t ^k D lim .1.1;"n / .Xs  a/ C 1."n ;1/ .Xs  a//

t ^k 0

 v.p.

1 ds: "n !1 0 Xs  a .5:5:19/ Since the convergence is uniform relative to a, integrating both sides of (5.5.19) by v.a/da with v 2 C01 .R1 /, we have Z Z

   Z t ^k  1 1 ds v.a/da D  v.Xs /ds v.p. v.p. Xs  a x 0 0  Z Z  1  v.x/L.t ^ k ; x/dx D  .H L .t ^ k ; // .a/v.a/da; D v.p. x t ^k



where H is the Hilbert transform: H f .a/ D 1 .v.p. x1 /  f .a/. Thus the CAF (5.5.19) is given by the Hilbert transform of the local time: Z t 0

 1 ds D  .H L.  ; t //.a/: v.p. Xs  a

.5:5:20/

Example 5.5.3. Let us consider the Dirichlet form E.u; v/ D

d Z X i;j D1

Rd

aij .x/

@u @v dx; @xi @xj

F D H 1 .Rd /;

.5:5:21/

on L2 .Rd / considered in Example 4.5.2, where we have seen that E admits an associated conservative diffusion M D .X t ; Px / on Rd whose transition function satisfies the absolute continuity condition (4.2.9) with respect to the Lebesgue measure.

282

5 Stochastic analysis by additive functionals

Assume that the distribution derivatives @x@ aij ; 1  i; j  d , are locally j bounded functions, say aij Ij . Then Theorem 5.5.5 well applies to coordinate functions 'i .x/ D xi , 1  i  d , in getting the decomposition of coordinate processes X ti D 'i .X t /; 1  i  d , as Z X ti

D

X0i

C

M ti

C

t

d X

0 j D1

aij Ij .Xs /ds Px -a.s.;

8x 2 Rd :

.5:5:22/

Hence M ti ; 1  i  d , are CAF’s of M in the strict sense which are local martingales with co-variations being CAF’s in the strict sense given by Z t hM i ; M j i t D 2 aij .Xs /ds Px -a.s.; 8x 2 Rd : .5:5:23/ 0

The last term of (5.5.22) is a CAF of bounded variation in the strict sense. We have seen in Theorem 5.5.1 that to any u 2 FPloc there corresponds a CAF 2 Nc;loc by the relation (5.5.5). In the next theorem we shall give a partial converse to it. We are still assuming the strong local property (5.5.1).

N Œu

Theorem 5.5.6. Assume that .E; F / is irreducible. If a CAF N belongs to Nc;loc , Œu  then there exist sequences ¹Gn º 2 „ and ¹un º  FPGn ;loc such that N t D N t n ; 8t < n , where n D Gn . Proof. If suffices to prove this theorem for N 2 Nc . Suppose first that N t 2 Nc 2 and Ex .N  / is bounded for q.e. x 2 X, where N t D supst jNs j. Put u.x/ D Ex .N / and M t D u.X t /u.X0 /N t . Since u is bounded , M t is well-defined and belongs to M, for 2

Ex .M t2 /  12Ex .N  /;

Ex .M t / D 0 q.e. x 2 X:

.5:5:24/

Denote by A the PCAF hM i t . Take a strictly positive Borel function f 2 L1 .XI m/ and put '.x/ D R1A f in accordance with Lemma 5.1.5. Since ' is finely continuous and strictly positive q.e., the sets Gn D ¹x 2 X W '.x/ > 1=nº define a sequence in „. In view of Lemma 5.5.2 (iii), Em .A t ^Gn / is subadditive in t and hence  Z G  n 1 lim Em .A t ^Gn / D lim ˛Em e ˛t dA t  n lim ˛.1; UA˛ '/: ˛!1 ˛!1 t !0 t 0 On the other hand, Lemma 5.1.5 implies ˛.1; UA˛ '/ D ˛.1; R˛ f  ' C .˛  1/R˛ '/ D ˛.R˛ 1; f /  ˛.1  ˛R˛ 1 C R˛ 1; '/  .1; f /:

5.5

Extensions to additive functionals locally of finite energy

283

Therefore

1 .5:5:25/ Em .A t ^Gn /  n.1; f /: t In Theorem 4.4.2, the space .FGn ; EGn / is identified with the Dirichlet space 1 of the part MGn of the process M on Gn . Put Fn1 D ¹xI '.x/  n1 º and let en be the (1-order) equilibrium potential of the set Fn1 relative to .FGn ; EGn /, that is, e e n .x/ D Ex .e  Fn1 I Fn1 < Gn /. We let un D u  en and show that n un 2 FGn . Denote by p G the transition function of the part process MGn . By t virtue of (5.5.25), lim

t !0

1 n .un ; un  p G t un / t !0 t 1 D lim Em ...uen /.X t /  .uen /.X0 //2 I t < Gn / t !0 2t 1 n C lim ..uen /2 ; 1  p G t 1/ t !0 t 1 n  kuk21 lim ¹Em ..en .X t /  en .X0 //2 I t < Gn / C .en2 ; 1  p G t 1/º t !0 t 1 C lim Em ..u.X t /  u.X0 //2 I t < Gn / t !0 t 1  2 kuk21 E.en ; en / C 2 lim Em .hM i t ^Gn C N t2 / t !0 t lim

 2 kuk21 E.en ; en / C 2n.1; f / < 1; proving that un 2 FGn  F . Since en D 1 q.e. on Gn1 we have that u 2 FPloc . Now let AŒu D M Œu C N Œu be the decomposition for u in Theorem 5.5.1. Œu  Œu  Then M t C N t D M t n C N t n ; t < Gn1 , and, using (5.5.25), we get that Œu  Œu  M t D M t n , N t D N t n , t < Gn1 , as in the proof of the uniqueness part of Œu Theorem 5.5.1. Hence N t D N t . Next, given a general N 2 Nc , we are going to prove that 2

9¹Gn º 2 „ such that Ex .N Gn / is bounded q.e.

.5:5:26/

Suppose (5.5.26) were true and denote Gn for a fixed n by G. We then set u.x/ D Ex .NG / and M t D u .X t ^G /  u.X0 /  N t ^G . By virtue of Lemma 5.5.2 (iii), we can regard M t as an AF in the strict sense of the stopped process MG D .X t ^G ; Px /x2X nN of the Hunt process MjX nN , N being a properly exceptional set containing .X n G/ \ .X n G/irr . In fact an analogous relation to (5.5.24) holds and M t is an MAF in the strict sense of MG . Let A be a PCAF in

284

5 Stochastic analysis by additive functionals

the strict sense of MG realizing the quadratic variation hM i t of M t in accordance with Theorem A.3.18. Taking a function f 2 L1 .XI m/ strictly positive on G and vanishing on X n G, we can apply the same argument as before to A; f and MG Œu in proving that u 2 FPG;loc and N t D N t ; t < G , where .EG ; FG / denotes the part of .E; F / on G. It only remains to prove (5.5.26) for N 2 Nc . To this end, we show that, for any relatively compact nearly Borel finely open set G with Px .G < 1/ > 0 q.e. on G, there exists a sequence of finely open sets ¹Un º such that Px



 lim n < G D 0 and

n!1

2

Ex .N n / is q.e. bounded;

.5:5:27/

where n D Un . Let '˛ .x/ D Ex .e ˛N G /;

˛ .x/

D Ex .e ˛H /;

for H D sup0s<G jNG  Ns j. Then 12 N G  H  2N G and hence 2˛ .x/  '˛ .x/  ˛=2 .x/. Since H ı  t D sup t s<G jNG  Ns j " H .t # 0/, 1  ˛ .x/ is an ˛-excessive function with respect to the part process MG . Hence the sets Un defined by Un D ¹x 2 GI ˛ .x/ > 1=nº are nearly Borel finely open and limk!1 k D G a.s. For any x 2 Un and T < 1, 1 < n

˛ .x/

 '˛=2 .x/  Px .N G  T / C e ˛T =2 D 1  Px .N G > T / C e ˛T =2 :

Therefore we get Px .N n > T /  Px .N G > T / < 1 

1 C e ˛T =2 ; n

x 2 Un :

By taking T large enough, we find a constant 0 < ˇ < 1 such that Px .N n > T /  ˇ for q.e. x 2 Un . Let k be stopping times defined by k D inf¹t > 0I N t D kT º. Then kC1  k C 1 ı k , and consequently we have Px .N n > .k C 1/T / D Px .kC1 < n /  Ex .PX k .1 < n /I k < n /  ˇPx .k < n /  ˇ kC1 ;

5.5

Extensions to additive functionals locally of finite energy

285

for q.e. x 2 Un ; yielding that 2

Ex .N n / 

1 X

..k C 1/T /2 ˇ k < 1

q.e. x 2 Un ;

kD0

the desired property (5.5.27). Finally to show (5.5.26), we take an increasing sequence ¹G .k/ º 2 „ such that Cap.X n G .k/ / > 0. Then Px .G .k/ < 1/ > 0 q.e. by the irreducibility. Hence, .k/ for each k, there exists a sequence ¹Un º satisfying (5.5.27) for G D G .k/ . Then Sn .k/ it suffices to put Gn D kD1 Un . If M is the 1-dimensional Brownian motion, then we can represent N 2 Nc;loc by using only a single function u 2 Floc . Corollary 5.5.3. When M is the 1-dimensional Brownian motion, for any N 2 Œu Nc;loc there exists a function u 2 Floc such that N t D N t . Proof. Let ¹Gn º 2 „ and ¹un º  Floc be the sequences determined by Theorem 5.5.6. Note that every finely open set is open and every non-empty set is nonpolar for 1-dimensional Brownian motion. Hence, owing to the construction of Gn and un in the proof of Theorem 5.5.6, we may suppose that each Gn is an open interval of the form .ln ; rn / for which limn!1 ln D 1 and limn!1 rn D 1 and un .x/ D Ex .Nn /, where n D Gn . Since unC1 .x/  un .x/ D Ex .unC1 .Xn // D

rn  x x  ln unC1 .ln / C unC1 .rn /; rn  l n rn  l n

for any x 2 .ln ; rn /, the function  k  X x  ln1 rn1  x un .ln1 / C un .rn1 / u.x/ D uk .x/  rn1  ln1 rn1  ln1 nD2

is well-defined independently of k satisfying x 2 .lk ; rk / and differs from un only by a harmonic function on .ln ; rn / for each n. Hence u 2 Floc and N Œu D N . Exercise 5.5.1. Suppose that M is a conservative diffusion process and let u 2 Œu FPloc . For the decomposition of A t in (5.5.5), show the following. V then u 2 Fe and N Œu 2 Nc . (i) If M Œu 2 M, (ii) Give an example of u such that N Œu 2 Nc but u … Fe .

286

5.6

5 Stochastic analysis by additive functionals

Martingale additive functionals of finite energy and stochastic integrals

A real-valued stochastic process Z t is said to be a semi-martingale if Z t is expressible as a sum of a local martingale and a process of bounded variation. We have Œu Œu u.X0 / D M t C N t for u 2 F is not necessarily seen that the process e u.X t /  e a semi-martingales. Accordingly the transformation rule for semi-martingales due to H. Kunita and S. Watanabe (a generalized Itô’s formula) does not apply directly. Nevertheless, we showed in Lemma 5.3.3 that the energy measure of M Œu;c is identical with chui , the local part of energy measure hui , while we established in Theorem 3.2.2 a transformation rule for chui in a purely analytical manner. By translating it in terms of a new notion of stochastic integrals, we can extend a due rule of transformation for the continuous part of the martingale M Œu . We shall formulate the stochastic integral based on Theorem 5.2.1 and the Hilbertian structure V e/ of MAF’s of finite energy. of the space .M; Our assumptions on X, m, and M are the same as in 5:1. First we introduce the notion of a stochastic integral. Consider the space 

Ac D ¹A  B W A; B 2 AC c ; Ex .A t / < 1; Ex .B t / < 1; 8t > 0; q.e. x 2 Xº: 

For M , N 2 M, hM; Li defined by (5.2.9) is a unique element of Ac satisfying (5.2.10).13 Let us consider the family M1 D ¹M 2 M W hM i .2 S / is a Radon measureº: M1 is a linear space containing the family MV of MAF’s of finite energy. For M; N 2 M1 , the measure hM;Li defined by (5.2.11) is a unique signed Radon measure such that Z t  Z 1 f .x/hM;Li .dx/ D lim Em f .Xs /d hM; Lis ; 8f 2 C0 .X/: t #0 t 0 X (5.6.1) Lemma 5.6.1. If M; L 2 M1 ; f 2 L2 .XI hM i / and g 2 L2 .XI hLi /, then f  g is integrable with respect to the absolute variation jhM;Li j of hM;Li and 2

Z X 13 Cf.

jf  gjjdhM;Li j

Appendix A.3.

Z 

Z 2

X

f dhM i

X

g 2 dhLi :

(5.6.2)

5.6

287

Martingale additive functionals of finite energy and stochastic integrals

Proof. Making the expressions dhM;Li D k1 d; dhM i D k2 d and dhLi D k3 d with  D hM i C hLi C jhM;Li j, we get from (5.6.1) dhaM CbLi D .a2 k2 C 2abk1 C b 2 k3 /d

for any a; b 2 R:

Therefore, the set [

B0 D

¹x 2 X W a2 k2 .x/ C 2abk1 .x/ C b 2 k3 .x/ < 0º

a;bW rational

is -negligible and we have for each ˛; ˇ 2 R ˛ 2 f .x/2 k2 .x/ C 2˛ˇjf .x/g.x/k1 .x/j C ˇ 2 g.x/2 k3 .x/  0 for every x 2 X n B0 . Integrating this with respect to , Z Z Z 2 2 2 ˛ f dhM i C 2˛ˇ jfgj jdhM;Li j C ˇ g 2 dhLi  0; X

X

X

proving (5.6.2). Theorem 5.6.1. Given M 2 M1 and f 2 L2 .XI hM i /, there exists a unique element f  M 2 MV such that e.f  M; L/ D

1 2

Z X

f .x/hM;Li .dx/;

V 8L 2 M:

(5.6.3)

The mapping f ! f  M is linear and continuous from L2 .XI hM i / into the V e/. space .M; Proof. By virtue of Lemma 5.6.1, ˇ Z ˇ p ˇ1 ˇ 1 ˇ f dhM;Li ˇˇ  p kf kL2 . hM i / e.L/; ˇ2 2 X

V L 2 M:

Hence, Theorem 5.2.1 implies Theorem 5.6.1 together with the inequality p p e.f  M /  .1= 2kf kL2 . hM i / /: The f  M 2 MV of Theorem 5.6.1 is called the stochastic integral. This terminology is legitimate by the following lemma.

288

5 Stochastic analysis by additive functionals

Lemma 5.6.2. Let M; f and f  M be as in Theorem 5:6:1. Then dhf M;Li D f  dhM;Li ;

V L 2 M:

(5.6.4)

Moreover, the following approximation holds for f 2 C0 .X/: ./

lim Ex .¹.f  M / t

jj!0

 .f  M / t º2 / D 0;

where ./

.f  M / t

D

n X

t > 0; q.e. x 2 X

(5.6.5)

f .X ti1 /.M ti  M ti1 /;

iD1

 denotes the partition 0 D t0 < t1 <    < tn D t and jj D max1in .ti  ti1 /. Proof. Let B be a common properly exceptional set for M and hM i such that the relation (5.2.7) holds for every x 2 X n B. Applying the argument in the paragraph preceding Theorem A.3.19 to the Hunt process MjX nB , we can see for f of MjX nB satisfying for f 2 C0 .X/ that there exists a unique martingale AF M each x 2 X n B, Z t  ./ f t º2 / D 0; Ex .M f2t / D Ex f .Xs /2 d hM is : lim Ex .¹.f  M / t  M jj!0

0

(5.6.6) It follows from this that f2t / D 1 f/ D lim 1 Em .M e.M 2 t #0 2t

Z X

f 2 dhM i < 1

V f 2 M. and, consequently, M On the other hand, (5.2.10) and (5.6.6) imply Z t  f f .Xs /d hM; Lis Ex .hM ; Li t / D Ex

(5.6.7)

0

V  for any L 2 M and q.e. x 2 X . When L 2 M, e ;Li is a bounded signed hM measure and hence Z 1 f lim Ehm .hM ; Li t / D h.x/hM e ;Li .dx/ t #0 t X for any bounded -excessive function h in view of the relation (5.1.11). Therefore, we have from (5.6.7) Z Z V h.x/hM .dx/ D h.x/f .x/hM;Li .dx/; L 2 M: (5.6.8) e ;Li X

X

5.6

289

Martingale additive functionals of finite energy and stochastic integrals

By setting h D ˛R˛e h; e h 2 C0 .X/, and letting ˛ tend to infinity, we conclude that V (5.6.8) holds for any e h 2 C0 .X/. Therefore, dhM e ;Li D f  dhM;Li , 8L 2 M. R 1 f; L/ D f In particular, e.M 2 X f dhM;Li which means M D f  M . We have proved Lemma 5.6.2 when f 2 C0 .X/. On account of Lemma 5.6.1 f D f  M readily extends to f 2 and Theorem 5.6.1, the relation (5.6.8) for M 2 L .XI hM i /. Corollary 5.6.1. (i) For M 2 M1 ; f 2 L2 .XI hM i / and g 2 L2 .XI f 2 hM i /, g  .f  M / D .gf /  M:

(5.6.9)

(ii) For M; L 2 M1 , f 2 L2 .XI hM i / and g 2 L2 .XI hLi /, Z 1 e.f  M; g  L/ D f .x/g.x/hM;Li .dx/: 2 X

(5.6.10)

By using Theorem 5.1.3, we can translate the formula (5.6.4) for M 2 M1 and 

Borel f 2 L2 .XI hM i / into the following relationship for elements of Ac : V 8L 2 M:

hf  M; Li D f  hM; Li;

(5.6.11)

In this sense our stochastic integral f  M can be regarded as a variant of the stochastic integral due to M. Motoo and S. Watanabe. Note that Z t  2 Ex f .Xs / d hM is < 1 q.e. x 2 X (5.6.12) 0

for M and f as above. In fact, we have by Lemma 5.1.9 that Z

t

E 0

 f .Xs /2 d hM is

Z  .1 C t/kU1 k1 

X

f .x/2 hM i .dx/ < 1; 8 2 S00 :

Let M be in the space Mad of MAF’s in the strict sense and f be a Borel function on X satisfying (5.6.12) for every x 2 X rather than for q.e. x 2 X: Then the Motoo–Watanabe integral f .X /  M exists as a unique element in Mad satisfying condition (A.3.19) by virtue of Theorem A.3.19. Theorem A.3.20 implies that the Motoo–Watanabe integral is a strict version of the present stochastic integral f  M provided that additionally M 2 M1 and f 2 L2 .XI hM i /: Let us extend the notion of the stochastic integral f  M to a wider class of AF’s. We say that a local AF M of the process M is locally in M1 (M 2 M1;loc

290

5 Stochastic analysis by additive functionals

in notation) if there exists a sequence ¹Gn º 2 „ and a sequence ¹M .n/ º of MAF’s .n/ in M1 such that M t D M t ; t < Gn . See the latter half of the preceding section for the relevant notions and notations. The class MV loc introduced in the preceding section is a subclass of M1;loc . If M 2 M1;loc , its quadratic variation hM i 2 AC c is well-defined by hM i t D hM .n/ i t ;

t < Gn ; n D 1; 2; : : : ;

(5.6.13)

by choosing an appropriate defining set and exceptional set of hM i. hM i does not depend (up to the equivalence) on the special choice of ¹Gn º and ¹M .n/ º for M . We can still define the energy measure of M 2 M1;loc as the Revuz measure hM i of the PCAF hM i. Due to Lemma 5.1.4, we then have Z Z f .x/hM i .dx/ D f .x/hM .n/ i .dx/; f 2 Bb .X/; suppŒf   Gn : X

X

(5.6.14) In particular, Lemma 5.6.1 extends to M; L 2 M1;loc . Therefore, the formula (5.6.3) still provides us with the stochastic integral f  M 2 MV for any M 2 M1;loc and f 2 L2 .XI hM i /. The relations (5.6.9), (5.6.10) and (5.6.11) extend to M1;loc as well. In particular, .f  M / t together with hf  M i t vanishes until the first leaving time G from a relatively compact open set G, Px -a.e. for q.e. x, whenever f vanishes on the set G. Keeping this observation in mind, we can finally define the stochastic integral f  M 2 MV loc for M 2 M1;loc and f 2 L2loc .XI hM i / by the relation g  .f  M / D .gf /  M

(5.6.15)

which holds for any bounded Borel function g with compact support. In fact, it suffices to let .f  M / t D ..1Gn  f /  M / t ; t < Gn ; ¹Gn º being any sequence of relatively compact open sets such that GN n  GnC1 and Gn " X; n ! 1. Here we mention a lemma which is useful in the subsequent development. Lemma 5.6.3. Let C1 and D be a uniformly dense subfamily of C0 .X/ and an E1 -dense subfamily of F . Then the family ¹f  M Œu W f 2 C1 ; u 2 Dº of V e/. stochastic integrals is dense in .M; V RProof. Suppose that an MAF M 2 M is e-orthogonal to the above family, namely, X f dhM;M Œu i D 0; 8f 2 C1 ; 8u 2 D. This identity extends Rto all u 2 F in u.x/2 k.dx/. view of Lemma 5.6.1 and the equality hM Œu i .X/ D 2E res .u; u/C X e Hence, hM; M Œu i D 0; 8u 2 F : (5.6.16)

5.6

Martingale additive functionals of finite energy and stochastic integrals

291

In particular, (5.6.16) holds for u D R˛ g; ˛ > 0; 8g 2 C0 .X/. According to the proof of Theorem A.3.20, we then have P .M D 0/ D 1; 8 2 S00 , from which follows M D 0. Lemma 5.6.4. If the difference of u1 ; u2 2 F is constant on finely open set G, then Œu ;c Œu ;c D M t 2 ; t < G : (5.6.17) Mt 1

Proof. By Lemma 5.3.1, chu1 u2 i D 0 on G and thus hM Œu1 ;c  M Œu2 ;c i t D 0; 8t < G , which means (5.6.17). Recall the spaces Fb;loc ; Floc and FPloc introduced in 3:2 and 5:5; Fb;loc  Floc  FPloc . For u 2 FPloc , Lemma 5.6.4 guarantees to set Œu;c

Mt

Œun ;c

D Mt

;

8t < Gn ;

(5.6.18)

where ¹Gn º 2 „ and ¹un º are functions in F such that u D un on Gn . In particular, we obtain from Lemma 5.6.4 u1 ; u2 2 FPloc ; u1  u2 D constant

)

M Œu1 ;c D M Œu2 ;c :

(5.6.19)

We can now present our transformation rule in the space MV loc . Theorem 5.6.2. (i) For any ˆ 2 C 1 .Rm / and any u1 ; u2 ; : : : ; um 2 Fb;loc , the composite function ˆ.u/ D ˆ.u1 ; u2 ; : : : ; um / is again locally in Fb and M

Œˆ.u/;c

D

m X

ˆxi .e u/  M Œui ;c :

(5.6.20)

iD1

(ii) For any ˆ 2 C 1 .Rm / with bounded derivatives and any u1 ; u2 ; : : : ; um 2 Floc , the composite function ˆ.u/ is again locally in F and the formula .5:6:20/ is satisfied. (iii) Suppose ˆ.u/ 2 F in the case (i) or (ii). Then E .c/ .ˆ.u/; ˆ.u// D

m Z 1 X ˆxi .e u/ˆxj .e u/dchui ;uj i : 2 X i;j D1

(5.6.21)

292

5 Stochastic analysis by additive functionals

Proof. By virtue of Theorem 3.2.2, we have ˆ.u/ 2 Fb and Z X

fgdchˆ.u/;vi D

m Z X iD1 X

fgˆxi .e u/dchui ;vi ;

8f; g 2 C0 .X/; 8v 2 Fb

for any ˆ 2 C 1 .Rm / with ˆ.0/ D 0 and for any u1 ; u2 ; : : : ; um 2 Fb . This identity readily extends to any ˆ 2 C 1 .Rm / and any u1 ; u2 ; : : : ; um 2 Fb;loc with the help of (5.6.14) and (5.6.19). The isometry (5.6.10) and the orthogonality of the decomposition (5.3.2) give e.f  M Œˆ.u/;c ; g  M Œv / D e

m X

 f ˆxi .e u/  M Œui ;c ; g  M Œv :

iD1

On account of Lemma 5.6.3, we have the following identity of stochastic integrals V in M: f  M Œˆ.u/;c D

m X ¹f ˆxi .e u/º  M Œui ;c ;

8f 2 C0 .X/;

iD1

which in turn means (5.6.20) by virtue of (5.6.15). We have proved (i). (ii) can be proved in the same way. Under the condition of (iii), the energy of M Œˆ.u/;c equals E .c/ .ˆ.u/; ˆ.u// by Theorem 5.3.1. Thus the expression (5.6.21) results from the calculation of energy of both sides of (5.6.20). Theorem 5.6.2 immediately leads us to the following presentation of the Beurling–Deny second formula: Corollary 5.6.2. Let X be an Euclidean domain D  Rd . Suppose that the coordinate functions xi are locally in F .1  i  d /. Then (i) C 1 .D/  Floc and, for any u 2 C 1 .D/, M Œu;c D

n X

uxi  M .i /;c ;

M .i /;c D M Œxi ;c :

(5.6.22)

iD1

(ii) C01 .D/  F and, for any u 2 C01 .D/, E

.c/

.u; v/ D

d Z X i;j D1 D

uxi vxj dij ;

1 ij D chxi ;xj i : 2

Each measure ij charges no set of zero capacity.

(5.6.23)

5.6

Martingale additive functionals of finite energy and stochastic integrals

293

V We finally state a representation theorem of the space M. Theorem 5.6.3. Let X be an Euclidean domain D  Rd and M be an m-symmetric diffusion on D and .F ; E/ its Dirichlet space. Assume that E possesses C01 .D/ as its core and that E is written as E.u; v/ D

d Z X i;j D1 D

aij .x/uxi .x/vxj .x/d m.x/;

u; v 2 C01 .D/;

(5.6.24)

where A D .aij / is a bounded non-negative definite symmetric matrix. Then the space MV can be represented by stochastic integrals based on M .i / D M Œxi  ; 1  i  d: 8 ® Pd ¯ R Pd .i / W <MV D iD1 fi  M i;j D1 X aij .x/fi .x/fj .x/d m.x/ < 1 R :e  Pd f  M .i /  D Pd iD1 i i;j D1 X aij .x/fi .x/fj .x/d m.x/: (5.6.25) V e/ because it conProof. The space of right-hand side of (5.6.25) is dense in .M; Pd Œu .i / tains the set ¹f  M D iD1 .f uxi /  M I f 2 C0 .Rd /; u 2 C01 .Rd /º; V e/ by Lemma 5.6.3. Hence, it is enough to show that the which is dense in .M; V e/. right-hand side of (5.6.25) is closed in .M; Let A1=2 .x/ D .ij .x//; B.x/ D lim A1=2 .x/.ACI /1 ; P .x/ D A1=2 .x/B.x/: !0

Then the matrices A1=2 .x/; B.x/; P .x/ commute with each other and P .x/ is the orthogonal projection on the range A.x/.Rd /. Suppose that limn!1 e.Mn  M / D 0 for Mn D

d X iD1

.n/ fi

M

.i /

;

d Z X i;j D1

Rd

.n/

aij fi

.n/

fj

d m < 1;

V M 2 M:

Put f n D .f1n ; : : : ; fdn /. Since Z .f n  f m ; A.f n  f m //Rd d m e.Mn  Mm / D D Z .P .f n  f m /; AP .f n  f m //Rd d m D D Z jAB.f n  f m /j2 d m; D D

294

5 Stochastic analysis by additive functionals

P .n/ we see that jdD1 .AB/ij fj converges in L2 .DI m/ to some hi for each i D 1; : : : ; d . Let h D .h1 ; : : : ; hd / and fi D .Bh/i ; 1  i  d . Set M 0 D Pd .i / , then iD1 fi  M Z Z 0 n 2 e.Mn  M / D jAB.f  f/j d m D jABf n  PA1=2 f/j2 d m D D Z jABf n  P 2 hj2 d m D D

which converges to zero as n ! 1 because P 2 h D h. Therefore we have M D R M 0 and e.M / D D jhj2 d m < 1. The representation of MV in the general case where the ij are not necessarily absolutely continuous with respect to m can be reduced to the above theorem by the time change method as we shall show in 6:2. .i /

Example 5.6.1. Consider the Brownian motion M D .X t ; Px / on Rd and B t D .i / .i/ .i / X t  X0 ; X t being the i-th coordinate of the sample path X t .1  i  d /. As was stated in Example 5.5.1, ´ M Œxi  D B .i / 2 M1 (5.6.26) hB .i / ; B .j / i t D ıij  t and hB .i/ ;B .j / i D ıij  dx. Consequently, the stochastic integral f  B .i / makes sense for every f 2 L2 .Rd /. Furthermore, ´ ® Pd ¯ .i / W f 2 L2 .Rd /; 1  i  d MV D i iD1 fi  B   Pd Pd 2 .i / D 1 e iD1 fi  B iD1 kfi kL2 .Rd / 2

(5.6.27)

in accordance with (5.6.25). V e/ is identical with the space of vector fields f D .f1 ; : : : ; Thus the space .M; 2 d fd /; fi 2 L .R /. f represents M Œu for some u in the extended space He1 .Rd / if and only if f D grad u: (5.6.28) He1 .Rd / coincides with the space G .Rd / in case that d D 1 or 2, but it does not contain non-zero constant functions in case that d  3 (see Example 1.5.3, Example 1.6.2 and Exercise 1.6.2). Thus (He1 .Rd /; 12 D) (its quotient space by V e/. constant functions in case that d D 1 or 2) is a proper closed subspace of .M;

5.7 Forward and backward martingale additive functionals

We see from (5.6.12) that  Z t 2 f .Xs / ds < 1 q.e. x 2 Rd ; Ex

295

(5.6.29)

0

for any f 2 L2 .Rd /. For instance, the function f .x/ D

f0 .x/ ; jxjˇ

ˇ<

d ; 2

f0 2 C0 .Rd /;

belongs to L2 .Rd /. When ˇ < 1, (5.6.29) holds for every x 2 Rd and the Motoo–Watanabe stochastic integral f  BR.i / is well-defined as an AF of M in t the strict sense. When ˇ  1 however, P0 . 0 f .Xs /2 ds D 1/ D 1 (see Example 5.1.1) and neither the Motoo–Watanabe integral nor the Itô integral is defined. The stochastic integral f  B .i / in our sense is still defined by taking the polar set ¹0º as its “essential” exceptional set.

5.7

Forward and backward martingale additive functionals

Let .E; F / be a regular Dirichlet form on L2 .XI m/ and M D .; M; X t ; Px / the corresponding Hunt process. We take D.Œ0; 1/ ! X / as the sample space , where D.Œ0; 1/ ! X / is the space of right continuous paths with left-hand limits. X t .!/ denotes the t -th coordinate of the path !. For a path ! with t < , the time reversal operator r t is defined by ´ !..t  s//; if 0  s  t; (5.7.1) r t .!/.s/ D !.0/; if s  t: Here, for r > 0, !.r/ D lims"r !.s/ and !.0/ D !.0/. Let us fix T > 0: Lemma 5.7.1. For any FT -measurable set A, Pm .rT1 AI T < / D Pm .AI T < /: Proof. We first prove this for A of the type A D ¹X t1 2 A1 ; X t2 2 A2 ; : : : ; X tk 2 Ak º; 0  t1  t2     tk  T; Ai 2 B.X/.i D 1; 2; : : : ; k/. Then Lemma 4.1.2 leads us to Pm .X t1 2 A1 ; X t2 2 A2 ; : : : ; X tk 2 Ak I T < / D Pm .X0 2 X; X t1 2 A1 ; X t2 2 A2 ; : : : ; X tk 2 Ak ; XT 2 X/ D Pm .X0 2 X; XT tk 2 Ak ; XT tk1 2 Ak1 ; : : : ; XT t1 2 A1 ; XT 2 X/ D Pm .XT tk 2 Ak ; XT tk1 2 Ak1 ; : : : ; XT t1 2 A1 I T < /:

296

5 Stochastic analysis by additive functionals

It follows from the quasi-left-continuity that for each t > 0, Pm .X t  ¤ X t / D 0. Hence the right-hand side equals Pm .X t1 .rT / 2 A1 ; X t2 .rT / 2 A2 ; : : : ; X tk .rT / 2 Ak I T < /. Let H D ¹F 2 FT W Pm .F ı rT I T < / D Pm .F I T < / º. Then H is closed under monotone increasing convergence. Hence FT0 is a subset of H , and so is FT . Theorem 5.7.1. For u 2 F Œu

At

1 Œu 1 Œu Œu D M t C .MT t .rT /  MT .rT //; 2 2

0  t  T < ; Pm -a.s. (5.7.2)

Proof. Operating rT to the formula (5.2.24), we have by Lemma 5.7.1 Œu

Œu

u.XT t /  u.XT / D M t .rT / C N t .rT / Pm -a.s. on T < :

(5.7.3)

Rt Œu When u D R1 f for f 2 L2 .XI m/, we have N t D 0 .u  f /.Xs /ds, Z t Z t Œu N t .rT / D .u  f /.Xs ı rT /ds D .u  f /.X.T s/ /ds 0

Z D

0

T

Œu

T t

Œu

.u  f /.Xs /ds D NT  NT t

Œu

Œu

Pm -a.s. on T < ; (5.7.4) Œu

and thus M t .rT / D u.XT t /  u.XT /  NT C NT t . Hence Œu

Œu

MT t .rT /  MT .rT / Œu

Œu

D u.X t /  u.XT /  NT C N t Œu

D u.X t /  u.X0 / C N t

Œu

 u.X0 / C u.XT / C NT Œu

D 2u.X t /  2u.X0 /  M t ;

which implies Theorem 5.7.1. In view of Corollary 5.2.1, the relation (5.7.4) can be extended to u 2 F . Substituting T  t to (5.7.3) for t , we have Œu

Œu

u.X t / D u.XT / C MT t .rT / C NT t .rT /:

(5.7.5)

By adding (5.7.5) to (5.2.24) 1 1 1 Œu Œu Œu Œu u.X t / D .u.X0 / C u.XT // C .M t C MT t .rT // C .N t C NT t .rT //: 2 2 2 Œu

Œu

On the other hand, since by (5.7.4) NT D N t

Œu

C NT t .rT /; we have

5.7 Forward and backward martingale additive functionals

297

Theorem 5.7.2. For u 2 F 1 1 1 Œu Œu Œu u.X t / D .u.X0 / C u.XT // C .M t C MT t .rT // C NT 2 2 2 0  t  T < ; Pm -a.s. (5.7.6) Assume that M is conservative and let G t D rT1 .F t /, 0  t  T . Then Œu M t .rT / is a .Pm ; G t /-martingale in the sense that for any A D rT1 .B/ 2 Gs , B 2 Fs and 0  s  t Œu

Œu

Œu

(5.7.7)

D Em .MsŒu 1B / D Em .MsŒu .rT /I A/:

(5.7.8)

Em .M t .rT /I A/ D Em .M t .rT /1B .rT // D Em .M t 1B /

Accordingly Theorem 5.7.1 can be viewed as a representation of the AF AŒu as the sum of a forward martingale and a backward one. In the remainder of this section, we suppose that .E; F / is a regular Dirichlet form on L2 .XI m/ with the strong local property and let M D .; M; X t ; Px / be the associated diffusion process. Under the strong locality, above formulae are extended to u 2 Floc . In view of Theorem 5.3.1 and Lemma 5.3.3 the Dirichlet form E can be represented as Z 1 E.u; v/ D dchu;vi ; u; v 2 F : (5.7.9) 2 X The diffusion process M may not be conservative; we apply Lyons–Zheng decompositions to the associated reflecting diffusions on compact sets. Let K denote the class of compact sets K satisfying that m.K/ > 0, supp.1K  m/ D K and that Z 1 K dchu;vi ; u; v 2 F ; E .u; v/ D 2 K is a closable bilinear form on L2 .XI 1K m/. Notice that, for K 2 K, E K .u; u/ is assumed to vanish whenever u 2 F vanishes m-a.e. on K. For K 2 K, we denote by F K the closure of F with respect to E K C.; /1K m . By identifying L2 .XI 1K m/ with L2 .KI m/, the pair .E K ; F K / can be regarded as a regular Dirichlet space on L2 .KI m/ which possesses the strong local property on account of Exercise 3.1.1. Then .E K ; F K / is conservative because 1K 2 F K and E K .1K ; 1K / D 0 by virtue of the strong local property of E. We set Floc;ac D ¹ 2 Floc W ch i is absolutely continuous with respect to mº

298

5 Stochastic analysis by additive functionals

and denote by . / the density of ch i with respect to the measure m for 2 Floc;ac . We set further A D ¹ 2 Floc;ac \ C.X/ W lim .x/ D 1 and the set ¹x 2 XI .x/  rº x!

belongs to K for any r > 0º:



Let Br D ¹x 2 X W .x/  rº and M .r/ D ess sup¹. /.x/ W x 2 Br º for

2 Floc;ac . We have then Lemma 5.7.2. It holds that for 2 A    2r ; sup . .X t /  .X0 //  r  6m.BRCr /` p 3 M .R C r/T 0t T (5.7.10) R 1  x2 1 2 dx. where mR D 1B  m and `.x/ D p x e PmR



R

2





Proof. Let us denote the pair .E Br ; F Br / for 2 A simply by .E r ; F r /. By virtue of Theorem 7.2.2, it admits an associated diffusion process Mr D .r ; Mr ; Pxr ; X t / on Br . Since Br is assumed to be in the class K, .E r ; F r / is conservative and we may take as Mr a conservative diffusion on account of Exercise 4.5.1. Then, we have, for R; r > 0, PmR



   sup . .X t /  .X0 //  r D PmRCr sup . .X t /  .X0 //  r R

0t T

0t T



PmRCr RCr



 sup . .X t /  .X0 //  r :

0t T

(5.7.11) The first equality follows from the fact that M and MRCr have a common part on the open set ¹x 2 X W .x/ < R C rº up to an equivalence (Theorem 4.2.7) because E and E RCr have a common part on it (Theorem 4.4.2). Since the diffusion process MRCr is conservative, it follows from the equation (5.7.2) that 1 Œ  1 Œ  Œ 

.X t /  .X0 / D M t C .MT t .rT /  MT .rT //; 2 2

PmRCr -a.e. RCr

5.7 Forward and backward martingale additive functionals

299

Hence the last expression of (5.7.11) is not greater than     2 2 Π Π RCr RCr sup M t  r C PmRCr sup M t .rT /  r PmRCr 3 3 0t T 0t T   2 Π (5.7.12) C PmRCr  MT .rT /  r RCr 3     2 2 Π Π RCr RCr  2 PmRCr sup M t  r C PmRCr sup .M t /  r 3 3 0t T 0t T by Lemma 5.7.1. According to a representation theorem for continuous martingales,14 there exists a one-dimensional Brownian motion B.t / on an extended Π e; P e F e RCr / such that M Π can be represented as M t D probability space .; x Rt B. 0 . /.Xs /ds/. Therefore we see that the last expression of (5.7.12) is dominated by   Z 2 RCr e 3 Px B.t /  r d mRCr sup  3 BRCr 0t M  .RCr/T   2r D 6m.BRCr /` p : 3 M .R C r/T Now we state a general criterion for the conservation of probability. We say that .E; F / is conservative if p t 1.x/ D 1;

m-a.e. x 2 X; t > 0:

(5.7.13)

Theorem 5.7.3. If there exist 2 A and T > 0 such that for any R > 0   r D 0; (5.7.14) lim inf m.BRCr /` p r!1 M .R C r/T then .E; F / is conservative. Proof. By Lemma 5.7.2 and the assumption (5.7.14), we get for T 0 D 49 T   PmR sup . .X t /  .X0 // D 1 0t T 0

D lim PmR r!1

6 14 Cf.



sup . .X t /  .X0 //  r

0t T 0



lim inf m.BRCr /` r!1

N. Ikeda and S. Watanabe [2].

p

r M .R C r/T



 D 0;

300

5 Stochastic analysis by additive functionals

and so pT 0 1.x/ D Px .T 0 < / D Px



 sup . .X t /  .X0 // < 1 D 1;

0t T 0

for m-a.e. x; where  is the life time of the diffusion process M. By virtue of the semi-group property we have (5.7.13). Example 5.7.1. Let E be a symmetric bilinear form on L2 .Rd I dx/ defined by E.u; v/ D

d Z @u @v 1 X aij .x/ dx; d 2 @x i @xj R

u; v 2 C01 .Rd /;

(5.7.15)

i;j D1

where the coefficients aij are locally integrable Borel functions satisfying that i) aij D aj i , ii) for any compact set K, there exists a positive constant ı.K/ such P that di;j D1 aij .x/i j  ı.K/jj2 , for any x 2 K and  2 Rd . Denote by .E; F / the closure on L2 . Rd I dx/ of the symmetric form (5.7.15). Then .E; F / is conservative if d X

aij .x/i j  k.2 C jxj/2 log.2 C jxj/jj2

i;j D1

for some constant k. In fact, by employing the function .x/ D log.2 C jxj/ 2 A,   r jBRCr j ` p M .R C r/T   r  j¹x W log.2 C jxj/  R C rºj` p k.R C r/T p d k.R C r/T  r 2 2  d e 2kT .RCr/ ! 0 as r ! 1;  e d.RCr/ r  C1 2

1 . Here jEj denotes the volume of a set E  Rd . This provided that T < 2kd sufficient condition for conservativeness is sharp in the sense that, if aij .x/ D .2 C jxj/2 .log.2 C jxj//ˇ ıij ; ˇ > 1, we see by applying Feller’s test15 to the radial part of this operator that the corresponding diffusion is not conservative.

We next state an application of Theorem 5.7.2. For 2 A put osc .Xs / D max .Xs /  min .Xs /:

0st 15 Cf.

H. P. McKean [1].

0st

0st

5.7 Forward and backward martingale additive functionals

Lemma 5.7.3. Let 2 A. Then PmR







osc .X t /  r 

0t T

16m.BRCr /`

p

301



r

(5.7.16)

M .R C r/T

Proof. For any r; R > 0,     PmR osc .X t /  r D PmRCr osc .X t /  r R 0t T

0t T

 PmRCr RCr



 osc .X t /  r :

0t T

By Theorem 5.7.2, the right-hand side is not greater than PmRCr RCr



Π

osc M t

0t T

   Π osc M t . T /  r  r C PmRCr RCr 0t T

D 2PmRCr RCr Π

is written as B.

0t T

Π

osc M t

0t T

 r :

Rt

. /.Xs /ds/, Z    e RCr P PmR osc .X t /  r  2 x

Since M t



0



BRCr

osc

0t M  .RCr/T

 B.t /  r d mRCr :

The right-hand side above is estimated by the right-hand side of (5.7.16) on account of Lemma 5.7.4 below. Lemma 5.7.4. Let .B t ; P / be the law of the Brownian motion starting at the origin. Then p   P osc Bs  r  8`.r= t /: 0st

Proof. Let s0 (resp. S0 ) be random times such that Bs0 D min0st Bs (resp. BS0 D max0st Bs ). Then     P osc Bs  r D 2P osc Bs  r; s0 < S0 0st

D 2P  2P

 

0st

sup .Bs  min Bu /  r; s0 < S0

0st

0us

0st

0us



 sup .Bs  min Bu /  r :

Since Bs  min0us Bu and jBs j have the same low, the right-hand side is dominated by 4P .sup0st Bs  r/ D 8P .B t  r/ because of the reflection principle.

302

5 Stochastic analysis by additive functionals

Example 5.7.2. Let .M; g/ be a complete Riemannian manifold and V .dx/ be its volume element. The Laplace–Beltrami operator  on M is defined by u D div.grad u/; u 2 C01 .M /. The minimal diffusion process corresponding to 12  is said to be the Brownian motion on M . Since Z div.grad u/vdV .u; v/V D  M Z .grad u; grad v/g dV ; for u; v 2 C01 .M /; D M

we can regard the Brownian motion as a diffusion process corresponding to the Dirichlet form Z 1 .grad u; grad v/g dV ; E.u; v/ D 2 M F W closure of C01 .M / with respect to E1 : For a compact set K, put .x/ D inf¹d.x; y/ W y 2 Kº, where d.x; y/ is the Riemannian distance. Then 2 Floc and 2 A because of the completeness of the Riemannian manifold M . Let Kr D ¹x 2 M W .x/  rº. Noting that .r ; r /.x/  1; m-a.e., we have from Lemma 5.7.3, Z Z     Px sup .X t /  r dV .x/ D Px osc .X t /  r dV .x/ K

0t T

K

0t T

p  16  V .Kr /  `.r= T /:

(5.7.17)

For a fixed point o 2 M , let Br D ¹x 2 M I d.o; x/  rº. We then see that if lim inf r!1

1 log V .Br / < 1; r2

(5.7.18)

then .E; F / is conservative. Indeed, the equation (5.7.18) says that for any compact set K andpa sufficiently small T there exists a sequence rn ! 1 such that V .Krn /  l.rn = T / ! 0 and thus Px .sup0t T .X t / D 1/ D 0, V -a.e. on K. Hence by the completeness of M , Px .XT 2 M / D 1 V -a.e. on M . In this case the transition function p t .x;  / is absolutely continuous with respect to V and thus the Brownian motion on M is conservative in the sense that its transition function p t satisfies (5.7.19) p t 1.x/ D 1; 8x 2 X; 8t > 0: N be the restrictions Exercise 5.7.1. Let D be an arbitrary domain in Rd and C01 .D/ to DN of functions in C01 .Rd /. Let m be the Lebesgue measure on D which is

5.7 Forward and backward martingale additive functionals

303

extended to DN by setting m.@D/ D 0. We consider the regular Dirichlet form N m/ obtained by closing the space C 1 .D/ N in the Dirichlet space .E; F / on L2 .DI 0 1 1 2 . 2 D; H .D// on L .D/. F could be a proper subspace of H 1 .D/ as was seen in Example 1.6.1. Prove that .E; F / is always conservative. As the second application of Theorem 5.7.1, we consider the tightness problem for a sequence of symmetric diffusion processes. Let E n .u; v/ be a sequence of symmetric bilinear forms on L2 .Rd ; mn / defined by E n .u; v/ D

d Z 1 X n @u @v aij d mn ; d 2 @xi @xj R

for u; v 2 C01 .Rd /;

(5.7.20)

i;j D1

where mn are everywhere dense positive Radon measures on Rd . Suppose that the n coefficients aij are Borel measurable functions such that n n (i) aij D aij

(ii) for each ball Br D ¹x 2 Rd W jxj  rº there exists a constant .r/ independent of n such that 0

d X

n aij .x/i j  .r/jj2 ;

for any .x; / 2 Br  Rd :

i;j D1

For each ball Br , let E n;r .u; v/ be the symmetric form E

n;r

d Z 1 X n @u @v .u; v/ D aij d mn ; 2 @xi @xj Br

for u; v 2 C 1 .Br /:

i;j D1

We assume the closability of .E n ; C01 .Rd // and .E n;r ; C 1 .Br // on L2 .Rd I mn / and L2 .Br I mn / respectively. The corresponding closures are denoted by .E n ; F n / and .E n;r ; F n;r /. We consider the following condition. Condition 5.7.1. There exists a constant T > 0 such that for any R > 0  ² ³ r lim sup mn .BRCr /` p D 0: r!1 n .R C r/T In view of Theorem 5.7.2, Condition 5.7.1 and Exercise 4.5.1 the diffusion processes Mn D .Pxn ; X t / corresponding to .E n ; F n / by Theorem 7.2.2 can be taken to be conservative. Hence we may assume that the probability measures Pxn live on C.Œ0; 1/ ! Rd /, the space of all continuous functions from Œ0; 1/ to Rd .

304

5 Stochastic analysis by additive functionals

For a sequence of probability measures n on Rd , we define probability measures P nn on C.Œ0; 1/ ! Rd / by Z n Pxn .  /dn .x/: P n .  / D Rd

We further consider the following conditions: Condition 5.7.2. (i) supn mn .K/ < 1 for any compact set K. (ii) n is absolutely continuous with respect to mn , say n D 'n mn , and a sequence ¹'n º satisfies that supn k'n k1;K .D supn ess supx2K j'n .x/j/ < 1 for any compact set K. (iii) ¹n º is tight. Theorem 5.7.4. Under Condition 5:7:1 and Condition 5:7:2, the sequence of probability measures ¹P nn º is tight on the space C.Œ0; 1// ! Rd / equipped with the local uniform topology. Proof. For ı > 0, let n .x/ D Pxn qh;L



sup 0s;t L;jt sjh

 jX ti  Xsi j > ı :

Here X ti is the i-th component of the diffusion process X t . Note that n n c ; 'n /mn k 'n k1;BR .qh;L ; 1BR /mn C n .BR /: .qh;L

Hence, if we can show that for any L; R > 0 n ; 1BR /mn D 0; lim sup.qh;L

h!0 n

(5.7.21)

then we arrive at the theorem on account of Condition 5.7.2 (ii) and (iii). Put T 0 D 49 T and mn;r D 1Br mn . Then   n n;RCr .qh;T sup jX ti  Xsi j > ıI ƒr (5.7.22) 0 ; 1BR /mn D Pmn;R 0s;tT 0 jtsjh

C Pmn n;R



sup

0s;tT 0 jtsjh

 jX ti  Xsi j > ıI ƒcr ;

where ƒr D ¹! W sup0t T 0 .jX t j  jX0 j/ < rº and Pxn;r is a conservative diffusion process corresponding to .E n;r ; F n;r /. Since it follows from the equation (5.7.2) that 1 1 Œx  Œx  Œx  X ti  Xsi D .M t i  MsŒxi  / C .MT 0 it .rT 0 /  MT 0 is .rT 0 //; 2 2

Pmn;RCr -a.e.; n;RCr

5.7 Forward and backward martingale additive functionals

305

the first term of the right-hand side of (5.7.16) is dominated by   Pmn;RCr sup jX ti  Xsi j > ı n;RCr 0s;tT 0 jtsjh

 Pmn;RCr n;RCr



sup

0s;tT 0 jtsjh

C Pmn;RCr n;RCr D 2Pmn;RCr n;RCr





Œxi 

jM t

sup

Œxi 

0s;tT 0 jtsjh

sup

0s;tT 0 jtsjh

 MsŒxi  j > ı

jM t

Œxi 

jM t



.rT 0 /  MsŒxi  .rT 0 /j > ı



  MsŒxi  j > ı :

(5.7.23)

Let B.t/ be a one-dimensional Brownian motion on an extended probability Rt Œx  e ;P eF e n;RCr / by which M Œxi  is represented as M t i D B. 0 ai i .Xu /du/, space .; x e n;RCr -a.e. Hence we have P x ® eW !2

sup

Œxi 

0s;tT 0 jtsjh

² eW D !2

® eW  !2

jM t

sup

0s;tT 0 jtsjh

 MsŒxi  j > ı

¯

ˇ Z t  Z ˇ ˇB ai i .Xu /du  B ˇ

sup

0s;t.RCr/T 0 jtsj.RCr/h

0

jB.t/  B.s/j > ı

¯

s 0

ˇ ³ ˇ ˇ ai i .Xu /du ˇ > ı

e n;RCr P -a.e., q.e. x: x

Therefore, denoting by .W D C.Œ0; 1/ ! Rd /; P w / the standard Wiener process and setting   w .h; R C r/ D P j!.t /  !.s/j > ı ; !2W W sup 0s;t.RCr/T 0 jtsj.RCr/h

the last term of (5.7.17) is not greater than 2mn .BRCr / .h; R C r/. On the other hand, according to Lemma 5.7.2 we have   r n c Pmn;R .ƒr /  6mn .BRCr /` p : .R C r/T Hence the right-hand side of (5.7.23) is dominated by   r ; 2mn .BRCr / .h; R C r/ C 6mn .BRCr /` p .R C r/T

306

5 Stochastic analysis by additive functionals

and consequently we get for any R > 0 n lim sup.qh;T 0 ; 1BR /mn D 0

(5.7.24)

h!0 n

by virtue of Condition 5.7.1 and Condition 5.7.2 (i). Note that by the Markov property   n i i n Pmn;R jX t  Xs j > ı D .pˇn .qh;T sup 0 /; 1BR /mn ˇs;tT 0 Cˇ jtsjh

(5.7.25)

n n n n D .qh;T 0 ; 1BR0 pˇ .1BR //mn C .qh;T 0 ; 1B c 0 pˇ .1BR //mn R



n .qh;T 0 ; 1BR0 /mn

C

.1BR ; pˇn .1B c 0 //mn : R

Therefore, it follows from (5.7.24) and Lemma 5.7.2 that, for 0  ˇ  T 0 ,   n i i jX t  Xs j > ı sup lim sup sup Pmn;R h!0

ˇs;tT 0 Cˇ jtsjh

n

n n  lim sup lim sup sup¹.qh;T 0 ; 1BR0 /mn C .1BR ; pˇ .1B c 0 //mn º D 0; R0 !1

h!0

n

R

n and consequently, limh!0 supn .qh;T 0 Cˇ ; 1BR /mn D 0, for any R > 0. By repeating this argument, (5.7.21) can be established.

Chapter 6

Transformations of forms and processes

We consider a Dirichlet form E and a symmetric Hunt process M as before. This chapter deals with three basic transformations of E corresponding to the transformations of M by means of the additive functionals studied in previous sections. Let  be a positive Radon measure on X charging no set of zero capacity and A be a PCAF of M with Revuz measure . The perturbed Dirichlet form .E ; F / is a Dirichlet form on L2 .XI m/ defined by e \ L2 .XI /; F DF

E .u; u/ D E.u; u/ C .u; u/ :

E corresponds to the canonical subprocess of M with respect to A (§6.1). L FL / of E on the support Y of  is a Dirichlet form on L2 .Y I / The trace .E; defined by e e \ L2 .Y I /; FL D F

L Y ; ujY / D E.H u; H u/ E.uj e e Y Y

e denotes the quasi support of . EL corresponds to the random time change where Y of M with respect to A (§6.2). In particular, the extended Dirichlet space is unchanged under the time change when  has the full quasi support. When M is the d -dimensional Brownian motion, its multiplicative functional ˇ Z t  Z ˇ 1 t ˇˇ r ˇˇ2 r Œ L t D exp .Xs /dXs  .X /ds s 2 0 ˇ  ˇ 0  gives rise to a drift transformation of M, by which the generator 12  (resp. the R Dirichlet form 12 jruj2 dx on L2 .Rd /) is transformed into 12  C r   r (resp. R 1 1 2 2 2 d 2 d 2 jruj  dx on L .R I  dx/). Here  is a positive function in Hloc .R /. When  decays near infinity, the drift r= forces the transformed process to move back inward. §6.3 shall treat this sort of transformation in the present general setting. Especially, if  is in the domain of L2 -generator, then the transformed process of M will be shown to be recurrent even when M is non-conservative, §6.4 presents the Donsker–Varadhan type large deviation principle for the occupation time distributions of (not necessarily conservative) symmetric Markov processes. The lower bound in it will be obtained by making use of the transformation in §6.3. Finally the large deviation from the ground state will be formulated.

308

6.1

6 Transformations of forms and processes

Perturbed Dirichlet forms and killing by additive functionals

Let .X; m/ be as in (1.1.7) and MD .X t ; Px / be an m-symmetric Hunt process whose Dirichlet space .E; F / is regular. We have introduced in §1.6 the regular Dirichlet space .E g ; F / on L2 .XI m/ for a function g in K of (1.6.14) and we have seen in §4.6 (Lemma 4.6.7) that the canonical subprocess Mg of M relative to Rt C t ; C t D 0 g.Xs /ds, is a Hunt process associated the multiplicative functional e with .E g ; F /. Clearly C t is the PCAF having  D g  m as its associated smooth measure. The aim of this section is to extend this result to general positive Radon measure  charging no set of zero capacity. In this section we shall fix such a measure  on X and consider the perturbed Dirichlet form .E ; F / on L2 .XI m/ defined by e \ L2 .XI /; F DF

E .u; v/ D E.u; v/ C .u; v/ ;

(6.1.1)

u; v 2 F :

(6.1.2)

We shall see in Theorem 6.1.1 that the transition function A t pA f .X t // t f .x/ D Ex .e

is associated with .E ; F /, where A is a PCAF with Revuz measure . We then show that .E ; F / is regular again. For any ˛ > 0 and u; v 2 F , let E˛ .u; v/ D E .u; v/ C ˛.u; v/ as usual. Lemma 6.1.1. Let  be a positive Radon measure charging no set of zero capacity. Then the symmetric form .E ; F / defined by .6:1:1/ and .6:1:2/ is a Dirichlet space relative to L2 .XI m/. If .X/ < 1, then .E ; F / is regular and more specifically any standard core of .E; F / becomes a standard core of .E ; F /. Proof. To prove the closedness of the symmetric form .E ; F /, let us take a Cauchy sequence ¹un º in .E ; F /. Then it is Cauchy in .E; F / and converges e relative to E1 -metric. Since ¹un º is Cauchy in L2 .XI / as to an element u 2 F well, there exists a function v 2 L2 .XI / such that limn!1 un D v in L2 .XI /. It suffices to prove that u D v -a.e.

6.1

Perturbed Dirichlet forms and killing by additive functionals

309

To this end, let F be a closed set such that 1F   2 S0 . Then, by taking a subsequence ¹unk º of ¹un º such that unk ! v -a.e., we have from (2.2.1) and Fatou’s lemma that Z Z ju.x/  v.x/j.dx/  lim ju.x/  unk .x/j1F .x/.dx/ F

k!1 X

 C lim

k!1

q

E1 .u  unk ; u  unk / D 0:

Hence u D v -a.e. on F and hence u D v -a.e. on X owing to Theorem 2.2.4. Therefore .E ; F / is closed and a Dirichlet space because it clearly satisfies the Markovian property. Now assume that .X/ < 1. We shall prove that any function u 2 F can be approximated by a sequence of functions of F \ C0 .X/ with respect to E1 -norm. By considering uC and u separately and making truncations if necessary, we may assume that 0  u  for some constant . According to the regularity of .E; F /, there exists a sequence ¹un º  F \ C0 .X/ such that limn!1 E1 .un  u; un  . / . / u/ D 0. Put un D .0 _ un / ^ for each n. Clearly un 2 F \ C0 .X/ and it is E1 -convergent to u as n ! 1 by Theorem 1.4.2 (v). By virtue of Theorem 2.1.4, . / . / there then exists a subsequence ¹unk º such that unk ! u q.e. and in particular . / -a.e. Therefore unk ! u in L2 .XI / by Lebesgue’s dominated convergence . / theorem, proving that unk ! u relative to E1 -norm. This proof still works in proving the assertion for a standard core C of .E; F / . / by taking a sequence ¹un º from C and setting un D " .un = / for any fixed " > 0. The finiteness condition of  for the regularity will be removed after the next theorem is proved. Let A be a PCAF of M whose Revuz measure is . We denote by N a properly exceptional set for A. By virtue of Theorem A.2.11, the canonical subprocess MA D .X tA ;  A ; Px / of MjX nN with respect to the multiplicative functional .e A t / is a Hunt process on X n N . Moreover, owing to Theorem A.2.9, MA is extended to a Hunt process on X by making each point of N a trap. The transition function p A of MA is given by A t pA f .X t //; t f .x/ D Ex .e

x 2 X n N ; f 2 B C .X/:

We denote by R˛A f the Laplace transform of p A t f:  Z 1 A A t ˛t e f .X t /dt ; R˛ f .x/ D Ex 0

x 2 X n N:

.6:1:3/

.6:1:4/

310

6 Transformations of forms and processes

Lemma 4.6.7 is extended to the following Theorem 6.1.1. Let  be a positive Radon measure charging no set of zero capacity and .A t / t 0 be its associated PCAF of M. Then MA is a Hunt process associated with the Dirichlet form .E ; F / on L2 .XI m/. Proof. As in the proof of Lemma 4.6.7, Theorem 6.1.1 reduces to the assertion R˛A f 2 F ;

E˛ .R˛A f; u/ D .f; u/;

f 2 L2 .XI m/; u 2 F :

.6:1:5/

Since kR˛A f kL2 .X Im/  kR˛ f kL2 .X Im/  ˛1 kf kL2 .X Im/ , we need to prove (6.1.5) only for bounded f 2 L2 .XI m/. We first prove that the relation (6.1.5) is valid when  2 S00 . According to Lemma 5.1.5, R˛A f  R˛ f C UA˛ R˛A f D 0;

˛ > 0; f 2 B C .X/:

.6:1:6/

If  is in S00 , and if f is a bounded function in L2 .XI m/ then kR˛A f k1 < 1 and UA˛ R˛A f is a quasi continuous version of the ˛-potential U˛ .R˛A f / 2 F by Lemma 5.1.3. Since kU˛ .R˛A f /k1  kR˛A f k1 kU˛ k1 < 1 and .X/ < 1, we have that e \ L2 .XI // R˛A f D R˛ f  UA˛ R˛A f 2 F .D F and that E˛ .R˛A f; u/ D E˛ .R˛ f; u/  E˛ .UA˛ R˛A f; u/ D .f; u/  .R˛A f; u/ ;

u 2 F ;

the assertion (6.1.5). For general positive Radon measure  charging no set of zero capacity, we can take by virtue of Theorem 2.2.4 and Lemma 5.1.6 an increasing sequence ¹Fn º of closed subsets of X such that Px .limn!1 X nFn < / D 0 q.e. x 2 X , and n D 1Fn   2 S00 . Since  charges no set of zero capacity, n .B/ increases to .B/ for any B 2 B.X/. Let An D 1Fn  A. Then An is a PCAF of M with Revuz measure n . Since n 2 S00 , we have for f 2 L2 .XI m/ R˛An f 2 F n ;

E˛ n .R˛An f; u/ D .f; u/;

8u 2 F n :

.6:1:7/

Clearly jR˛An f j  R˛ jf j < 1 q.e. and hence limn!1 R˛An f .x/ D R˛A f .x/ for q.e. x 2 X. For n < m, we get from (6.1.7) E˛ n .R˛An f  R˛Am f; R˛An f  R˛Am f /  .f; R˛An f  R˛Am f /;

.6:1:8/

6.1

Perturbed Dirichlet forms and killing by additive functionals

311

which converges to zero as n; m ! 1. Therefore ¹R˛An f º is E1 -convergent in F e . On the other hand, we also get from (6.1.7) and the limit function R˛A f is in F 2 An kR˛An f kL 2 .X I /  .f; R˛ f /L2 .X Im/  n

1 2 kf kL 2 .X Im/ ˛

and by Fatou’s lemma 1 kR˛A f kL2 .X I /  p kf kL2 .X Im/ ; ˛ getting R˛A f 2 F . Finally observe the estimate j.R˛An f; u/ n  .R˛A f; u/ j  kR˛An f  R˛A f kL2 .X I n /  kukL2 .X I / C j.R˛A f; u/  n j holding for u 2 L2 .XI /. The second term of the right-hand side tends to zero as n ! 1. The first term also tends to zero because we have from (6.1.8) 2 An Am kR˛An f  R˛Am f kL 2 .X I /  .f; R˛ f  R˛ f / n

and it suffices to let first m ! 1 and then n ! 1. By letting n ! 1 in (6.1.7), we arrive at the desired equation (6.1.5). Theorem 6.1.2. Let  be a positive Radon measure charging no set of zero capacity. The Dirichlet space .E ; F / on L2 .XI m/ is then regular. Furthermore, if C is a special standard core for .E; F /, then so it is for .E ; F /. Proof. This was proved in Lemma 6.1.1 when .X/ < 1. Due to Theorem 6.1.1, we now have the canonical subprocess M of M relative to the multiplicative functional e A t as a Hunt process associated with .E ; F /. Let .M /G be the part of M on a relatively compact open set G and ..E /G ; .F /G / be its Dirichlet form on L2 .GI m/. By virtue of Theorem 4.4.5, 8 <.F /G  F ; .E /G D E on .F /G  .F /G .6:1:9/ :S .F / is E -dense in F : G G 1 Note that the operations on M of killing by PCAF A t and killing by the exit time G commute with each other, i.e. .M /G D .MG / , producing the same transition function ;G

pt

f .x/ D Ex .e A t f .X t /; t < G /:

This means that ..E /G ; .F /G / D ..EG / ; .FG / /.

312

6 Transformations of forms and processes

Let C be a special standard core of .E; F /. We let CG D ¹u 2 CI suppŒu  Gº: On account of Theorem 4.4.3, the Dirichlet form .EG ; FG / of MG coincides with the part of .E; F / on G and possesses CG as its core. SinceS.G/ is finite, CG is also a core of ..EG / ; .FG / / by Lemma 6.1.1. Since C D G CG and .EG /1 D E1 on .FG /  .FG / , we can conclude from (6.1.9) that C is E1 -dense in F . We shall designate by the superscript  the notions associated with the Dirichlet space .E ; F /. The following lemma, which can be proved similarly as the proof of Lemma 2.1.8, means that the regular Dirichlet form .E ; F / on L2 .XI m/ shares the quasi-notions in common with .E; F /. Lemma 6.1.2. (i) For any decreasing sequence ¹An º of relatively compact open subsets of X , Cap.An / # 0 if and only if Cap .An / # 0. (ii) S D S . ˛ Given  and A as before, we next consider a related kernel Up;A defined by

Z ˛ Up;A f .x/

D Ex

1

e

˛t pA t

 p  0:

f .X t /dA t

0

.6:1:10/

˛ D UA˛ and we can show similarly to (6.1.6) the resolvent equation U0;A ˛ ˛ UA˛ f .x/  Up;A f .x/  pUA˛ Up;A f .x/ D 0:

.6:1:11/

The following lemma will be used in the next section. ˛ Lemma 6.1.3. For any ˛; p > 0 and f 2 L2 .XI /, Up;A f is quasi continuous and ˛ f 2 F ; Up;A

˛ E˛p .Up;A f; u/ D .f; u/ ;

u 2 F :

.6:1:12/

Proof. We can proceed analogously to the proof of Theorem 6.1.1. We first note that, if (6.1.12) were true, then ˛ f kL2 .X I /  kUp;A

1 kf kL2 .X I / : p

.6:1:13/

Hence we need to show the quasi continuity and (6.1.12) only for those functions f which are dense in L2 .XI /.

6.1

Perturbed Dirichlet forms and killing by additive functionals

313

˛ f ) is Suppose that  2 S00 and f is bounded. Then UA˛ f (resp. UA˛ Up;A bounded and a quasi continuous version of the ˛-potential of the finite measure ˛ ˛ f ). Consequently we conclude from (6.1.11) that Up;A f 2F f  (resp. Up;A and ˛ ˛ E˛ .Up;A f; u/ D E˛ .U˛ .f  /; u/  pE˛ .U˛ .Up;A f  /; u/ ˛ D .f; u/  p.Up;A f; u/ ;

u2F

the validity of (6.1.12). In general, we find as in the proof of Theorem 6.1.1 an increasing sequence ¹Fn º of quasi-closed sets such that n D 1Fn   2 S00 and n " . Let us assume that f is bounded and vanishing on X n Fn0 for some n0 . Then ˛ f 2 F n ; Up;A n

˛ E˛p n .Up;A f; u/ D .f; u/ n0 ; n

u 2 F n ;

.6:1:14/

for n  n0 , where An D 1Fn  A. Since we see from the expression (6.1.10) ˛ ˛ that Up;A f converges to Up;A f pointwise and boundedly, we can conclude from n ˛ ˛ f is (6.1.8) that Up;An f is E1 -convergent and accordingly the limit function Up;A ˛ e in F . Similarly as in the proof of Theorem 6.1.1, we can observe that Up;A f 2 ˛ ˛ L2 .XI / and that .Up;A f; u/ n converges to .Up;A f; u/ for any u 2 L2 .XI /. n Corollary 6.1.1. For any ˛  0; p  0 and f; g 2 B C , ˛ ˛ .Up;A f; g/ D .f; Up;A g/ ;

(6.1.15)

˛ .R˛pA f; g/ D .f; Up;A g/m :

(6.1.16)

The second identity follows from (6.1.5) and (6.1.12). Let us denote by Fe the extended Dirichlet space of .E ; F /. According to the regularity of .E ; F /, any function u 2 Fe possesses an .E ; F /-quasi continuous modification e u. e u is .E; F /-quasi continuous as well by Lemma 6.1.2. By virtue of Theorem 5.2.2, e u.X t /  e u.X0 / is expressed as Œu

e u.X t /  e u.X0 / D M t

V N Œu 2 Nc : M Œu 2 M;

Œu

C Nt ;

Applying the same theorem to .E ; F / and its associated canonical subprocess MA , we get the corresponding decomposition ;Œu

u.X0A / D M t e u.X tA /  e ;Œu

;Œu

C Nt

;Œu

into an MAF M t of finite energy and a CAF N t of zero energy relative to g;Œu ;Œu g;Œu ;Œu MA . We put M t D Mt and N t D Nt when  D g  m.

314

6 Transformations of forms and processes

In the proof of Theorem 5.4.1, we have already made use of the relation of N Œu and N g;Œu which is a special case of the following lemma.

Lemma 6.1.4. For any u 2 Fe , Œu Nt

D

;Œu Nt

Z C

t

0

e u.XsA /dAs ;

t <  A:

.6:1:17/ ;Œu

D Proof. We first assume that u D R1A f; f 2 L2 .XI m/ \ Bb . Then N t Rt 1 A 0 .u  f /.Xs /ds by (5.2.26). On the other hand, since UA u is a quasi continuous version of U1 .u  / and u D R1 f  UA1 u by Lemma 5.1.5, we have from Lemma 5.4.1 Z t Z t Z t Œu 1 Nt D .R1 f  f /.Xs /ds  UA u.Xs /ds C u.Xs /dAs 0

Z D

0

0

Z

t

.u  f /.Xs /ds C

0

t

u.Xs /dAs : 0

Now it suffices to note that XsA D Xs ; 8s <  A to get (6.1.17). For general u 2 Fe take a sequence un D R1A fn ; fn 2 L2 .XI m/ \ Bb which is E -convergent to u. Since ¹un º is E-convergent to u as well, the final part of the proof of Theorem 5.2.2 then implies that there exists a subsequence ¹unk º such ;Œun 

;Œu

Œun 

Œu

k u.X tA /, N t ! Nt and N t k ! N t uniformly on that unk .X tA / ! e any finite t -interval a.s. The relation (6.1.17) now follows from the same relation for un .

Exercise 6.1.1. Suppose that M satisfies the absolute continuity condition (4.2.9). Let  be a positive Radon measure on X smooth in the strict sense and A be the PCAF in the strict sense with Revuz measure  (Theorem 5.1.7). Let M be the canonical subprocess of M with respect to the multiplicative functional (in the strict sense) e A t . Show that M also satisfies (4.2.9) and Z A r˛ .x; z/r˛A .z; y/.dz/; x; y 2 X; r˛ .x; y/ D r˛ .x; y/ C X

r˛A

where r˛ and are the resolvent densities of M and M respectively in the sense of Lemma 4.2.4.

6.2

Traces of Dirichlet forms and time changes by additive functionals

Let X; m, M and .E; F / be as in the preceding section. By virtue of Theorem 2.1.7, every function in Fe has a quasi continuous modification. Throughout

6.2

Traces of Dirichlet forms and time changes by additive functionals

315

this section every function in the space Fe is considered to be quasi continuous already. As in the preceding section, let us fix an arbitrary positive Radon measure  on X charging no set of zero capacity and A be a PCAF with Revuz measure . We denote by Y the support of ; i.e., Y is the smallest closed set outside of which e of A is defined by (5.1.21). As was seen in §5.1, Y e is  vanishes. The support Y e e e a quasi-support of , Y is nearly Borel measurable and Y  Y q.e. Y n Y is -negligible but not necessarily of zero capacity. When .E; F / is transient, we can consider the following orthogonal decomposition of the Hilbert space .Fe ; E/: 8
' 2 FL ; ' D u -a.e. on Y; u 2 Fe :

.6:2:4/ L FL / the trace Owing to Lemma 6.2.1, the definition makes sense. We may call .E; of the space .E; F / on Y relative to . Let us also define the time changed process L D .XL t ; Px / M of M with respect to the PCAF A by x2e Y XL t D X t ;

 t D inf¹s > 0I As > t º:

.6:2:5/

316

6 Transformations of forms and processes

L L Show that A D . Exercise 6.2.1. Let L be the life time of M. L is a normal right continuous strong Markov In view of Theorem A.2.12, M e. We shall designate by superscript check the notions associated with process on Y L FL / and the time changed process M. L In particular, the transition this form .E; L function and the resolvent of M are given respectively by e; pL t '.x/ D Ex .'.X t // ; x 2 Y  Z 1 Z pt RLp '.x/ D Ex e '.X t /dt D Ex 0

1

e

pA t

 .6:2:6/ '.X t /dA t :

0

0 e. Since .Y nY e/ D 0 and U 0 is -symmetric by Corol' on Y Thus RLp ' D Up;A p;A lary 6.1.1, RLp and pL t are -symmetric. Furthermore lim t #0 pL t '.x/ D '.x/; 8x 2 e; 8' 2 Cb .Y /. On account of Lemma 1.4.3 pL t gives a strongly continuous semiY group TLt on L2 .Y I /. The main theorem of this section is the following.

Theorem 6.2.1. Let M D .X t ; Px / be the Hunt process associated with a regular Dirichlet space .E; F / relative to L2 .XI m/. L is -symmetric (i) The transition function .pL t / t >0 of the time changed process M L and determines a strongly continuous semigroup .T t / t >0 on L2 .Y I /. (ii) The Dirichlet space on L2 .Y I / associated with .TLt / t 0 coincides with L FL / of .6:2:4/. .E; L FL / is regular. Furthermore, if C is a special standard core of .E; F /, (iii) .E; L FL /, where CjY D ¹ujY W u 2 Cº. then CjY is a core of .E; L FL / in the L is properly associated with .E; e is of EL 1 -capacity zero. M (iv) Y n Y L L sense that pL t ' is an E1 -quasi continuous version of T t ' for any t > 0 and ' 2 L2 .Y I /. The first assertion of this theorem was just proven. We shall prove the rest in two cases separately; the transient case and the general case.

6.2.1

Transient case

We shall first suppose that the process M is transient, i.e., Z Rf .x/ D

0

1

p t f .x/dt < 1 m-a.e.; 8f 2 B C \ L1 .XI m/:

6.2

Traces of Dirichlet forms and time changes by additive functionals

Let us consider a function space defined by 8
' 2 FLe ; ' D ujY -a.e., u 2 Fe :

317

.6:2:7/

Here P denotes the orthogonal projection on H e Y in the Hilbert space .Fe ; E/: L P u D He Y u. We regard two functions in Fe to be identical if they coincide -a.e. To describe this definition in a clearer way, we introduce the space L of equivalence classes of all -measurable functions on Y , two functions being regarded equivalent if they coincide -a.e. Introduce the linear mapping from H e Y to L by 

He Y 3 u 7! ujY 2 L: Recall that any function in Fe is assumed to be quasi continuous. Lemma 6.2.1 assures that the mapping is injective. We can then let FLe be the image of H e Y and EL be the image form of E by . Since .H e ; E/ is a real Hilbert space, so is Y L L L L the image space .Fe ; E/. We may call .Fe ; E/ the trace of .Fe ; E/ on the support of . Lemma 6.2.2. .TLt / t >0 is transient. There is a reference function gL for the transient semigroup .TLt / t >0 such that, for any ' D   gL with bounded , L 2 FLe ; G'

L G'; L E. / D .'; / ;

8

2 FLe :

Since FLe \ L2 .Y I / equals the space FL of (6.2.4), this lemma combined with Theorem 1.5.5 leads us to Theorem 6.2.1 (ii). L is the extended transient Dirichlet space of .TLt / t >0 . .E; L FL / Corollary 6.2.1. .FLe ; E/ 2 L is the Dirichlet space on L .Y I / associated with .T t / t >0 . Proof of Lemma 6:2:2. According to the 0-order version of Theorem 2.2.4, there exists a generalized nest ¹Fn º satisfying (2.2.16) and 1Fn   is of finite 0-order energy integral for each n, i.e. Z p jv.x/j1Fn .x/.dx/  Cn E.v; v/; 8v 2 Fe Y

P n 1 for some positive constant Cn . Put g.x/ L D 1 nD1 2 .Cn _ .Fn / _ 1/ 1Fn .x/: Then gL is a -a.e. positive bounded -integrable function on Y such that Fe  L1 .Y I gL  / and Z p jv.x/jg.x/.dx/ L  E.v; v/; v 2 Fe : .6:2:8/ Y

318

6 Transformations of forms and processes

Set B1 .Y / D ¹' D   gL W  is a bounded Borel function on Y º. (6.2.8) implies the existence of the 0-order potential U.'  / 2 Fe of the signed Radon measure '   satisfying .6:2:9/ E.U.'  /; v/ D .'; v/ ; v 2 Fe ; for any ' 2 B1 .Y /. As an element of Fe , U.'  / is assumed to be quasi continuous. Then we have .6:2:10/ U.'  /.x/ D UA0 '.x/; q.e. x 2 X; ' 2 B1 .Y /: R 1  Recall that UA0 '.x/ D Ex 0 '.X t /dA t . In fact, for ' 2 B1 .Y / and for h with .h; R0 h/m < 1, we let ˛ # 0 in (5.1.14) with f D ' to get .h; UA0 '/m D h'  ; R0 hi D E .U.'  /; R0 h/ D .h; U.'  //m : Since UA0 ' is excessive and finely continuous q.e., we arrive at the identity (6.2.10). On the other hand, (6.2.6) and the strong Markov property of M imply e; UA0 '.x/ D RL 0 '.x/; x 2 Y Z 1  0 L UA '.x/ D Ex '.X t /dA t D H e Y .R0 '/.x/;

e Y

x 2 X n N:

.6:2:11/

Therefore inequality (6.2.8) combined with equation (6.2.9) for ' D gL and v D U.gL  / means that .g; L RL 0 g/ L  1, namely, .TLt / t >0 is transient and gL is its reference function. Further, rewriting the relation (6.2.9) and (6.2.10), we have L that RL 0 ' D U.'  /j e Y 2 Fe and L RL 0 '; / D E.P .U.'  // ; P v/ E. D E.U.'  /; P v/ D .'; H e Y v/ D .'; / for any

2 FLe with

D vjY -a.e., v 2 Fe .

L FL /. We make use of the reguProof of Theorem 6:2:1 (iii). The regularity of .E; larity of .E ; F / proven in Theorem 6.1.2 and an approximation of the potential 0 ˛ ' by potentials Up;A ' 2 F appearing in Lemma 6.1.3. Up;A First notice the obvious relations 8
6.2

Traces of Dirichlet forms and time changes by additive functionals

319

a family C  F \ C0 .X/ is E˛ -dense in F , then CjY . FL \ C0 .Y // is EL1 -dense in F jY . On account of Theorem 6.1.2, it therefore suffices to show the next lemma.

Lemma 6.2.3. F jY is EL1 -dense in FL .   Proof. The range of the resolvent RL 1 L2 .Y I / is EL 1 -dense in FL . Take any non˛ negative ' 2 L2 .Y I /. Since U1;A ' 2 F for ˛ > 0 by Lemma 6.1.3, it is enough to prove that ˛ 'jY ! RL 1 ' U1;A ˛#0

in EL1 -metric:

˛ 0 ' increases pointwise to U1;A ' which equals RL 1 ' -a.e. on Y . But, as ˛ # 0, U1;A ˛ Hence we need only to check that U1;A 'jY is EL1 -Cauchy. We now utilize Lemma 6.1.3 to get for ˇ > ˛ > 0 ˇ

ˇ

˛ ˛ E .U1;A '  U1;A '; U1;A '  U1;A '/ ˇ



ˇ

ˇ

˛ ˛ ˛ '; U1;A '/  2E˛ .U1;A '; U1;A '/ C Eˇ .U1;A '; U1;A '/  E˛ .U1;A ˇ

˛ '/  .'; U1;A '/ ; D .'; U1;A ˛ which tends to zero as ˛; ˇ ! 0. Hence U1;A 'jY is EL 1 -Cauchy in view of (6.2.12).

The last assertion (iv) of Theorem 6.2.1 will be proved by using the next lemma. Lemma 6.2.4. (i) For any decreasing sequence ¹An º of open subsets of X, L L Cap.An / # 0 implies Cap.A n \ Y / # 0: Conversely, Cap.An \ Y / # 0 e implies Cap.An \ Y / # 0 provided that A1 is relatively compact. L continuous (ii) If u is an E-quasi continuous function on X then ujY is E-quasi on Y . Proof. (ii) is obvious from (i). (i) Suppose that A is an open subset of X with Cap.0/ .A/ < 1. Let e .0/ 2 Fe be the 0-order equilibrium potential of A relative to .Fe ; E/ and set ' D e .0/ jY . Then ' D 1 -a.e. on A \ Y and L L .0/ .A \ Y /  E.'; Cap '/ D E.P e .0/ ; P e .0/ /  E.e .0/ ; e .0/ / D Cap.0/ .A/: Hence the 0-order version of the first assertion of (i) holds.

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6 Transformations of forms and processes

L .0/ .A \ Y / < 1 for an open A  X. Let Conversely suppose that Cap L FL /. We can choose be the 0-order equilibrium potential of A \ Y relative to .E; a function v 2 Fe such that H e -a.e. on Y . Since H e Yv D Y v is a quasi e continuous function which equals 1 -a.e. on A and Y is a quasi-support of , we e have H e Y v.x/ D 1 q.e. x 2 A \ Y by Exercise 4.6.2. Therefore, by virtue of Theorem 2.1.5, L ; / D Cap e/  E.H v; H v/ D E. L .0/ .A \ Y /: Cap.0/ .A \ Y e e Y Y We have now proved the 0-order version of the assertion (i) which obviously implies (i) by Lemma 2.1.8. L FL / is a regular Dirichlet space, there exists Proof of Theorem 6.2.1 (iv). Since .E; L FL / by Theorem 7.2.1. Let .Sp /p>0 a Hunt process MY on Y associated with .E; L be the resolvent kernel of MY . Then Sp ' is an E-quasi continuous version of the 2 L FL / for any Borel ' 2 L .Y I / by Theorem 4.2.3. resolvent GL p ' for .E; 0 On the other hand, Up;A ' is E-quasi continuous in view of Lemma 6.1.3 and L continuous on the proof of Lemma 6.2.3, and consequently U 0 'jY is E-quasi p;A

0 account of Lemma 6.2.4. Therefore noting the identity Up;A '.x/ D RLp '.x/; 0 L e, we get that Sp ' D U ' E-q.e. x2Y on Y . In particular, if ' 2 C0C .Y / then p;A L for E-q.e. x2Y 0 '.x/ D lim nSn '.x/ D lim nUn;A '.x/ n!1 n!1 Z 1  nA t D lim nEx e '.X t /dA t n!1

D lim Ex n!1

0 1

Z

e 0

s

 '.Xs=n /ds

D Ex .'.X //; e Y L because  e Y D 0 D inf¹t > 0 I A t > 0º Px -a.s. for E-q.e. and hence E-q.e. x 2 Y . Therefore the family of functions on Y defined by L x 2 Yº ¹' 2 B C .Y / W '.x/ D Ex .'.X //; E-q.e. e Y is a monotone class containing C0C .Y /, and consequently it coincides with B C .Y /. By taking ' D 1Y ne Y we have L // E-q.e. x 2 Y: 1Y ne Y .x/ D Ex .1Y ne Y .X e Y

6.2

Traces of Dirichlet forms and time changes by additive functionals

321

e a.s. Px for E-q.e. x 2 X. Now, applying e is finely closed, X 2 Y Since Y e Y L Lemma 6.2.4 (i), we get 1Y ne Y .x/ D 0 E-q.e. x 2 Y which is equivalent to L e/ D 0. Cap.Y nY L continuous version of GL p ' for Borel It is now clear that RLp ' is an E-quasi 2 ' 2 L .Y I /. An analogous statement is valid for pL t ' in view of the proof of Theorem 4.2.3.

6.2.2

General case

We now consider the general case that .E; F / is not necessarily transient. We shall prove Theorem 6.2.1 by the method of the perturbations. First we give a general remark. Let  be an arbitrary positive Radon measure charging no set of zero capacity. We denote by .Fe ; E  / the extended Dirichlet space of the perturbed Dirichlet form .E  ; F  / on L2 .XI m/: Lemma 6.2.5. It holds that Fe \ L2 .XI / D Fe :

.6:2:13/

Proof. If u 2 Fe ; then u 2 Fe : Take an E  -approximating sequence ¹un º  F  of u: By the proof of Lemma 1.6.7, there exists g 2 K such that u 2 Feg and ¹un º is E g -convergent to u: By the 0-order version of Theorem 2.1.4, a subsequence of ¹un º converges to u q.e. Since ¹un º is L2 .XI /-Cauchy, u 2 L2 .XI /: Conversely, suppose u 2 Fe \ L2 .XI /: We may assume that u  0: By Theorem 2.1.7, we find ¹un º  F which is E-Cauchy and convergent to u q.e. Let vn D .0 _ un / ^ u .2 F /; n  1: vn then converges to u q.e. as n ! 1; and kvn k2E  kun k2E C kuk2E which is uniformly bounded in n: By the Banach-Saks theorem, the Cesàro mean sequence ¹wn º of a suitable subsequence of ¹vn º is E-Cauchy. Since 0  wn  u; ¹wn º is an E  -approximating sequence of u, yielding u 2 Fe . Returning to the setting in the beginning of §6.2, let us consider the set D D ¹x 2 X n N W Px .A1 > 0/ > 0º;

.6:2:14/

N being the properly exceptional set for the additive functional A. Obviously e; D Y

.D c / D 0:

.6:2:15/

Lemma 6.2.6. (i) There is a properly exceptional set N1 containing N such that both D n N1 and D c n N1 are M-invariant. (ii) The Dirichlet space .F C1Dc m ; E C1Dc m / on L2 .XI m/ is regular and transient.

322

6 Transformations of forms and processes

Proof. (i) In view of Lemma 5.1.11, D D ¹x 2 X W Px . < 1/ > 0º

.6:2:16/

q.e.

e. Therefore (i) is a consequence of Lemma 4.6.4. where  is the hitting time of Y (ii) By virtue of Theorem 6.1.1, q t f .x/ D Ex .e A t 

Rt 0

1D c .Xs /ds

f .X t //

.6:2:17/

is the transition function associated with the Dirichlet form .E C1Dc m ; F C1Dc m / on L2 .XI m/. By (i), we have for x 2 X n N 1 , q t f .x/ D 1D .x/Ex .e A t f  1D .X t // C 1D c .x/e t Ex .f  1D c .X t //: In particular, q t 1.x/ < 1 for some t > 0 if x 2 D n N1 and q t 1.x/  e t if x 2 D c n N1 . Therefore this Dirichlet space is transient by Lemma 1.6.5. Its regularity was shown by Theorem 6.1.2. We have already seen in Lemma 4.6.6 and Theorem 4.6.5 that the space H e Y defined by (6.2.3) constitutes a linear subspace of Fe and it is E-orthogonal to the linear subspace Fe;X ne Y defined by (6.2.1). C1

c m

D 2 . Lemma 6.2.7. (i) H e Y \ L .XI /  Fe 2 C1D c m .u; v/; 8v 2 F C1D c m . (ii) If u 2 H e e Y \L .XI /, then E .u; v/ D E

2 Proof. Fix any u 2 H e Y \ L .XI /. Then, u D H e Y w; 9w 2 Fe , and we see from (6.2.16) that u D 1D  u q.e. This implies that 1

u 2 Fe D

c m

\L2 .XI /;

E.u; v/ D E 1Dc m .u; v/;

1

8v 2 Fe D

c m

: .6:2:18/

Indeed, we get from Theorem 1.6.1 and Lemma 6.2.6 (i), E.w; w/ D E.1D  w; 1D  w/ C E.1D c  w; 1D c  w/;

w 2F:

.6:2:19/

Choose un 2 F such that ¹un º is E-Cauchy and un ! u .2 Fe / m-a.e. Then 1D  un 2 F 1Dc m ; ¹1D  un º is E 1Dc m -Cauchy, 1D  un ! 1D  u D u m-a.e. 1 c m 1 c m and further E.u; v/ D E 1Dc m .u; v/; 8v 2 Fe D : The Hence u 2 Fe D assertions then follow from (6.2.18) and Lemma 6.2.5. L FL / on Proof of Theorem 6:2:1 in the general case. To prove the closedness of .E; 2 L sequence 'n 2 FL and choose wn 2 Fe such that L .Y I /, take any E-Cauchy wn jY D 'n . By setting un D H e Y wn , we see from Lemma 6.2.7, k'n  'm kEL D kun  um kE D kun  um kE C1Dc m : 1

6.2

Traces of Dirichlet forms and time changes by additive functionals C1

323

c m

Therefore ¹un º is a Cauchy sequence in .Fe D ; E C1Dc m / which is a Hilbert space by virtue of Lemma 6.2.6 (ii). un is then E C1Dc m -convergent to C1 c m an element u 2 Fe D . C1D c m For any v 2 Fe vanishing q.e. on Y , we have from Lemma 6.2.7 that E C1Dc m .u; v/ D lim E C1Dc m .un ; v/ n!1

D lim E .un ; v/ D lim E.un ; v/ n!1

n!1

which vanishes because v is an element of Fe;X ne Y and Theorem 4.6.5 applies. 2 Applying the same theorem to H e Y u 2 He Y \ L .XI /, E C1Dc m .H e Y u; v/ D E .H e Y u; v/ D E.H e Y u; v/ D 0: C1

c m

D vanishes q.e. on Y , we then obtain that Since v D u  H e Y u 2 Fe c m C1 D .u  H e E Y u; u  H e Y u/ D 0, proving that u D H e Y u 2 He Y . If we L set ' D ujY , then ' 2 F and

k'n  'kEL D kun  ukE D kun  ukE C1Dc m ! 0; 1

which means that FL is complete with respect to EL 1 . We next show that, for any ' 2 L2 .Y I /, RLp ' 2 FL ;

ELp .RLp '; / D .'; / ;

2 FL :

.6:2:20/

As in the last paragraph of the proof of Lemma 6.2.3, we can use Lemma 6.1.3 ˛ 0 '.x/ increases to Up;A '.x/ as to get, for non-negative ' 2 L2 .Y I /, that Up;A ˛ 0 p ˛ # 0 and that ¹Up;A 'º  F are E -Cauchy as ˛ # 0. Hence Up;A ' 2 Fe . ˛ By noting that the uniform bound (6.1.13) of L2 ./-norm of Up;A ' and by letting ˛ # 0 in the equation (6.1.12), we get 0 E p .Up;A '; v/ D .'; v/ ;

8v 2 F ;



.6:2:21/

which readily extends to v 2 Fe . 0 L ' D He The strong Markov property readily yields that Up;A Y .Rp '/ and RLp ' 2 FL . If w 2 Fe , wjY D 2 L2 .Y I /, then v D H e Y w 2 Fe by Theorem 4.6.5, and we are led from (6.2.21) to 0 ELp .RLp '; / D E p .Up;A '; v/ D .'; v/ D .'; / ;

arriving at (6.2.20).

324

6 Transformations of forms and processes

L FL / is a Dirichlet space on L2 .Y I / associated with We have proved that .E; L FL / is L The regularity and the core property for .E; the time changed process M. clear from Theorem 6.1.2 and Lemma 6.2.5. Finally we shall prove that Lemma 6.2.4 holds true in general case. Indeed, L .0/ and L and Cap C1Dc m in place of Cap its proof in transient case works for Cap .0/ C1D c m

Cap.0/ , respectively. Let A be an open subset of X with Cap.0/

.A/ < 1 and C1

c m

be its 0-order equilibrium potential. is an element of the space F.0/ D which coincides with Fe \ L2 ./ \ L2 .1D c  m/ by virtue of Lemma 6.2.5. We ˇ .0/ let ' D e ˇY : Since ' D 1 -a.e. on A \ Y and ' 2 FL ;

e .0/

e .0/

L Cap.A \ Y /  EL 1 .'; '/ D E.HYQ '; HYQ '/ C .e .0/ ; e .0/ / C1D c m

 E.e .0/ ; e .0/ / C .e .0/ ; e .0/ / C1Dc m D Cap.0/

.A/:

L Next, suppose Cap.A \ Y / < 1 for an open set A  X: For the associated equilibrium potential 2 FL ; there exists v 2 Fe such that w DW HYQ v D -a.e. on Y . As in the proof for the transient case, w D 1 q.e. on A \ YQ on C1 c m because w D 0 on D c : It account of Exercise 4.6.2. Furthermore w 2 Fe D then follows from Theorem 2.1.5 that C1D c m

Cap.0/

L .A \ YQ /  E C1Dc m .w; w/ D EL 1 . ; / D Cap.A \ Y /:

We can now combine the two inequalities obtained above with Lemma 6.1.2 and Lemma 2.1.8 to get Lemma 6.2.4 in the present general case. We can then repeat exactly the same argument as in transient case to prove (iv) of Theorem 6.2.1 in general case. L FL / on L2 .Y I / defined by (6.2.4) is called the time The Dirichlet space .E; changed Dirichlet space with respect to . e D X q.e. and H D Fe by (6.2.3) Suppose  has full quasi support, then Y e Y L L and the time changed Dirichlet form .E; F / in (6.2.4) is reduced to FL D Fe \ L2 .XI /;

L E.u; u/ D E.u; u/

u 2 FL :

.6:2:22/

Since Lemma 6.2.4 holds in general, we can further conclude that the quasi notions L FL / coincide. Consequently, Theorem 2.1.7 leads us to for .E; F / and .E; FLe D Fe ;

L E.u; u/ D E.u; u/

u 2 FLe :

.6:2:23/

Here we regard the space FLe as well as Fe to consist of quasi continuous functions. Thus .Fe ; E/ is the common extended Dirichlet space for all the time changed

6.2

Traces of Dirichlet forms and time changes by additive functionals

325

Dirichlet spaces with respect to positive Radon measures charging no set of zero capacity and possessing full quasi support. If  has full quasi support, Lemma 6.2.4 holding in the general (not necessarily transient) case implies that any  2 S is again a smooth measure relative to L FL /. We prove in the next lemma that the the time changed Dirichlet space .E; L is given by changing the time of the original PCAF of M associated PCAF of M associated with . Lemma 6.2.8. Suppose that  2 S has full quasi support. Let B t be a PCAF of M L with Revuz measure  is given with Revuz measure  2 S. Then the PCAF of M L by B t D B t . Proof. Clearly BL t is a PCAF of YL . Hence it suffices to prove, for any bounded L h and f 2 B C \ L1 ./ such -integrable -excessive function (relative to M) ˛ that UB f is bounded for some ˛ > 0, the relation Z 1  .pC /t e f .XL t /d BL t D h; hf i: lim p EL h .6:2:24/ p!1

0

In fact, any -excessive function h and f 2 B C can be written as increasing limits of functions satisfying the above conditions, and consequently Theorem 5.1.3 is applicable. Put Z 1  .pC /A .pC /A t f .x/ D Ex e f .X t /dB t : RB 0

.pC /A

.pC /A

coincides with R0 defined by (6.1.4). Similarly to If B t D t , then RB (4.4.9), we have the generalized resolvent equation: .pC /A

RB

.pC /A

0 f D UB˛ f  .p C /UpC;A UB˛ f C ˛R0

UB˛ f:

Since UB˛ f ! 0 boundedly as ˛ ! 1, this equation and the relation (6.1.16) lead us to Z 1  .pC /A EL h e .pC /t f .XL t /d BL t D hh; R f i B

0

.pC /A

D lim hh; ˛R0 ˛!1

D

0 UB˛ f i D lim ˛hR˛ UpC;A h; f i ˛!1

0 hUpC;A h; f i :

Now (6.2.24) follows from 0 lim phUpC;A h; f i D Ef  .h.X0 // D h; hf i:

p!1

326

6 Transformations of forms and processes

Applying this lemma, we can extend the representation theorem of the space MV formulated in Theorem 5.6.3 to the general case that the energy measure ij D hxi ;xj i is not necessarily absolutely continuous with respect to m. Theorem 6.2.2. Let M be an m-symmetric diffusion process on a domain D  Rd . Suppose that the associated Dirichlet space .E; F / has C01 .D/ as core and E is expressed as E.u; v/ D

d Z X i;j D1 D

uxi .x/vxj .x/dij .x/;

u; v 2 C01 .D/:

Then MV D

d °X

fi  M

.i /

W

iD1

e

d X

fi  M

iD1

.i /



D

d Z X

d Z X i;j D1 D

fi .x/fj .x/dij .x/ < 1

fi .x/fj .x/dij .x/:

±

(6.2.25)

i;j D1 D

P Proof. Put  D m C di;j D1 jij j and let A t be a PCAF of M with Revuz measure . Clearly A t  t and  has full quasi support. Making a time change of M L D .XL t ; Px / on D. If relative to A t , we get a -symmetric diffusion process M M 2 M, then the time changed functional ML t D M t is a square integrable MAF L In fact, since  t  t for any t  0, the optional sampling theorem relative to M. and the submartingale inequality imply that Ex .ML t / D 0 and Ex ..ML t /2 / < 1 for q.e. x 2 D. By virtue of Lemma 6.2.9, the Revuz measures of hM i t relative to L are the same, and consequently ML is of finite energy M and of hML i t relative to M V Furthermore, since the Revuz measure 2ij L for any M 2 M. integral relative to M .i / .i / of the CAF hML .i / ; ML .j / i t of the time changed MAF ML t D M t is absolutely V there L for any M 2 M, continuous relative to , Theorem 5.5.3 is applicable to M; exist fi ; i D 1; : : : ; d such that d Z X i;j D1

d Z X dij fi .x/fj .x/ fi .x/fj .x/dij .x/ < 1; .x/d.x/ D d D D i;j D1

ML D

d X iD1

The assertion of the theorem is clear from this.

fi  ML .i / :

6.2

Traces of Dirichlet forms and time changes by additive functionals

327

The first assertion of the next theorem has been proved in Lemma 6.2.2. L is transient. Theorem 6.2.3. (i) If M is transient, then M L is recurrent. (ii) If M is recurrent, then M Proof. Suppose that M is recurrent. Since RLp '.x/ given by (6.2.6) is a EL1 L FL /, it suffices to prove that quasi continuous modification of the resolvent of .E; 1 L R0 ' D 0 or 1 -a.e. for any ' 2 LC .Y I / by (1.6.2). Let ' 2 L1C .Y I / \ Bb . For any f 2 L1C .XI m/ \ Bb , it then follows from (6.1.16) that pA

0 0 .f; U0;A '/ D lim .f; Up;A '/ D lim .R0 f; '/ D .R0 f; '/ : p!0

p!0

By the recurrence property (1.6.2) of M; m.B/ D 0 for the set B D ¹x W 0 < R0 f .x/ < 1º: Since B is finely open, B is exceptional by Lemma 4.1.4. Hence .B/ D 0 and the right end of the above identity equals 0 or 1: Therefore the set 0 '.x/ < 1º is m-negligible and .C / D 0 by the same reason C D ¹x W 0 < U0;A L follows because RL 0 ' D U 0 ' on Y e: as above. The recurrence of M 0;A

Example 6.2.1 (Time change and killing of Brownian motion). Let M be the Brownian motion on Rd . Its Dirichlet form on L2 .Rd / equals . 12 D; H 1 .Rd //, whose extended Dirichlet space He1 .Rd / is described in relation to the space G .Rd / of (1.2.14) in Example 1.5.3, Example 1.6.2 and Exercise 1.6.2. Consider positive Radon measures m and k on Rd charging no set of zero capacity such that m is of full quasi support. Then e 1e .Rd / \ L2 .Rd I m/ \ L2 .Rd I k/; F DH E.u; v/ D

1 D.u; v/ C .u; v/k ; 2

.6:2:26/ u; v 2 F ;

is the Dirichlet space on L2 .Rd I m/ associated with the Markov process on Rd obtained from the Brownian motion M by time change and killing with respect to PCAF’s with Revuz measures m and k respectively on account of (6.2.22) and Theorem 6.1.1. The stated conditions on m are met when d m D f  dx for some Borel function f strictly positive a.e. on RRd (Corollary 4.6.1). In this case, the corresponding t PCAF is given by A t D 0 f .Xs /ds. When d D 1, we can take any positive Radon measures m and k with suppŒm D R1 in view of Example 5.1.1. Then (6.2.26) reduces to Example 1.2.2.

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6 Transformations of forms and processes

Example 6.2.2 (A trace of the Sobolev space and symmetric Cauchy process on the boundary). Consider the half space D D ¹x D .x 0 ; xd / W x 0 2 Rd 1 ; xd > 0º  Rd , d  2, and the reflecting Brownian motion M on D (see Example 4.4.2). Its Dirichlet form E on L2 .D/ is regular and identical with . 12 D; H 1 .D//, whose extended Dirichlet space He1 .D/ is described in relation to the space G .D/ of (1.2.14) in Example 1.5.3 and Example 1.6.2. e 1e .D/ the collection of functions in He1 .D/ which are E1 If we denote by H quasi continuous on D, then we see exactly in the same manner as in Exame 1 .D/ of BLD functions of potential type on D can ple 2.3.2 that the space H 0;e be described as e 1e .D/ W u D 0 q.e. on @Dº: e 10;e .D/ D ¹u 2 H H

.6:2:27/

Let H be the space of harmonic functions (in the ordinary sense) on D with finite Dirichlet integrals and put HV D H \ He1 .D/: When d  3, we get from (1.5.26), (6.2.27) and Theorem 4.3.2 that e 1e .D/ W u D H@D uº: HV D ¹u 2 H

.6:2:28/

When d D 2, we also get from (1.6.11), (6.2.27) and Exercise 4.6.4 that e 1e .D/ W u D H@D uº: H D ¹u 2 H

.6:2:29/

Let us consider the .d  1/-dimensional Lebesgue measure  on the boundary @D; .dx/ D dx 0 ı0 .dxd /. Then  charges no set of E1 -capacity zero according to Exercise 2.2.1. The topological support of  in Rd equals @D. @D is also a quasi support of  because the hitting distribution H@D .x;  / is absolutely continuous with respect to  (with the density function being the Poisson kernel) for every x 2 D and Exercise 4.6.1 applies. On account of (1.5.26) and (1.6.11), the trace (6.2.4) of . 12 D; H 1 .D// on @D relative to  now takes the expression FL D H \ L2 .@DI /; 1 L E.'; / D D .H@D '; H@D / ; 2

';

2 FL

.6:2:30/

where denotes the restriction of a function on D to @D. We shall derive an L FL / by making use of Theorem 6.2.1. explicit description of .E; Recall that the reflecting Brownian motion M is obtained from the d -dimen.d / bt D sional Brownian motion ..X t0 ; X t /; Px / by means of the reflection X

6.2

Traces of Dirichlet forms and time changes by additive functionals

329

.d /

.X t0 ; jX t j/. Denote by L.t / the local time at 0 of the one dimensional Brow.d / nian motion X t (see Example 5.1.1). Let A be the PCAF of the process M with Revuz measure . Then .6:2:31/ A t .!/ D 4L.t; !/: In fact, by making use of the equality

Z

E.x 0 ;xd / .2L .t // D

t 0

1 2 e .xd / =2s ds; p 2 s

we can see that the formula (5.1.13) holds for 4L.t /; , and the transition function of M. L D ..X 0 ; 0/; Px /x2@D of M with respect to the The time changed process M t L is a spatially AF A t is a symmetric Cauchy process on @D. More specifically, M homogeneous Hunt process on @D (see Example 4.1.1) such that   jj 0 E0 .exp.i  X t // D exp  t ;  2 Rd 1 : .6:2:32/ 2 In fact, the inverse function  t of 4L.t/ p obeys the one-sided stable law of exponent 1=2, i.e. E0 .exp.˛ t // D exp. ˛=2t/.1 Consequently, Z 1 0 0 E0 .e i Xs /P0 . t 2 ds/ E0 .exp.i  X t // D 0

Z D

1

e jj

2 s=2

0

P0 . t 2 ds/ D e jjt =2 :

L FL / is the Dirichlet space of M L on L2 .@D/ D L2 .@DI / by TheoSince .E; rem 6.2.1, we get from (6.2.32) and (1.4.26) and the identity  Z  d2 1 d .1  cos .; // d D jj d 2 jj 2 2 Rd 1 that ² ³ Z 2 1 2 L .' . C /  '.// F D ' 2 L .@D/ W d d < 1 ; Rd 1 Rd 1 jjd  Z  d2 1 L .' . C /  './/2 d d d: E.'; '/ D d jj 4 2 Rd 1 Rd 1 .6:2:33/ An analogous relation has been derived in (1.2.19) of Example 1.2.3, when D is the unit disk on R2 . 1 Cf.

N. Ikeda and S. Watanabe [2; Chap. III.4.3].

330

6 Transformations of forms and processes

Exercise 6.2.2. Let .X t ; Px / be the 1-dimensional Brownian motion and  a positive Radon measure on R1 with suppŒ D Z .D ¹: : : ; 2; 1; 0; 1; 2; : : :º/: DeL FL / the Dirichlet form corresponding to the time changed process of note by .E; X t by the PCAF associated with . Show that ´ ˇ FL D He1 .R1 /ˇZ \ L2 .ZI /; P L E.u; v/ D 14 i;j 2Z; jij jD1 .u.i/  u.j //.v.i/  v.j //; u; v 2 FL : Exercise 6.2.3. Let .X t ; Px / be the 1-dimensional Brownian motion and  a posL FL / itive Radon measure on R1 with suppŒ D RC .D Œ0; 1//. Denote by .E; the Dirichlet form corresponding to the time changed process of X t by the PCAF associated with . Let r.x; y/ D jxj C jyj  jx  yj be the function given in Exercise 4.8.1 and let r.x; L y/ be its restriction to RC  RC . L L v/ D 12 DC .u; v/; where (i) Show that F R D He1 .RC / \ L2 .RC I / and E.u; 1 0 0 DC .u; v/ D 0 u .x/v .x/dx: (ii) For any R 1bounded measurable function on Œ0; 1/ with compact support such .y/.dy/ D 0, show that the function that 0 Z 1 L R .x/ D r.x; L y/ .y/.dy/ 0

belongs to He1 .RC / and satisfies 12 DC .RL ; '/ D . ; '/ for any ' 2 FL : Let M D .X t ; Px / be the 3-dimensional Brownian motion. For a positive L1 function V , define its scattering length by Z Rt 1 .V / D lim .1  Ex .e  0 V .Xs /ds //dx: t !1 t R3 Then for a compact set K  R3 with regular boundary, .˛1K / converges to the Newtonian capacity of K as ˛ ! 1.2 Using the time change argument, we can extend this fact to a general symmetric Hunt process and smooth measures. Let X; m; M D .X t ; Px ; / and  be as in §6.1. We denote by A D A the PCAF of M with Revuz measure : Define Z Ex .e A /.dx/: .6:2:34/ ./ D X

If M is conservative, we can then rewrite ./ as Z 1 .1  Ex .e A t //m.dx/: ./ D lim t !1 t X 2 M.

Kac [1].

.6:2:35/

6.2

Traces of Dirichlet forms and time changes by additive functionals

331

Indeed, the subprocess MA D .X tA ; PxA / of M with respect to the MF e A t is an m-symmetric Hunt process with an associated regular Dirichlet form .E ; F / by Theorem 6.1.1. A t can be also regarded as a PCAF of MA with the property3 Z t  As Ex e dAs D ExA .A t /: 0

Further the Revuz measure of A with respect to MA is equal to  again.4 Hence by Theorem 5.1.3 (iii), Z t  Z t A t As A hm; 1  E .e /i D Em e dAs D Em .A t / D h; psA 1ids: 0

0

A1 / as t ! 1, we have the equation (6.2.35). Since p A t 1.x/ converges to Ex .e

Theorem 6.2.4. Assume that M is transient and .X/ < 1: It then holds that e/; .˛/ " Cap.0/ .Y

˛ " 1;

.6:2:36/

e is a quasi support of the measure  and Cap.0/ denotes the 0-order cawhere Y pacity. L FL / be L be the time changed process of M with respect to A and .E; Proof. Let M 2 2 eI / where Y is the topological support of its Dirichlet form on L .Y I / D L .Y L L e : First note that the life time  of M is A by Exercise 6.2.1. For x 2 Y Ex .e

˛A

/ D EL x .e

˛ L

/ D 1  ˛ EL x

Z

L

e 0

˛t

 dt

D 1  ˛ RL ˛ 1.x/;

e, 1 D 1 .x/. Hence the left-hand side of where 1 is the identity function on Y e Y (6.2.36) equals ˛.1; 1  ˛ RL ˛ 1/ . 2 e L The function 1 e Y is in L .Y I / by the finiteness of . Thus if 1 e Y 2 F , then L 1/ as ˛ " 1. We see ˛.1; 1  ˛ RL ˛ 1/ is non-decreasingly convergent to E.1; from Theorem 6.2.1 that L 1/ D E.H 1; H 1/; E.1; e e Y Y

He /I  e Y 1.x/ D Ex .1 e Y .X e Y < 1/; Y

e where  e Y D inf¹t > 0 W X t 2 Y º. Hence L 1/ D E.H 1; H 1/ .˛/ D ˛.1; 1  ˛ RL ˛ 1/ " E.1; e e Y Y 3 Cf. 4 P.

Blumenthal–Getoor [1; IV, (1.18)]. Fitzsimmons and R. Getoor [2].

332

6 Transformations of forms and processes

e as ˛ " 1. Since H e Y 1.x/ D Px . e Y < 1/; E.H e Y 1; H e Y 1/ equals Cap.0/ .Y / by Theorem 4.3.3. L L Assume next that 1 e Y … F . Then lim˛!1 ˛.1; 1˛ R˛ 1/ " 1 as ˛ " 1. DeL FL /: Since .FL /e \L2 .Y I / D FL , note by .FL /e the extended Dirichlet space of .E; L L 1e u  Y does not belong to the space .F /e so that the set DYL D ¹u 2 .F /e W e L e e 1 E-q.e. on Y º is empty. Therefore Cap.0/ .Y / D 1 in view of the 0-order version of Theorem 2.1.5. The proof of the theorem is complete.

6.3

Transformations by supermartingale multiplicative functionals

Let M D .; M; X t ; Px / be an m-symmetric Hunt process whose Dirichlet form .E; F / on L2 .XI m/ is regular as in the preceding section. In this section we take as the sample space  the space D.Œ0; 1/ ! X / of right continuous functions ! W Œ0; 1/ ! X possessing left limits. Here X is the one point compactification of X: X t .!/ then denotes the t -th coordinate of !. Let us denote the resolvent ¹R˛ º˛>0 of M by  Z 1 ˛t R˛ f .x/ D Ex e f .X t /dt ; f 2 Bb .X/; 0

and define the generator A by Au D ˛u  f;

u D R˛ f:

For f 2 Bb .X/ let f Mt

Z D R˛ f .X t /  R˛ f .X0 / 

0

t

.˛R˛ f  f /.Xs /ds:

f

f

Then M t is a MAF. If R˛ f .x/ D R˛ g.x/ q.e., then M t and M tg are equivalent by the uniqueness of the semimartingale R t decomposition of R˛ f .X t /  R˛ f .X0 / Rt because 0 .˛R˛ f  f /.Xs /ds and 0 .˛R˛ g  g/.Xs /ds are continuous, in parRt Rt ticular, predictable. As a result, 0 f .Xs /ds and 0 g.Xs /ds are also equivalent. Hence for  D R˛ f; f 2 Bb .X/; the MAF Z t Œ M t D .X t /  .X0 /  A.Xs /ds 0

is well defined.

333

6.3 Transformations by supermartingale multiplicative functionals

Let .x/ D R˛ f .x/; f 2 Bb .X/ with .x/ > 0 q.e. For   0 and t  0, define   Z t .X t / C  A ; ; ; e L t 1¹t < º : (6.3.1) exp  .Xs /ds ; L t D e Lt D .X0 / C  0 C ;

Note that since  is finely continuous, .Xs / is right continuous in s a.s. L t ; ; t  0 is a multiplicative functional (a MF in abbreviation) of M, that is, L t is ; ; F t -adapted, L0 D 1I L t is finite on t < , right continuous on Œ0; / and with left-hand limit on .0; /, L t D 0; t  , and ;

;

L t Cs D L t

 L; s ı t ;

t; s  0:

(6.3.2)

For n 2 N let Fn D ¹x 2 X W .x/  n1 º and Fn0 the fine interior of Fn . Since  is nearly Borel and finely continuous, Fn .Fn0 / is nearly Borel fine closed set (fine open set). Moreover, ¹ > n1 º  Fn0 and ¹Fn0 º " X as n ! 1. Hence the exit time n from Fn0 , n D inf¹t > 0 W X t … Fn0 º, satisfies n D X nFn0 ^  and limn!1 n D  a.s. by the quasi-left-continuity of M. Lemma 6.3.1. For each t  0 and n,   Z s Z t ^n 1 A ; e L t ^n  1 D exp  .Xu /du dMsŒ : .X0 / C  0 C 0

(6.3.3)

Proof. The right-hand side of (6.3.3) is equal to  Z s  Z t ^n 1 A exp  .Xu /du .d.Xs /  A.Xs /ds/: .X0 / C  0 0 C Since   Z d ..Xs / C / exp 

0

s

 Z D exp 

A .Xu /du C

0

s



 A .Xu /du .d.Xs /  A.Xs /ds/ a:s: C

we have (6.3.3). If  > 0, the equality holds for t instead of t ^ n . By (6.3.1), ; ; ; Ex .L t /  lim inf Ex .L t ^n /  lim inf Ex .e L t ^n / D 1; n!1

;

n!1

x 2 X:

(6.3.4)

Thus we see that ¹L t º t 0 is a supermartingale with respect to .Px ; ¹F t º/ for each x 2 X. In the remainder of this section, we simply write L for L;0 .

334

6 Transformations of forms and processes 





Let p t f .x/ D Ex .L t f .X t //; t  0; x 2 X; f 2 Bb .X/. Then ¹p t º t 0  is a transition function. The Markov process M on X generated by ¹p t º t 0 is said to be the transformed process by the supermartingale MF L : Denote by  M D .; X t ; ; Px / the transformed process. Note that the transformed process has the elements ; X t ;  in common with M. This is possible because  equals D.Œ0; 1/ ! X /. M is a right process possessing the following property:5 

Ex .F 1¹t < º / D Ex .L t F /;

8F 2 F t0 ; t  0; x 2 X:

(6.3.5)

Let us define the function space on X by DC D ¹ D R˛ f W  > 0 q:e:; ˛ > 0; f 2 L2 .XI m/ \ BbC º: If f 2 Cb .X/ \ L2 .XI m/ is strictly positive on X, then  D R˛ f belongs to  DC . For  D R˛ f 2 DC , let M D .; X t ; ; Px / be the transformed process  by ¹L t º t 0 . Lemma 6.3.2. Let  2 DC . Then M is  2 m-symmetric. Proof. For f; g 2 bBC .X/  Z t    .X t / A  exp  .Xs /ds f .X t /g.X0 / .p t f; g/ 2 m D E 2 m .X0 / 0   Z t    A D Em .X0 /g.X0 / exp  .Xs /ds .X t /f .X t / : 0  R t A Rt Since 0 A  .Xs ı r t /ds D 0  .Xs /ds, the right-hand side equals  Z t    A  Em .X t /g.X t / exp  .Xs /ds .X0 /f .X0 / D .f; p t g/ 2 m  0 by Lemma 5.7.1. We can now consider the Dirichlet form on L2 .XI  2 m/ generated by M . Let us denote it by .E  ; F  /. According to the Beurling–Deny formula, E can be represented for u; v 2 F as Z ZZ 1 c E.u; v/ D dhu;vi C .e u.x/  e u.y//.e v .x/  e v .y//J.dxdy/ 2 X X X nd (6.3.6) Z C e u.x/e v .x/k.dx/: X

5 Cf.

M. J. Sharpe [1; (62.19)].

335

6.3 Transformations by supermartingale multiplicative functionals

The next lemma shows how E changes into E  : In particular, the killing term of the latter disappear. Theorem 6.3.1. F is contained in F  and Z 1  E .u; u/ D  2 dchui 2 X ZZ C .e u.x/  e u.y//2 .x/.y/J.dx; dy/; X X nd

u2F: (6.3.7)

Proof. For n 2 N, let .E n ; F n / be the part of the Dirichlet form of .E; F / on the fine open set Fn0 : e D 0 q:e: on X n F 0 º; F n D ¹f 2 F W f n

E n .f; g/ D E.f; g/;

f; g 2 F n :

Let Mn be the part process of M on Fn0 and ¹p nt I t  0º the transition function of Mn : First Step. First we shall show that for u 2 F n \ L1 .XI m/, the limit 1  D lim .u  E .L t u.X t /I t < n /; u/ 2 m t #0 t

(6.3.8)

exists and equals the right-hand side of (6.3.7). Put 

.u  E .L t u.X t /I t < n /; u/ 2 m 

D .u  E .u.X t /I t < n /; u/ 2 m  .E ..L t  1/u.X t /I t < n /; u/ 2 m D .I/ t  .II/ t :

(6.3.9)

By Theorem 4.4.2 1 1 lim .I/ t D lim .u  p nt u; u 2 /m D E.u; u 2 /; t #0 t t #0 t and by (6.3.3), .II/ t equals  Z  Z t exp  Eu m u.X t / 0

Put

0

 Z .III/ t D Eu m u.X t /

0

t ^n

s

(6.3.10)

  A Π.Xu /du dMs I t < n : 

 Z exp 

0

s

  A Π.Xu /du dMs I t  n : 

336

6 Transformations of forms and processes

We then have    Z t ^n  Z s A 1 1 2 Π.III/ t  Ejuj m exp  2 .Xu /du d hM is (6.3.11) t2 t 0  0 1  Ejuj m .u.X t /2 I t  n /: t By noting that A= is bounded on Fn , the first factor of the right-hand side of (6.3.11) is not greater than 1 exp.ct/  Ejuj m .hM Πi t /  exp.ct /kuk1 hi .X/; t where Z hi .dx/ D chi .dx/ C 2

X

..x/  .y//2 J.dx; dy/ C .x/2 k.dx/:

The second factor of the right-hand side of (6.3.11) equals 1t .juj; p t u2  u2 /m  1 n 2 2 2 n 2 t .juj; p t u  u /m , which converges to E.juj; u /  E .juj; u / D 0 as t ! 0, because juj and u2 are elements of F n and we can appeal to Theorem 4.4.2. Hence we see that lim t !0 .1=t/(II) t is equal to  Z s    Z t ^n 1 A Œ exp  .Xu /du dMs lim Eu m u.X t / t !0 t 0  0 and also to  Z s    Z t ^n 1 A exp  Eu m .u.X t /  u.X0 // .Xu /du dMsŒ : t !0 t 0  0 (6.3.12) Œu Œu We use the decomposition u.X t /  u.X0 / D M t C N t : Since lim

 ³2 A Œ .Xu /du dMs 0  0  Z s  Z t ^n  A Œu 2 Œ  Eu m ..N t / /  Eu m exp 2 .Xu /du d hM is ; 0  0

 ² Z Œu Eu m N t

t ^n

 Z exp 

s

we get  Z s    Z t ^n 1 A Œu Œ .Xu /du dMs exp  lim Eu m N t D 0: t !0 t 0  0

6.3 Transformations by supermartingale multiplicative functionals

337

Therefore (6.3.12) equals  Z s  Z t ^n  1 A Œu Œ exp  .Xu /du d hM ; M is : lim Eu m t !0 t 0  0 Moreover, since  Rs ˇ ˇ exp  0 ˇ ˇ

 A  .Xu /du s

ˇ  1ˇ ˇ  M < 1; ˇ

s  t ^ n ;

(6.3.12) coincides with 1 lim Eu m t !0 t

Z 0

t ^n

 d hM

Œu

;M

Œ

is

Z D

X

udhu;i :

Here, hu;i is given by dhu;i .x/ D dchu;i .x/ Z C 2 .u.x/  u.y//..x/  .y//J.dx; dy/ C u.x/.x/k.dx/: X

Combining the result obtained above with (6.3.9), (6.3.10) , we see that the limit of (6.3.8) exists and Z 2 udhu;i : D E.u; u /  X

can be written as a sum of the diffusion part Z Z Z 1 1 c c dhu;u 2 i  udhu;i D  2 dchu;ui 2 X 2 X X and the jump part ZZ ¹.u.x/  u.y//.u.x/ 2 .x/  u.y/ 2 .y// X X nd

ZZ D

 2u.x/.x/.u.x/  u.y//..x/  .y//ºJ.dx; dy/

X X nd

.u.x/  u.y//2 .x/.y/J.dx; dy/:

(6.3.13) (6.3.14) (6.3.15)

Indeed, subtracting .u.x/  u.y//2 .x/.y/ from the inside of the brace ¹  º of the left-hand side, we have .u.x/  u.y//..x/  .y//u.y/.y/  .u.x/  u.y//..x/  .y//u.x/.x/

338

6 Transformations of forms and processes

and its integral with respect to the symmetric measure J equals zero. Finally, noting the killing part disappear, equals the right-hand side of (6.3.7) for u 2 F n \ L1 .XI m/. Second Step. We shall show that, for u 2 F n \ L1 .XI m/, u 2 F  and D E  .u; u/. We see from (6.3.5) that for any  2 P .X /, P -null set of  is    also P -null. Hence F t  F t ; F  F  , where F t ; F  are the minimal  (augmented) admissible filtration of M . (6.3.5) extends to F 2 F t : In addition,  Px D Px on F0 . Therefore a nearly Borel set B of M is also nearly Borel set of M , and x 2 X is regular point of B with respect to M if and only if it is so with respect to M . In particular, Fn and Fn0 are a finely closed set and its fine interior ;0 with respect to M . Let M;0 be the part process M on Fn0 and ¹p t I t  0º its transition semigroup. From (6.3.5) for F D f .X t /1¹t <n º .2 F t /, we have ;0



p t f .x/ D Ex .L t f .X t /I t < n /;

x 2 Fn0 ; f 2 B.X/:

By virtue of Theorem 4.4.5 (i), M;0 is  2 m-symmetric and its Dirichlet form .E ;0 ; F ;0 / on L2 .Fn0 I  2 m/ satisfies F ;0  F  ;

E ;0 .f; f / D E  .f; f /;

f 2 F ;0 :

Hence by (6.3.8) 1 ;0 D lim .u  p t u; u/ 2 m t #0 t and the first step says that uS2 F ;0  F  ; D E ;0 .u; u/ D E  .u; u/. Finally, by noting that . n F n / \ L1 .XI m/ is dense in F in view of Theorem 4.4.5 (ii) and that E  .u; u/  kk21 E.u; u/;

u 2 F n \ L1 .XI m/;

we finish the proof of the theorem. Recall from Theorem 1.6.3 that the Dirichlet form .E  ; F  / on L2 .XI  2 m/ is recurrent if and only if 9hn 2 F  ;

lim hn D 1;

n!1

lim E  .hn ; hn / D 0:

n!1

(6.3.16)

Theorem 6.3.2. Let  2 DC . Then the Dirichlet form .E  ; F  / is recurrent.

6.3 Transformations by supermartingale multiplicative functionals

339

Proof. R  We shall derive (6.3.16). Let  D R˛ g be an element of DC and n .x/ D Ex . 0 n e ˛t g.X t /dt/. Then, for .x; y/ 2 Fn0  Fn0 ; ˇ ˇ ˇ n ˇ ˇ .x/ˇ  nn .x/; ˇ ˇ ˇ ˇ ˇ n ˇ ˇ .x/  n .y/ˇ  2njn .x/  n .y/j C n2 j.x/n .x/  .y/n .y/j; ˇ ˇ  and thus hn WD n = 2 Fbn . Fb /. By the derivation property of c (Lemma 3.2.5), dchn i D dchhn i D 2hn dchhn ;i C  2 dchhn i  h2n dchi ; and thus Z Z Z Z 2 c c 2 c  dhhn i D dhn i C hn dhi  2 hn dchhn ;i : (6.3.17) X

X

X

X

Since hn ! 1 and E.n  ; n  / ! 0 as n ! 1 and ˇZ ˇ Z 1=2 Z 1=2 Z ˇ ˇ c c 2 c c ˇ hn d dhi ˇˇ  hn dhi dhn i hn ;i  ˇ X X X X Z C jhn  1jdchi ; X

the right-hand side of (6.3.17) converges to chi .X/ C chi .X/  2chi .X/ D 0. On the other hand, if we put ZZ ;j .u.x/  u.y//2 .x/.y/J.dx; dy/; E .u; u/ D X X nd

then

ZZ E

;j

.hn ; hn / D

X X nd

..y/n .x/  .x/n .y//2 J.dx; dy/: .x/.y/

As .y/n .x/  .x/n .y/ D .x/.n .x/  n .y//  n .x/..x/  .y// D .y/.n .x/  n .y//  n .y/..x/  .y//; E ;j .hn ; hn / equals ZZ .n .x/  n .y//2 J.dx; dy/ X X nd

ZZ

C ZZ 

X X nd

X X nd

hn .x/hn .y/..x/  .y//2 J.dx; dy/ Œhn .x/ C hn .y/.n .x/  n .y//..x/  .y//J.dx; dy/:

340

6 Transformations of forms and processes

Since hn ! 1 and E.n  ; n  / ! 0 as n ! 1, E ;j .hn ; hn / converges to E ;j .; / C E ;j .; /  2E ;j .; / D 0. Theorem 6.3.3. Assume that the Dirichlet form E of M is irreducible. Then the Dirichlet form .E  ; F  / of the transformed process M on L2 .XI  2 m/ is irreducible recurrent and the following ergodic theorem is valid for M : (i) For any Borel measurable function f 2 L1 .XI  2 m/, the convergence 1 lim t !1 t

Z 0

Z

t

f .Xs /ds D cf ;



cf D

2

1 Z

 dm X

f  2 d m (6.3.18)

X



holds P 2 m -a.s. and in L1 .I P 2 m /. 

(ii) (6.3.18) holds Px -a.s. for q.e. x 2 X: (iii) Assume that the transition function of M satisfies the absolute continuity condition (4.2.9). Let f be a Borel measurable  2 m-integrable function such that for any x 2 X there exists its neighbourhood U.x/ with supy2U.x/ jf .y/j 

< 1. Then (6.3.18) holds Px -a.s. for every x 2 X: 

Proof. Denote by T t the L2 .XI  2  m/-semigroup for E  : If, for B 2 B.X/  and t > 0; 1B p t .x; B c / D 0; then it follows from (6.3.5) that 1B p t .x; B c / D 0   because L t > 0; t < ; Px -a.s. for every x 2 X: This means that any ¹T t ºinvariant set is ¹T t º-invariant and accordingly m-trivial by the assumption. Thus E  is irreducible. M is a  2 m-symmetric right process on X whose Dirichlet form .E  ; F  / is not only irreducible but also recurrent by Theorem 6.3.2. M is not necessarily a Borel right process but Theorem 4.7.3 can be applied to it according to the remark at the end of §4.7. In view of (6.3.5), we readily see that the absolute continuity condition (4.2.9) for the transition function of M is inherited to the same property for the transition function of M : In a similar way, we can see that a set N  X is exceptional for M if and only if so it is for M : So the statements (i), (ii) and (iii) follow from Theorem 4.7.3. Œ

We have considered the multiplicative functional L t for  belonging to the class R. But the class R is rather restrictive. In the rest of this section, we assume that .E; F / is a regular Dirichlet space with the strong local property and that the corresponding diffusion process M is conservative. We shall then extend the class R to the class FPloc . Floc / introduced in 5:5. We aim at proving that the  transformed process of M by L t never approaches in finite time to the nodal set of  2 FPloc :

6.3 Transformations by supermartingale multiplicative functionals

341

Lemma 6.3.3. It holds that for u 2 FPloc (i) hM Œu i t .r t / D hM Œu i t ; Pm -a.e., Œu

Œu

(ii) N t .r t / D N t ; Pm -a.e., where r t is the time reversal operator defined by (5.7.1). Proof. In view of (5.2.14) we get for u 2 F hM Œu i t D lim

n X

n!1

.e u.X k t /  e u.X .k1/ t //2 n

n

in L1 .Pm /:

kD1

Hence we obtain (i). We have already proved (ii) for u 2 F in (5.7.5). By the conservativeness of M these identities can be extended to u 2 FPloc . Take any  2 FPloc such that   0; m-a.e. and m.¹ > 0º/ > 0. Set X .n/ D ¹x 2 X W n1 < e .x/ < nº and define stopping times n and  by n D inf¹t > 0 W X t … X .n/ º; Œlog 

Let M t

Œlog 

Mt

Œlog 

and N t

n!1

be functionals on ¹t <  º defined by

Œlog  .n/ 

D Mt

 D lim n :

;

Œlog 

Nt

Œlog  .n/ 

D Nt

Œlog  .n/ 

;

t  n ; n  1;

(6.3.19)

Œlog  .n/ 

where  .n/ D .. n1 _ / ^ n/. Note that M t and N t are well-defined by Theorem 5.5.1 because log  .n/ 2 FPloc . The definition (6.3.16) makes sense on account of log  .n/ D log  on X .n/ and Lemma 5.5.1. We introduce a superŒ martingale multiplicative functional L t by Œ

Lt

  1 Œlog  D exp M t  hM Œlog  i t 1¹t < º 2

(6.3.20)

e the transformed process of M by L t , which lives on ¹x 2 and we denote by M e  will be denoted by E ex : e X W 0 < .x/ < 1º: The expectation with respect to M CC P Let us further consider the space Floc defined by 

Œ

CC FPloc D ¹ 2 FPloc W there exists a constant  > 0 such that 0 < 1     < 1º: Œlog   CC , log  2 FPloc and L t of (6.3.17) equals exp.M t  12 hM Œlog  i t / by For FPloc the conservativeness assumption on M:

342

6 Transformations of forms and processes

CC e  by LŒ is Lemma 6.3.4. If  2 FPloc , then the transformed diffusion process M 2  m-symmetric. Œ

Proof. Let e p t f .x/ D Ex .L t f .X t //. We get for any non-negative functions f and g, .e p t f; g/ 2 m     1 Œlog  2 Œlog  D Em  .X0 / exp M t  hM i t f .X t /g.X0 / 2    1 Œlog  2 Œlog  D Em  .X0 / exp log .X t /  log .X0 /  N t  hM it 2   f .X t /g.X0 /     1 Œlog  Œlog  D Em .X0 /.X t / exp N t  hM i t f .X t /g.X0 / : 2 By Lemma 5.7.1 and Lemma 6.3.3, the last expression equals     1 Œlog  Œlog  Em .X t /.X0 / exp N t  hM i t f .X0 /g.X t / 2 D .f; e p t g/ 2 m : Lemma 6.3.5. Let A t be a PCAF of M with Revuz measure . Then, Z CC e e  2 d; 8 2 FPloc : E  2 m .A t /  t X

Proof. In view of Theorem 2.2.4 (iii), there exists a sequence ¹Fn º of increasing compact sets satisfying (2.2.16) and (2.2.17) and 1Fn   2 S00 for each n. .n/ Consider the potential U1 .1Fn / of 1Fn  and put g` D `.U1 .1Fn /  `G`C1 .U1 .1Fn ///. Take f 2 F \ C0 .X/ such that 0  f  1 and f D 1 on Fn . Then, for any v 2 F , Z .n/ .n/ fe v 1Fn d; E1 .G1 .fg` /; v/ D .g` ; f v/ ! E1 .U1 .1Fn /; f v/ D `!1

X

R .n/ v 1Fn d D E1 .U1 .1Fn /; v/. Thus G1 .fg` / converges E1 which equals X e weakly to U1 .1Fn /, and so by the Banach–Saks theorem, we can construct a se.n/ .n/ quence h` 2 F such that G1 .f h` / converges E1 -strongly to U1 .1Fn /. Then in view of Lemma 5.1.2, Corollary 5.2.1 and Lemma 5.4.1 we see that

6.3 Transformations by supermartingale multiplicative functionals

343

Rt .n/ h` /.Xs /ds converges to 0 1Fn .Xs /dAs uniformly in t on each finite interval by taking a subsequence if necessary. Since Lemma 6.3.4 implies that the measure  2 m is excessive with respect e  , we obtain to M Z t  e e 1Fn .Xs /dAs E  2 m .A t / D lim E  2 m

Rt

0 .f

n!1

0

Z

t

e2m .f  lim lim inf E n!1 `!1 0 Z .n/  lim lim inf t  2 f h` d m: n!1 `!1

.n/ h` /.Xs /ds



X

.n/

Noting that  2 f belongs to F and that G1 h` converges E1 -weakly to U1 .1Fn / by Lemma 2.2.2, we can rewrite the last expression as .n/

lim lim inf t E1 .. 2 f /; G1 h` / D lim t E1 .. 2 f /; U1 .1Fn // n!1 Z Z 2 e e D lim t  1Fn d D t  2 d:

n!1 `!1

n!1

X

X

CC Let  2 FPloc . Let ¹Gn º 2 „ and ¹n º  F be sequences such that  D n m-a.e. on Gn as before. Then log  belongs to FPloc and Z Z Z Z 2 2 e e  dhlog i D lim  n dhlog n i D lim dhn i D dhi : n!1 G n

X

n!1 G n

X

The third equality follows from LeJan’s formula (Theorem 3.2.2). Hence, by applying Lemma 6.3.5 to A t D hM Œlog  i t , we get CC for  2 FPloc :

e  2 m .hM Œlog  i t /  thi .X/; E

(6.3.21)

CC e Lemma 6.3.6. For  2 FPloc with hi .X/ < 1, the transformed diffusion M is conservative in the sense that

e  2 m . < 1/ D 0: P

(6.3.22)



Further ¹L t º t 0 is a Px -martingale for m-a.e. x 2 X: Proof. Let us define a sequence of ¹F t º-stopping times Tn and T by Tn D inf¹t W hM Œlog  i t  nº; Then

ΠL t ^Tn

6 Cf.

T D lim Tn : n!1

Œ

is a martingale by Novikov’s theorem,6 and so Ex .Ln^Tn / D 1.

N. Ikeda and S. Watanabe [2].

344

6 Transformations of forms and processes

e x .S < / D Ex .L / holds for any ¹F t º-stopping time,7 Since the identity P S e x .n ^ Tn < / D 1: we obtain P e  2 m .T < t / D 0 for On the other hand, the inequality (6.3.18) tells us that P e e any t > 0. Therefore  D 1; P  2 m -a.e. and P x -a.e. for m-a.e. x 2 X: For  such x, the second assertion holds because Ex .L t / D 1 by (6.3.5). CC For  2 FPloc with hi .X/ < 1, put  Œlog  fŒlog M D Mt  hM Œlog  i t ; t

 Œlog  e Œlog N D Nt C hM Œlog  i t : t

Then in view of Lemma 6.3.3 and Lemma 6.3.6 we can show in the same way as in the proof of Theorem 5.7.1 that  e Œlog   N e Œlog  ; e Œlog .rT / D N N t T T t

and 1 fŒlog  1 fŒlog  fŒlog  .rT // log e .X t /  log e .X0 / D M C .M T t .rT /  M t T 2 2 e  2 m -a.e. 0  t  T; P (6.3.23) CC Lemma 6.3.7. It holds that for  2 FPloc

e2m P



e   .X t / e .X0 / 8  _ e sup  2 Thi .X/:  .X0 / e .X t / 0t T e

(6.3.24)

Proof. According to the relation (6.3.23) and the reversibility of the measure e2m, P e   .X t / e .X0 /  _ e sup .X0 / e .X t / 0t T e   e e e D P  2 m sup j log .X t /  log .X0 /j  

e2m P



e2m P



e2m D 2P 7 Cf.

0t T

sup 0t T



 fŒlog jM j t

  2



e2m CP

  Œlog  f j : sup jM t 2 0t T

M. J. Sharpe [1; (62.20), (23.20)].

 sup 0t T

 fŒlog jM .rT /j t

  2



6.3 Transformations by supermartingale multiplicative functionals

345

 fŒlog is a continuous loBy virtue of Lemma 6.3.6 and Girsanov’s theorem,8 M t cally square integrable martingale relative to PQx for m-a.e. x with sharp bracket being identical with hM Œlog  i t : Doob’s inequality implies that the last exprese  2 m .hM fŒlog  iT / which equals 42 E e  2 m .hM Œlog  iT /: sion is dominated by 42  E  Therefore the inequality (6.3.21) leads us to the lemma.

Theorem 6.3.4. Consider  2 FPloc such that   0; m-a.e., m.¹ > 0º/ > 0 e  D .; F ; X t ; P e x / by LŒ and hi .X/ < 1. Then the transformed process M t of (6.3.17) is  2  m-symmetric, conservative in the sense of (6.3.22) and it never approaches in finite time to the set N D ¹x 2 X W e .x/ D 0 or e .x/ D 1º in the following sense: e  2 m . < 1/ D 0: P (6.3.25) e .n/ D .; F ; X t ; P e .n/ / the transformed diffusion proProof. Let us denote by M Œlog  .n/ 

cess by L t , where  .n/ D .. n1 _ / ^ n/. Then, for any 0 < ` < n,     1 e 1 e .n/ e e  .X0 /  `; n < t D P .n/ 2  .X0 /  `; n < t P 2m  m ` ` e x on the -field Fn . e x is equal to P because P On the other hand, the right-hand side is dominated by .n/

e .n/

P

2  .n/ m

 sup 0st

 e.n/  .X t / e 

.n/

_

.X0 /

 .n/ e  .X0 / e 

.n/

.X t /

n  `

  

8 log n`

2 t hi .X/

because of Lemma 6.3.7 and Lemma 3.2.3. Hence, by letting n tend to infinity we get   1 e e  .X0 /  `;  < t D 0: P 2m ` e  2 m . < t / as ` ! 1, we attain (6.3.25), Since the left-hand side tends to P .n/

.n/

pt f D e p t f  2  m-a.e. where p t denotes the which implies that limn!1 e .n/ e e . Therefore the symmetry and conservativeness of M transition function of M follow from Lemma 6.3.4 and Lemma 6.3.6. Example 6.3.1. Let M D .; M; X t ; Px / be the Brownian motion on Rd . The associated Dirichlet form E on L2 .Rd / is given by . 12 D; H 1 .Rd //. If  is a nontrivial non-negative element in the space G .Rd / defined by (1.2.14), then  satisfies the conditions in Theorem 6.3.4. Hence we can conclude from Theorem 6.3.4 8 Cf.

N. Ikeda and S. Watanabe [2].

346

6 Transformations of forms and processes

e by that the transformed process M ΠLt

 D exp

r B 

 t

1  2

ˇ  Z tˇ ˇ r ˇ2 ˇ ˇ ˇ  ˇ .Xs /ds 1¹t < º ; 0

never hits the set ¹x 2 Rd W e .x/ D 0º and is conservative in the sense of (6.3.22). Here B t D X t  X0 ; t  0; and r   B is an extension of the stochastic integral in Example 5.6.1.

6.4

Donsker–Varadhan type large deviation principle

The Donsker–Varadhan large deviation theory for the occupation time distributions of Markov processes is considerably tractable in symmetric situations. M. Donsker and S. R. S. Varadhan introduced the so-called I-function as the rate function in their large deviation principle. While the evaluation of the I-function is generally hard, it becomes easier for symmetric Markov processes; the I-function has been identified with the Dirichlet form.9 Moreover, we can derive the Donsker– Varadhan type large deviation principle for a general, not necessarily conservative symmetric Markov processes by invoking an original idea in Donsker–Varadhan [1], where the one-dimensional Brownian motion was treated. Let X be a locally compact separable metric space and m a positive Radon measure on X with full support. As before, let M D .; X t ; Px ; / be an m-symmetric Hunt process on X whose Dirichlet form .E; F / on L2 .XI m/ is regular. But  is specifically taken to be the space of all right continuous functions from Œ0; 1 into the one point compactification X D X [ ¹º of X possessing the left limits such that !.t/ D  for any t  .!/ D inf¹s  0 W w.s/ D º and !.1/ D .  is called the lifetime which can be finite and X t is defined by X t .!/ D !.t / for and the re! 2 ; t  0: Let us denote by ¹p t º t 0 and ¹R˛ º˛>0 the semigroup R1 solvent of M, i.e., p t f .x/ D Ex .f .X t // and R˛ f .x/ D Ex . 0 e ˛t f .X t /dt /. Throughout this section, we make the following assumptions: I. (Irreducibility) If a Borel set A is p t -invariant, i.e., p t .1A f /.x/ D 1A p t f .x/ m-a.e. for any f 2 L2 .XI m/ \ Bb .X/ and t > 0, then A satisfies either m.A/ D 0 or m.X n A/ D 0. Here Bb .X/ is the space of bounded Borel functions on X. II. (Strong Feller Property) R1 .Bb .X//  Cb .X/, where Cb .X/ is the space of bounded continuous functions. 9 Donsker–Varadhan

[2].

6.4

Donsker–Varadhan type large deviation principle

347

III. (Tightness) For any  > 0, there exists a compact set K such that sup R1 1K c .x/  : x2X

Here 1K c is the indicator function of the complement of K. We notice that the irreducibility I follows from the assumption II whenever the state space X is connected in view of Exercise 4.6.3. It follows from the assumption II that the resolvent kernel R˛ .x;  /; ˛ > 0; is absolutely continuous with respect to m for each x 2 X and so is the transition function p t .x;  / on account of Theorem 4.2.4. We shall take a density function r˛ .x; y/; x; y 2 X; satisfying the properties of Lemma 4.2.4 R and define, for any Borel measure  on X, R˛ ; ˛  0; by R˛ .x/ D X r˛ .x; y/.dy/, R0 .x/ D sup˛>0 R˛ .x/; x 2 X: Let us state some remarks on the assumption III. Remark 6.4.1. (i) If R1 1 2 C1 .X/, then the assumption III holds. Here C1 .X/ is the set of continuous functions vanishing at infinity. Indeed, since Z 1 Z 1   t t R1 1K c .x/ D Ex e 1K c .X t /dt D Ex e 1K c .X t /dt 0

K c

D Ex .e  K c R1 1K c .X K c //; we have sup R1 1K c .x/ D sup R1 1K c .x/  sup R1 1.x/: x2X

x2K c

x2K c

In this case, M is non-conservative because, if p t 1 D 1 for all t > 0, then R1 1 D 1: (ii) If m.X/ < 1 and kR1 k1;1 < 1, then kR1 1K c k1  kR1 k1;1 m.K c / and the assumption III is fulfilled. Here k  k1;1 is the operator norm from L1 .XI m/ to L1 .XI m/. (iii) Let M D .X t ; Px ; / be a one-dimensional diffusion process on an open interval I D .r1 ; r2 / such that Px .X  D r1 or r2 ;  < 1/ D Px . < 1/, x 2 I; and Pa .b < 1/ > 0 for any a; b 2 I: We refer the readers to Chapter 5 of K. Itô [4] for the following properties of M: It is symmetric with respect to its canonical measure m and it satisfies I trivially and II as well. The boundary point ri of I is classified into four classes: regular boundary, exit boundary, entrance boundary and natural boundary.

348

6 Transformations of forms and processes

(a) If r2 is a regular or exit boundary, then limx!r2 R1 1.x/ D 0. (b) If r2 is an entrance boundary, then limr!r2 supx2.r1 ;r2 / R1 1.r;r2 / .x/ D 0. (c) If r2 is a natural boundary, then limx!r2 R1 1.r;r2 / .x/ D 1 and thus supx2.r1 ;r2 / R1 1.r;r2 / .x/ D 1. As a result, III is satisfied if and only if no natural boundaries are present. Exercise 6.4.1. Prove that, if the transition function ¹p t º of a Hunt process satisfies p t 1 D 1 and p t .C1 .X//  C1 .X/ for t > 0, then supx2X R1 1K c .x/ D 1 for any compact set K and thus III is not fulfilled. We set D C .A/ D ¹R˛ f W ˛ > 0; f 2 L2 .XI m/ \ CbC .X/ and f 6 0º: Here CbC .X/ denotes the set of non-negative bounded continuous functions. We note that any function in D C .A/ is strictly positive. Indeed, for a given  D R˛ f 2 D C .A/, the open set ¹ > 0º is of positive capacity because it contains the set ¹f > 0º of positive m-measure. Therefore, for some > 0, the open set O D ¹ > º is of positive capacity and hence non-exceptional. By Exercise 4.7.1, Px .O < 1/ > 0 for every x 2 X and we get .x/  Ex .e ˛ O .X O //  Ex .e ˛ O / > 0;

x 2 X:

Let L be the supermartingale multiplicative functional of M introduced in §6.3:   Z t A .X t /  Lt D (6.4.1) exp  .Xs /ds  1¹t < º ; t  0: .X0 / 0  Let P be the set of probability measures on X equipped with the weak topology and define the function IE on P by ´ p p p E. f ; f / if  D f  m; f 2 F IE ./ D 1 otherwise: We define the normalized occupation time distribution L t .!/ by Z 1 t 1A .Xs .!//ds; A 2 B.X/: L t .!/.A/ D t 0 For each ! 2  with .!/ > t , L t .!/ is a member of P : We then have

6.4

Donsker–Varadhan type large deviation principle

349

Theorem 6.4.1. (i) For each open set G of P lim inf t !1

1 log P .L t 2 G; t < /   inf IE ./; 2G t

 2 P:

(ii) For each closed set K of P lim sup t !1

1 log sup Px .L t 2 K; t < /   inf IE ./: 2K t x2X 

Proof. (i) Let  D R˛ f 2 D C .A/ and  2 m 2 G. Let M D .; X t ; Px / be the transformed process of M by the functional L defined by (6.4.1). Put ˇZ ˇ ² ³ Z ˇ A ˇ S.t; / D ! 2  W ˇˇ Ad mˇˇ <  ; .x/L t .!; dx/  X  X S 0 .t; / D S.t; / \ ¹! 2  W L t .!/ 2 Gº;

 > 0;

and ²

1 ‚1 D ! 2  W lim t !1 t

Z 0

t

A .Xs .!//ds D 

Z

³ Ad m ;

X

‚2 D ¹! 2  W L t .!/ converges to  2 mº: R R A 2 Observe that X j A  j d m D X jAjd m < 1;  is bounded on each compact subset of X and Cb .X/  L1 .XI  2 d m/: Therefore we can conclude by  Theorem 6.3.3 (iii) that for i D 1; 2, Px .‚i / D 1 for any x 2 X by taking the absolute continuity of the transition function of M into account. Hence, for any x 2 X and  > 0; Px .S 0 .t; // ! 1 as t ! 1: Since 

  Z t .X0 / A P .L t 2 G; t < / D exp .Xs /ds I L t 2 G .X t / 0   Z     .X0 / 0 Ad m   E I S .t; /  exp t .X t / X  Z   0 P .S .t; //  exp t Ad m   kk1 X E

350

6 Transformations of forms and processes 

and P .S 0 .t; // !

R

d as t ! 1, we obtain Z 1 Ad m  ; lim inf log P .L t 2 G; t < /  t !1 t X

and thus lim inf t !1

X

1 log P .L t 2 G; t < /   inf E.; /: t

2D C .A/  2 m2G

The right-hand side is equal to  inf E.; / D  inf IE . 2 m/ .D  inf IE .//;

2F C

2F C

 2 m2G

 2 m2G

2G

where F C is the space of non-negative functions in F . Indeed, for f 2 F C , ˛R˛ f is E1 -convergent to f as ˛ ! 1; and for a sequence ¹fn º1 nD1  C 2 2 L .XI m/\Cb .X/ which is L -convergent to f as n ! 1, R˛ fn ! R˛ f in E1 . Thus D C .A/ is E1 -dense in F C . (ii) Define Qx;t .C / D Px .L t 2 C I t < / for any set C 2 B.P /: For u 2 C D .A/ and  > 0; we get from (6.3.4)   Z t   Au u.x/ C  Ex exp  .Xs /ds I t <   ;  0 uC R Au and the left-hand side dominates exp.t sup 2C X uC .z/.dz//Qx;t .C /. Hence Z 1 Au lim sup log sup Qx;t .C /  inf sup d: (6.4.2) C u2D .A/ 2C X u C  t !1 t x2X >0

Let K be a compact set of P and set ` D sup

inf

Z

2K u2D C .A/ X >0

Au d: uC

Then, for any ı > 0 and  2 K, there exist u 2 D C .A/ and  > 0 such that R Au Au X u C d  ` C ı: Since the function u C belongs to Cb .X/, there exists R Au a neighbourhood N./ of  such that X u C d  ` C 2ı for any  2 N./: S Since 2K N./ is an open covering of K, there exist 1 ; : : : ; k in K such S that K  jkD1 N.j /. Put Nj D N.j /. We then have for 1  j  k Z Au j sup d  ` C 2ı; 2Nj X u j C  j

6.4

Donsker–Varadhan type large deviation principle

Z

and so max

1j k

inf

u2D C .A/

>0

Au d  ` C 2ı: uC

sup 2Nj

351

X

Therefore, by (6.4.2) lim sup t !1

1 1 log sup Qx;t .K/  max lim sup log sup Qx;t .Nj / t 1j k t !1 t x2X x2X Z Au  max inf sup d  ` C 2ı: C 1j k u2D .A/ 2Nj X u C  >0

(6.4.3) Z

Since 

inf

u2D C .A/

>0

X

Au d D IE ./ uC

(6.4.4)

by Theorem 6.4.2 below, the proof is completed for any compact set K. For 0 <  < 1=3, let K be a compact set such that supx2X R1 1Kc .x/  . Let V .x/ D 

AR1 1Kc .x/ 1Kc .x/  R1 1Kc .x/ D : R1 1Kc .x/ C  R1 1Kc .x/ C 

On account of (6.3.4) we have  Z  Z R1 1Kc C  exp t V .x/.dx/ dQx;t   2:  P X

(6.4.5)

In addition, noting that ´  1 2 > V .x/  0;

1 3 ;

and V .x/ > 1, we have Z Z Z V .x/.dx/ D V .x/.dx/ C Kc

X

K

x 2 Kc ; x 2 K ;

V .x/.dx/ 

and thus  Z   Z Z Z exp t V .x/.dx/ dQx;t  exp t P

P

X

e

t

Z P

Kc



1 .Kc /  1; 3 

V .x/.dx/  t dQx;t

 t c .K / dQx;t : exp 3

352

6 Transformations of forms and processes

Let Pı D ¹ 2 P W .Kc / > ıº. We then see from (6.4.5) that   tı ı Qx;t .P /  2 exp t  : 3 For any  > 3 let J D

1 [ nD1

Then Qx;t .J /  2e t  and so lim sup t !1

3

P n1 :

(6.4.6)

n2

e t 1  e t

1 log sup Qx;t .J /  1  : t x2X

Since Jc is compact by Lemma 6.4.1 below, we have for any closed subset K of P lim sup t !1

1 log sup Qx;t .K/ t x2X

   1 1 c D lim sup log sup Qx;t .K \ J / _ lim sup log sup Qx;t .K \ J / t !1 t t !1 t x2X x2X       inf c IE ./ _ .1  /   inf IE ./ _ .1  /: 

2K

2K\J

By letting  to 1, the proof is completed. Lemma 6.4.1. The complement Jc of the set J defined by (6.4.6) is compact in P . Proof. First note that the set .Pı /c is closed. Indeed, if n 2 .Pı /c weakly converges to , .Kc /  lim inf n .Kc /  ı n!1

because the set

Kc

is closed. Since ³ 1  3 c 1 ² \ \ 3 ; P n1  2 P W .K c 1 /  D Jc D n n2 n2 nD1

nD1

Jc

the set is closed. For any  > 0, we take n0 so large that 3=n0 < . Then sup .K c / 

2Jc

which implies that the set Jc is tight.

3 < ; n0

KDK

1 n2 0

;

6.4

353

Donsker–Varadhan type large deviation principle

Let us define the I -function on P by Z I./ D  inf u2D C .A/

>0

X

Au d; uC

 2 P:

(6.4.7)

Denote by BbC .X/ the set of non-negative bounded Borel functions on X . Let us define a function on P by 

 ˛R˛ u C  d: log uC X

Z I˛ ./ D 

inf

C u2B .X/ b

(6.4.8)

>0

Lemma 6.4.2. I˛ ./ 

I./ ; ˛

 2 P:

Proof. For u D R˛ f 2 D C .A/ and  > 0, set 

Z .˛/ D  Then, d.˛/ D d˛

Z X

log X

 ˛R˛ u C  d: uC

R˛ u  ˛R˛2 u d D ˛R˛ u C 

Z X

AR˛2 u d: ˛R˛ u C 

Since ˛R2 u  R˛ u ˛R˛2 u  R˛ u  2˛ 2 ; ˛R˛ u C  ˛ R˛ u C    Z Z Z AR˛2 u AR˛2 u AR˛2 u 1 d d  d D    2 2 2 ˛2 X ˛R˛ u C  X ˛ R˛ u C  X R˛ u C ˛ 2  Hence,

1 I./: ˛2 

 ˛R˛ u C  I./ log d   ; .1/  .˛/ D uC ˛ X Z

and thus

 ˛R˛ u C  I./ d  : log u C  ˛ X

Z 

inf

u2D C .A/

>0



354

6 Transformations of forms and processes

Since for any f 2 CbC .X/, kˇRˇ f k1  kf k1 and 0  ˇRˇ f .x/ ! f .x/ as ˇ ! 1,     Z Z ˛R˛ .ˇRˇ f / C  ˇ !1 ˛R˛ f C  log d ! log d: (6.4.9) ˇRˇ f C  f C X X Define the measure ˛ by ˛ .A/ D

Z ˛R˛ .x; A/d.x/; X

A 2 B.X/:

(6.4.10)

C 2 Given v 2 BbC .X/, take a sequence ¹gn º1 nD1  Cb .X/ \ L .XI m/ such that Z jv  gn jd.˛ C / ! 0 as n ! 1: X

We then have Z Z Z j˛R˛ v  ˛R˛ gn jd  ˛R˛ .jv  gn j/d D jv  gn jd˛ ! 0 X

X

X

as n ! 1, and thus     Z Z ˛R˛ gn C  ˛R˛ v C  n!1 log d ! log d: gn C  vC X X

(6.4.11)

Hence, combining (6.4.9) and (6.4.11)     Z Z ˛R˛ u C  ˛R˛ u C  log log d D inf d; inf uC uC u2D C .A/ X u2BbC X which implies the lemma. Lemma 6.4.3. If I./ < 1, then  is absolutely continuous with respect to m. Proof. For a > 0 and A 2 B.X/, set u.x/ D a1A .x/ C 1 2 BbC .X/, where 1A is the indicator function of the set A. Then for  > 0     Z Z ˛R˛ u C  a˛R˛ .x; A/ C ˛R˛ .x; X/ C  log d D log d uC a1A .x/ C 1 C  X X  I˛ ./: Let ˛ be the measure in (6.4.10) and c˛ D ˛ .X/. By Lemma 6.4.2 and Jensen’s inequality log .a˛ .A/ C c˛ C /  .A/ log.a C 1 C / C .Ac / log.1 C /  I./=˛;

6.4

Donsker–Varadhan type large deviation principle

355

and by letting  ! 0; log .a˛ .A/ C c˛ /  .A/ log.a C 1/  I./=˛: Since log x  x  1 for x > 0, a˛ .A/ C c˛  1  .A/ log.a C 1/  I./=˛; and thus ˛ .A/  .A/ 

I./=˛ C .A/.log.a C 1/  a/ C 1  c˛ : a

Noting that log.a C 1/  a < 0, we have for any A 2 B.X/ ˛ .A/  .A/ 

I./=˛ C .log.a C 1/  a/ C 1  c˛ ; a

and .A/  ˛ .A/ D 1  c˛ C .˛ .Ac /  .Ac // 

I./=˛ C .log.a C 1/  a/ C .1  c˛ /.a C 1/ : a

Thus we see that sup j.A/  ˛ .A/j  A2B.X /

a  log.a C 1/ C I./=˛ C .1  c˛ /.a C 1/ : a

Noting that c˛ ! 1 as ˛ ! 1, we have lim sup sup j.A/  ˛ .A/j  ˛!1 A2B.X /

a  log.a C 1/ : a

Since the right-hand side converges to 0 as a ! 0 and ˛ is absolutely continuous with respect to m; we get the lemma. Theorem 6.4.2. I./ D IE ./;  2 P : Proof. I./  IE ./: Suppose that I./ D ` < 1. Lemma 6.4.2 implies that  is absolutely continuous with respect to m. Let us denote by f its density and put p f n D f ^ n. From log.1  x/  x for 1 < x < 1; we have     Z Z ˛R˛ f n C  f n  ˛R˛ f n log log 1  fdm D fdm fnC fnC X X Z f n  ˛R˛ f n  f d m; fnC X so that

Z X

f n  ˛R˛ f n f d m  I˛ .f  m/: fnC

356

6 Transformations of forms and processes

Note that f n  ˛R˛ f n f D fnC

p f  ˛R˛ f n n  ˛R˛ f n f 1¹pf nº C f 1¹pf >nº : p n C  f C

Since the absolute value p of thepfirst (resp. second) term of the right-hand side is p dominated by . f C ˛R˛ f / f 2 L1 .XI m/ (resp. f 2 L1 .XI m/), we have p Z Z p f  ˛R˛ f f n  ˛R˛ f n fdm D f d m: lim p n!1 X fnC f C X Moreover, by letting  ! 0, Z p p p I.f  m/ f . f  ˛R˛ f /d m  I˛ .f  m/  ; ˛ X p p p which implies that f 2 D.E/ and E. f ; f /  I.f  m/. 

I./  IE ./: For  2 D C .A/ define the semi-group P t by   Z t    .X t / C  A  P t f .x/ D Ex exp  .Xs /ds f .X t / : .X0 / C  0 C 



Then P t is . C /2 m-symmetric and satisfies P t 1  1 in p view of Lemma 6.3.1 and the proof of Lemma 6.3.2. Given  D f m 2 P with f 2 D.E/, let p    Z t p A  .Xs /ds f .X t / : S t f .x/ D Ex exp  0 C Then we see by the same argument as in Theorem 6.1.1 that Z p p p p 1 p A  f d m: lim . f  S t f ; f /m D E. f ; f / C t !0 t X C Since

(6.4.12)

2  s f  2 2 f / d m D . C / P t dm C X X 2  s Z f 2   . C / P t dm C X 2 s Z Z f 2 dm D f d m;  . C / C X X p p the right-hand side of (6.4.12) is non-negative, and thus we have E. f ; f /  I.f  m/. Z

 .S t

p

Z

6.4

Donsker–Varadhan type large deviation principle

357

Exercise 6.4.2. Let E be a discrete countable set and .E; F 0 / be a regular Dirichlet form on L2 .EI m/ in Example 1.2.5. Show that for a probability measure  D ¹i ºi2E on E, q 2 X m0 1 X q 0 i qij  j0 qj i C 0i ki ; where 0i D i i ; i 2 E: I./ D 2 mi i;j 2E

i2E

Since the symmetric Markov process M is allowed to be non-conservative, the proof of the lower bound (i) in Theorem 6.4.1 becomes more difficult; for the proof, it is crucial that an irreducible symmetric Markov process can be always transformed into an ergodic Markov process by the supermartingale MF L . Donsker–Varadhan [1] utilized the Feller test to verify that an absorbing onedimensional Brownian motion can be transformed into an ergodic diffusion by a drift transform. The proof of the upper bound (ii) in Theorem 6.4.1 says that without the assumption III, the upper bound holds for any compact subset of P . Let ¹fn º 2 C0 .E/ be a non-negative increasing sequence such that fn .x/ converges to f 2 CbC .E/ for each point x. Then Z Z f  ˛R˛ f fn  ˛R˛ fn d D lim d; ˛ > 0: n!1 X R˛ fn C  X R˛ f C  Set

® ¯ O D R˛ f W ˛ > 0; f 2 C C .E/; f 6 0 ; DC .A/ b

O Then we have O D ˛R˛ f  f for  D R˛ f 2 DC .A/. and define A Corollary 6.4.1 (Extended variational formula for Dirichlet forms). For f 2 F Z O Au f 2 d m: (6.4.13) E.f; f / D sup u C  C X O u2D .A/ >0

This formula has been shown to hold for any symmetric right process in Shiozawa–Takeda [1]. Let 2 be the bottom of spectrum: ² ³ Z f 2d m D 1 : (6.4.14) 2 D inf E.f; f / W f 2 F ; X

O and  > 0, Using Corollary 6.4.1, we have for any u 2 D C .A/ 2  m- ess inf x2X

O Au .x/; uC

where m-ess inf denotes the m-essential infimum.

(6.4.15)

358

6 Transformations of forms and processes

Exercise 6.4.3. Using Corollary 6.4.1, show that 2 

1 supx2X Ex ./

:

Exercise 6.4.4. Let  be a -finite measure on X with kR˛ k1 < 1 for an ˛  0. Using Exercise 4.2.2 and (6.4.15), show that  is then smooth and satisfies a Poincaré type inequality Z e u2 d  kR˛ k1  E˛ .u; u/; u 2 F : X

Setting G D K D P in Theorem 6.4.1, we have Corollary 6.4.2. For  2 P 1 1 log P .t < / D lim log sup Px .t < / t !1 t t !1 t x2X ³ ² Z u2 d m D 1 : D  inf E.u; u/ W u 2 F ; lim

(6.4.16)

X

Let us denote by kp t kp;p the operator norm of p t from Lp .XI m/ to Lp .XI m/ and put 1 p D  lim log kp t kp;p ; 1  p  1: t !1 t Note that supx2X Px .t < / D kp t k1;1 and the right-hand side of (6.4.16) is equal to 2 by the spectral theorem. We then see from Corollary 6.4.2 that 1 D 2 :

(6.4.17)

Since kp t k2;2  kp t kp;p  kp t k1;1 , 1 < p < 1 by the symmetry of p t and Riesz–Thorin interpolation theorem10 , we can obtain Theorem 6.4.3. Under the assumptions I–III, p (1  p  1) is independent of p. Exercise 6.4.5. Prove that kp t k2;2  kp t k1;1 . The probabilistic interpretation of 1 is known. 10 Cf.

E. B. Davies [2].

6.4

Donsker–Varadhan type large deviation principle

359

Exercise 6.4.6. Prove that ° ± 1 D sup   0 W sup Ex .e  / < 1 : x2X

Moreover, we can show that supx2X Ex .e 1  / D 1. Indeed, let R f .x/ D R1 R1  Ex . 0 e t f .X t // with   0. Since 0 Ex .e t I t < /dt D Ex .e /1 , kR k1;1 < 1 if and only if supx2X Ex .e  / < 1. Assume that supx2X Ex .e 1  / < 1. Then for 0 <  < 1=kR1 k1;1 , kR1  k1;1 < 1 and 2 3 R1  D R1 C R C  2 R C    :11 1 1

Hence supx2X Ex .e .1 C/ / < 1, which is contradictory to Exercise 6.4.6. We now obtain Theorem 6.4.4.12 supx2X Ex .exp.// < 1 if and only if  < 1 : Therefore, we obtain Corollary 6.4.3. Assume I–III. Then sup Ex .exp.// < 1

if and only if

2 > 1:

x2X

Exercise 6.4.7. Let M D .X t ; Px ; / be a one-dimensional diffusion process on an interval .r1 ; r2 / as is specified in Remark 6.4.1. Show that if no natural boundaries are present, then sup

Ex .exp.// < 1 if and only if

 < 2 :

x2.r1 ;r2 /

Example 6.4.1. Let M D .X t ; Px / be the Brownian motion on Rd . Let V be a non-negative continuous function on Rd with limx!1 V .x/ DR 1. Denote by t MV D .X t ; ; PxV / the subprocess of M with respect to exp. 0 V .Xs /ds/. It 2 follows from Theorem 6.1.1 that the Dirichlet form of M on L .XI m/ is expressed by ´ R E V .u; v/ D 12 D.u; v/ C Rd uv  Vdx F V D ¹u 2 H 1 .Rd / W u 2 L2 .Rd I Vdx/º: 11 Cf. 12 Cf.

T. Kato [1; III, §6]. S. Sato [1].

360

6 Transformations of forms and processes

MV satisfies the assumptions I–III. Indeed, I follows from the irreducibility of the Rt Brownian motion and the positivity of exp. 0 V .Bs /ds/: The assumption II is satisfied.13 Denote by R1V the 1-order resolvent of MV . Then since Z R1V 1.x/ D Z D

1

   Z t e t Ex exp  V .Xs /ds dt

1

   Z t t e Eo exp  V .x C Xs /ds dt;

0

0

0

0

limx!1 R1V 1.x/ D 0 by the assumption on V and the bounded convergence theorem. Hence III follows from the remark (i) on it. Noting    Z t V .Xs /ds D PxV .t < /; Ex exp  0

we see from Corollary 6.4.2 that    Z t 1 V .Xs /ds D  inf¹E V .u; u/ W u 2 F V º: lim log Ex exp  t !1 t 0 (6.4.18) We can extend (6.4.18) to any continuous function V with limx!1 V .x/ D 1 by adding a positive constant. Let M D .X t ; Px / be the Brownian motion on Rd . A positive Radon measure  on Rd is said to be in the Kato class Kd if Z .dy/ D 0; d  3; lim sup ˛#0 x2Rd jxyj<˛ jx  yjd 2 Z lim sup .log jx  yj1 /.dy/ D 0; d D 2; ˛#0 x2Rd

jxyj<˛

Z sup x2Rd

jxyj1

.dy/ < 1;

d D 1:

We suppose that d  3. A measure  2 Kd is said to be Green tight if Z d.y/ D 0: lim sup R!1 x2Rd jyjR jx  yjd 2 We denote by Kd1 the set of all Green tight Kato measures. 13 Cf.

B. Simon [1; Theorem B.3.1].

6.4

Donsker–Varadhan type large deviation principle

361

Exercise 6.4.8. (i) Show that, if  2 Kd is finite, then  2 Kd1 . (ii) Show that, for ˇ > 2, the measure ˇ .dx/ D .jxjˇ ^ 1/dx belongs to Kd1 . A measure  2 Kd is smooth in the strict sense14 and regarded as the Revuz L Px / be the time changed L D .XL t ; ; measure of PCAF A in the strict sense. Let M L L process of M with respect to A and .E; F / be its Dirichlet form on L2 .Y I / e be the support of A defined by where Y is the topological support of : Let Y (5.1.21). We assume that e: Y DY (6.4.19) We prove that for  2 Kd1 the associated time changed process satisfies the assumptions I–III. We first show that for  2 Kd1 Z G.  ; y/d.y/ 2 C1 .Rd /: (6.4.20) Rd

Note that the Green function is written by G.x; y/ D v.x  y/ (v: the Newton kernel in (1.5.21)). Then G n .x; y/.WD G.x; y/ ^ n/ D G.x; y/; n D 1; 2; : : : ; on jx  yj  ˛n , where ˛n D .c=n/1=.d 2/ > 0. Since ˛n ! 0 as n ! 1, ˇZ ˇ Z ˇ ˇ n ˇ sup ˇ G.x; y/1¹jyj
x2Rd

Rd

Z  2 sup x2Rd

jxyj<˛n

G.x; y/1¹jyj
n ! 1:

Moreover, Rsince G n .x; y/ is continuous on Rd  Rd and limx!1 G n .x; y/ D 0 for any y, Rd G n .  ; y/1¹jyj
Since

ˇZ ˇ sup ˇˇ d

x2R

Z Rd

G.x; y/d.y/ 

Rd

ˇZ ˇ  sup ˇˇ d x2R

Rd

ˇ ˇ G.x; y/1¹jyj
R!1

by the definition of Kd1 , we get (6.4.20). Noting f   2 Kd1 for f 2 BbC .Rd /, we have Z G.  ; y/f .y/d.y/ 2 C1 .Rd /: Rd

14 Cf.

S. Albeverio, P. Blanchard and Z. M. Ma [1].

362

6 Transformations of forms and processes

L Then, for f 2 Bb .Y /, Let ¹RL ˛ .x; dy/º˛0 be the resolvent kernel of M.   Z 1 Z 1 Z 1  Ex f .XL t /dt D Ex f .X t /dt D Ex f .X t /dA t ; x 2 Y: 0

0

0

On account of the identity (6.2.10) holding q.e. on Rd , excessiveness of the functions involved and the absolute continuity of the transition function, we obtain Z Z Z RL 0 .x; dy/f .y/ D G.x; y/f .y/.dy/ D G.x; y/f .y/.dy/; x 2 Y: Y

Rd

Y

Hence RL 0 f D G.f  / 2 C1 .Y /; f 2 Bb .Y /; because the restriction of a function in C1 .Rd / to Y belongs to C1 .Y /. Since RL 1 f .x/ D RL 0 f .x/  RL 0 RL 1 f .x/ D RL 0 .f  RL 1 f / 2 C1 .Y / L satisfies the assumptions II, III. by the resolvent equation, M For any A 2 B.F / with .A/ > 0 Z L R0 1A .x/ D G.x; y/1A .y/d.y/ > 0; Y

L which implies the irreducibility of M. L satisfies Lemma 6.4.4. For  2 Kd1 the corresponding time changed process M the assumptions I–III. L is A1 . Hence Corollary 6.4.2 tells us that for Recall that the life time L of M 1  2 Kd satisfying (6.4.19)

² ³ Z 1 2 L L lim e u d D 1 ; log Px .A1 > ˇ/ D  inf E.u; u/ W u 2 F ; ˇ !1 ˇ Y

x 2 Y: (6.4.21)

Note that on Y < 1 A 1 D A Y C A1 . Y / D A1 . Y /;

Px -a.s.

Then by the strong Markov property Px .A 1 > ˇ/ D Px .A1 > ˇI Y < 1/ D Ex .PXY .A 1 > ˇ/I Y < 1/ D P .A1 > ˇ/;

6.4

Donsker–Varadhan type large deviation principle

363

where .B/ D Px .X Y 2 BI Y < 1/; B 2 B.Y /. Therefore the equation (6.4.21) holds for any x 2 Rd . We further see from Theorem 4.3.2 and Theorem 6.2.1 that the equation (6.4.21) can be rewritten as 1 lim log Px .A 1 > ˇ/ ˇ !1 ˇ ² ³ Z 1 1 d 2 e u d D 1 ; x 2 Rd : D  inf D.u; u/ W u 2 H .R /; 2 Rd (6.4.22) Since the space C01 .Rd / is D-dense in H 1 .Rd /; we can make use of Exercise 6.4.4 to conclude that the right-hand side of (6.4.22) equals ² ³ Z 1 1 d 2  inf D.u; u/ W u 2 C0 .R /; u d D 1 : 2 Rd Thus we have the next theorem. Theorem 6.4.5. For  2 Kd1 satisfying (6.4.19) 1 log Px .A 1 > ˇ/ ˇ !1 ˇ ³ ² Z 1 1 d 2 u d D 1 ; D  inf D.u; u/ W u 2 C0 .R /; 2 Rd lim

x 2 Rd :

(6.4.23)

We see from the next exercise that the infimum in (6.4.23) is the principal eigenvalue of the time changed process. Exercise 6.4.9. Let  2 Kd1 . Show that the embedding of . 12 D; He1 .Rd // into L2 .Rd I / is compact. Example 6.4.2. Let K be a compact set of Rd .d  3/ with smooth boundary and .dx/ D 1K .x/dx. We then have  Z 1 1 1K .X t /dt > ˇ lim log Px ˇ !1 ˇ 0 ² ³ Z 1 1 d 2 D  inf D.u; u/I u 2 C0 .R /; u dx D 1 : (6.4.24) 2 K In particular, for d D 3, .dx/ D 1B.0;1/ dx, B.0; 1/ D ¹x 2 R3 W jxj  1º, ³ ² Z 2 1 1 d 2 u dx D 1 D : (6.4.25) inf D.u; u/I u 2 C0 .R /; 2 8 B.0;1/

364

6 Transformations of forms and processes

Define the operator GK on L2 .K/ by Z G.x; y/f .y/dy; GK f .x/ D K

f 2 L2 .K/:

Then the right-hand side of (6.4.24) equals the reciprocal of the principal eigenvalue of GK . Exercise 6.4.10. Prove the identity (6.4.25). Exercise 6.4.11. Let E be a discrete countable set and M D .X t ; ; Pi / be the Hunt process on E associated with the regular Dirichlet form .E; F 0 / in Example 1.2.5. Suppose that M is transient. Prove that !   Z  Cap.0/ .¹iº/ Pi 1¹iº .X t /dt > ˇ D exp  ˇ : mi 0 where Cap.0/ denotes the 0-order capacity relative to .E; F 0 /: Example 6.4.3. Let M D .X t ; Px / be the Brownian motion on Rd (d  3). Let  2 Kd1 satisfying (6.4.19). Then applying Lemma 6.4.4 and Corollary 6.4.3, we see that sup Ex .exp.A (6.4.26) 1 // < 1 x2Rd

if and only if ²

1 inf D.u; u/ W u 2 C01 .Rd /; 2

³

Z 2

Rd

u d D 1 > 1:

Combining this with Theorem 2.4.1 and Corollary 2.4.1, we can see that (6.4.26) holds if ‚1 ./ < 1=4, where ‚1 ./ is defined by (2.4.6). Exercise 6.4.12. Let M D .X t ; Px / be the Brownian motion on Rd (d  3). Let K  Rd be a compact set and define mK .  / D m.K \  /, m is the Lebesgue measure. (i) Prove that ‚1 .mK / 

2 2rK , d.d 2/

where rK is the radius of the ball with the p same volume as K, that is, rK D .m.K/. d2 C 1//1=d = .

(ii) Prove that if m.K/ < . 18 d.d  2/ /d=2 =. d2 C 1/, then ‚1 .mK / < 1=4. Exercise 6.4.13. Let M D .X t ; Px / be the Brownian motion on Rd (d  3). Let ır be the surface measure of the sphere Sr D ¹jxj D rº and `r .t / the PCAF corresponding to ır .

6.4

Donsker–Varadhan type large deviation principle

(i) Show that supx2Rd Ex .exp.`r .1/// < 1 if and only if r <

365

d 2 2 .

(ii) Prove that supx2Rd Ex .`r .1// < 1 if and only if supx2Rd Ex .exp.`r .1/// < 1. S. R. S. Varadhan [1] gave an abstract formulation for the large deviation principle. Theorem 6.4.1 is slightly different from the lower estimate and the upper estimate in his formulation; since the Markov process is not supposed to be conservative, we can not regard Theorem 6.4.1 as the large deviation principle from the invariant measure. By this reason, we consider the normalized probability e x;t on P defined by measure Q e x;t .B/ D Px .L t 2 B; t < / ; Q Px .t < /

B 2 B.P /:

(6.4.27)

e x;t º t >0 then satisfies the large deviaThe family of probability measures ¹Q tion principle with the rate function J./ WD IE ./  2 ;  2 P , as t ! 1 in Varadhan’s formulation, where 2 is the bottom of the spectrum of the L2 e x;t º t >0 obeys the full generator A for E defined by (6.4.14). In other words, ¹Q large deviation principle with the good rate function J./. In addition, we shall see that the ground state 0 of the operator A exists and 02  m is a unique probability measure for which J./ D 0. On account of these facts, we shall reinterpret Theorem 6.4.1 as a large deviation principle from the ground state. A function 0 on X is called a ground state of the L2 -generator A for E if 0 2 F ; k0 k2 D 1 and E.0 ; 0 / D 2 . Lemma 6.4.5. Assume that M satisfies I–III. Then there exists a ground state 0 of A uniquely up to a sign. 0 can be taken to be strictly positive on X: Proof. In our proof of the existence of the minimizer in the right-hand side of (6.4.14), the identification of the I-function with the Dirichlet form (Theorem 6.4.2) plays a crucial role. In fact, let ¹un º1 nD1  F be a minimizing sequence, that is, kun k2 D 1 and 0 D limn!1 E.un ; un /. In a usual proof of the existence of the minimizer, the E1 -weak compactness of ¹un º1 nD1 in F and the lower semi-continuity of E are used, while we use the tightness of ¹u2nmº1 nD1  P and the lower semi-continuity of the function I with respect to the weak topology. We see from Theorem 2.4.2 that for any  > 0 there exists a compact set K such that Z   u2n  d m  4kR1 IK c k1  sup E.un ; un / C 1 < ; sup n

Kc

n

that is, the subset ¹u2n mº of P is tight. Hence there exists a subsequence ¹u2nk mº such that u2nk m converges weakly to a probability measure . It follows from

366

6 Transformations of forms and processes

Theorem 6.4.2 that the function IE is lower semi-continuous with respect to the weak topology, IE ./  lim inf IE .u2nk  m/ D lim inf E.unk ; unk / < 1: k!1

k!1

Therefore we see that  is expressed as  D 02  m; 0 2 F ; 0  0. 0 is just a ground state of A: It follows from the inequality k0 C gk2E  2 k0 C gk22 holding for any g 2 F and any  > 0 that E.0 ; g/ D 2 .0 ; g/: Hence ˛R˛2 0 D 0 ; ˛ > 2 ; which implies that 0 is strictly positive in view of the paragraph above (6.4.1). To prove the uniqueness of the ground state, we introduce a closed symmetric form .E 0 ; F 0 / on L2 .XI 02 m/ by ´ E 0 .u; v/ D E.u0 ; v0 /  2 .u0 ; v0 / (6.4.28) F 0 D ¹u 2 L2 .XI 02  m/ W u0 2 F º: 



Since 1 2 F 0 ; E 0 .1; 1/ D 0 and the associated resolvent G˛ 0 satisfies G˛ 0 f D 01 R˛2 .f 0 /; ˛ > 2 ; we see that .E 0 ; F 0 / is an irreducible recurrent Dirichlet form so that 15 f is constant whenever f 2 F 0 ; E 0 .f; f / D 0: Let 0 be another ground state of A: Then 0 D f 0 with f D 0 =0 2 F 0 ; E 0 .f; f / D E. 0 ; 0 /  2 D 0; which yields that f is constant and 0 D ˙0 : .X t / Exercise 6.4.14. Define the MF L t by L t D e 2 t 00.X . Show that the trans0/ formed process of M by L t generates the Dirichlet form (6.4.28).

Exercise 6.4.15. Let .X t ; Px / be the Brownian motion on Rd and D  Rd a domain with regular boundary. If D is unbounded, assume that limx2D;jxj!1 Px .D > 0/ D 0. (i) Show that the Dirichlet form .1=2D; H01 .D// admits the ground state. (ii) Denote by 0 the ground state in (i). Show that for a set B R D of finite Lebesgue measure, lim t !1 e 2 t Px .X t 2 B; t < D / D 0 .x/ B 0 .x/dx. Exercise 6.4.16. Let M D .X t ; Px / be the Brownian motion on Rd , d  3. Suppose a finite measure  2 Kd satisfies (6.4.19) and the inequality (2.4.20) for L be the time changed process of M by the ˛ D 2 and for some q > 2. Let M L Show that PCAF A with Revuz measure  andR 0 be the ground state for M:  ˇ 2 limˇ !1 e Px .A1 > ˇ/ D 0 .x/ Y 0 d; 8x 2 Y: 15 Cf.

Fukushima–Takeda [2; Th. 4.2.4] or Chen–Fukushima [1; Th. 2.1.11].

6.4

Donsker–Varadhan type large deviation principle

367

Lemma 6.4.6 below states that the function J./ D IE ./  2 ;  2 P ; enjoys the properties as a rate function in the large deviation principle. Lemma 6.4.6. The function J satisfies: (i) 0  J./  1. (ii) J is lower semicontinuous. (iii) For each ` < 1, the set ¹ 2 P W J./  `º is compact. (iv) J.02  m/ D 0 and J./ > 0 for  6D 02  m. We note that the identity J./ D IE 0 ./;

 2 P;

(6.4.29)

holds true, where IE 0 is defined in terms of the Dirichlet form (6.4.27) by ´

p p p E 0 . f ; f / if  D f 02  m; f 2 F 0 IE 0 ./ D 1 otherwise:

(6.4.30)

From Theorem 6.4.1 and Corollary 6.4.2, we obtain the next large deviation principle: e x;t º t >0 be a family of probability measures defined by Theorem 6.4.6. Let ¹Q e x;t º t >0 obeys the large deviation principle with (6.4.27). Then the sequence ¹Q rate function J : (i) For each open set G  P lim inf t !1

1 e x;t .G/   inf J./: log Q 2G t

(ii) For each closed set K  P lim sup t !1

1 e x;t .K/   inf J./: log Q 2K t

e x;t converges weakly to ı 2 as t ! 1. Corollary 6.4.4. The measure Q  m 0

Proof. If a closed set K does not contain 02  m, then inf2K J./ > 0 by e x;t .K/ D 0 Lemma 6.4.6 (iii), (iv). Hence Theorem 6.4.6 (ii) says that lim t !1 Q

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6 Transformations of forms and processes

e x;t .K c / D 1. For a positive constant ı and a bounded continand lim t !1 Q uous function f on P , define the closed set K  P by K D ¹ 2 P W jf ./  f .02  m/j  ıº. Then we have ˇ Z ˇZ ˇ ˇ 2 ˇ e x;t .d/  f .0  m/ˇ  e x;t .d/ f ./Q jf ./  f .02  m/jQ ˇ ˇ P P Z Z 2 e e x;t .d/ jf ./  f .0  m/jQx;t .d/ C jf ./  f .02  m/jQ D Kc

K

e x;t .K/ C ı Q e x;t .K / ! ı  2kf k1 Q c

as t ! 1. Since ı is arbitrary, the proof of the corollary is complete. On account of Corollary 6.4.4, we can regard Theorem 6.4.6 as a genuine large deviation principle from the ground state.

Chapter 7

Construction of symmetric Markov processes

It is shown in this chapter that, given a regular Dirichlet space, there exists a Hunt process whose Dirichlet space is the given one. Such a process is unique up to an equivalence in accordance with Theorem 4.2.8. Furthermore, the local property of the given regular Dirichlet space is equivalent to the sample path continuity of the constructed Hunt process by virtue of Theorem 4.5.1. As is stated in Theorem A.2.2 of the Appendix, any transition function possessing the Feller property admits a Hunt process. In general, it is hopeless to construct a Feller transition function from the L2 -semigroup T t associated with the given Dirichlet space. However, if the Dirichlet space is regular, then the potential theory of Chapter 2 provides us with quasi continuous versions T t f and a sequence tn # 0 such that T tn f .x/ ! f .x/; n ! 1; q.e., for sufficiently many f . Going along a similar line as in the case of the Feller transition function, but ignoring successively the sets of capacity zero on which things might go wrong, we finally get a Hunt process outside some set of capacity zero.

A

7.1

e

Construction of a Markovian transition function

Let the pair .X; m/ be as in (1.1.7) and E be a regular Dirichlet form on L2 .XI m/. By making use of the potential theory developed in Chapter 2 in connection with the form E, we now construct an m-symmetric Hunt process on X whose Dirichlet form is the given one E. As a preliminary step, we construct in this section a transition function on X with time parameter QC . Here we denote by Q and QC the set of all rational numbers and positive rational numbers respectively. Lemma 7.1.1. Let F be a set. Consider a countable subset G of F and a countable collection S of mappings from F  F into F . Then there exists a countable set H such that (i) G  H  F , (ii) s.H  H /  H; 8s 2 S: Proof. Construct a sequence ¹s1 ; s2 ; : : :º such that each s 2 S appears infinitely often in the sequence. Let G0 D G; GnC1 D Gn [snC1 .Gn Gn /; n D 1; 2; : : : :

370

7 Construction of symmetric Markov processes

S Then the set H D 1 nD1 Gn is a desired one. In fact, for any x; y 2 H and s 2 S , we can find n such that x; y 2 Gn and s D snC1 . Let ¹T t ; t > 0º and ¹G˛ ; ˛ > 0º be the semigroup and resolvent on L2 .XI m/ associated with the form E (§1.4) and ¹eA ; A 2 O0 º be the 1-equilibrium potentials associated with E (§2.1). We use those notions and notations employed in Chapter 2. e1 (I) Markovian kernels ¹e p t ; t 2 QC º and R Lemma 7.1.2. There exist a regular nest ¹Fk0 º and Markovian kernels ¹e pt ; t 2 e QC º and R1 on X satisfying the following conditions: e1 .C1 .X//  C1 .¹F 0 º/, (i) e p t .C1 .X//  C1 .¹F 0 º/; R k

k

e1 u are quasi continuous version of T t u and G1 u respectively for (ii) e p t u and R any non-negative Borel u 2 L2 .XI m/. Proof. Let us first show that there exists a countable set B0  F \ C0 .X/ such that B0 is dense in C0 .X/ u; v 2 B0

a 2 Q ) juj 2 B0 ;

u C v 2 B0 ;

au 2 B0 :

In order to construct B0 , let ¹u1 ; u2 ; : : :º be a countable dense subfamily of C0 .X/. For each k and n, choose a uk;n 2 F \ C0 .X/ such that kuk  uk;n k1  1=n. The set G D ¹uk;n W k; n D 1; 2; : : :º is dense in C0 .X/. Define the mappings s0 ; s1 ; s2 ; : : : from ¹F \ C0 .X/º2 into F \ C0 .X/ by s0 .u; v/ D juj, s1 .u; v/ D u C v and si .u; v/ D ai u; i D 2; 3; : : : ; where ¹a2 ; a3 ; : : :º D Q. Applying the preceding lemma to G and S D ¹s0 ; s1 ; s2 ; : : :º, we get a countable set B0 satisfying (7.1.1) and (7.1.2). Let  [  H0 D T t .B0 / [ G1 .B0 /: .7:1:3/ t 2QC

Then H0 is countable and H0  F in view of Lemma 1.3.3. Choose a quasi e 0 D ¹e uW continuous version e u (in the restricted sense) of each u 2 H0 and let H 0 u 2 H0 º. By Theorem 2.1.2 (i), there exists then a regular nest ¹Fk º on X such e 0  C1 .¹F 0 º/. that H Sk 0 Now let Y0 D 1 pt kD1 Fk and fix t 2 QC . We construct a Markovian kernel e e1 can be constructed in the same way. By on X satisfying (i) and (ii). The kernel R

7.1

Construction of a Markovian transition function

371

Theorem 2.1.2 (ii),

B e e x 2 Y ; u; v 2 B ; B T .au/.x/ D aTe u.x/; x 2 Y ; u 2 B ; a 2 Q:

T t .uCv/.x/ D T t u.x/ C T t v.x/; t

t

0

0

0

0

(7.1.4) (7.1.5)

By the Markovian nature of T t , we also have 0  u  1;

e

u 2 B0 ) 0  T t u.x/  1;

8x 2 Y0 :

.7:1:6/

(7.1.4), (7.1.5) and (7.1.6) imply

e

jT t u.x/j  kuk1 ;

x 2 Y0 ; u 2 B0 :

.7:1:7/

It is then easy to see that, for each x 2 Y0 , there exists a unique positive linear functional lx on C1 .X/ satisfying

e

lx .u/ D T t u.x/;

8u 2 B0 ;

jlx .u/j  kuk1 ;

8u 2 C1 .X/:

.7:1:8/

To see this, choose for any u 2 C1 .X/ a sequence un 2 B0 such that kun  uk1 ! 0; n ! 1. Then T t un .x/ is uniformly convergent on Y0 . Denote the limit by lx .u/: lx is positive because lx .juj/ D limn!1 lx .jun j/  0. The linearity and the inequality in (7.1.8) are also clear. Note that lx .u/ belongs to the space C1 .¹Fk0 º/ as a function of x 2 Y0 because T t un 2 C1 .¹Fk0 º/. p t .x;  / on X such that lx .u/ D e p t u.x/ and lx admits a positive measure e e p t .x; X/  1; x 2 Y0 . Extend e p t by setting e p t .x; A/ D 0; 8x 2 X n Y0 ; 8A 2 B.X/. Then e p t becomes a Markovian kernel on X satisfying the condition (i). It now suffices to derive the property (ii) for u 2 C0C .X/ because then, by the same argument as in the proof of Theorem 4.2.3, we can get (ii) for non-negative u 2 L2 .XI m/. Choose un 2 B0 which is uniformly convergent to u 2 C0C .X/. Take w 2 B0 such that w  u. Then the functions vn D .0 _ un / ^ w belong to B0 and converge to u uniformly and in L2 -sense as well. Since e p t vn is a quasi continuous version of T t vn , a standard argument already adopted in the proof of Theorem 4.2.3 implies that e p t u is a quasi continuous version of T t u.

A

A

(II) A regular nest ¹Fk º Let ¹An º be a countable basis of open sets of X . We may assume that each An is relatively compact. Let O1 D ¹A W A is a finite union of An ’sº;

.7:1:9/

eA of then O1  O0 . For each A 2 O1 , choose a quasi continuous Borel version e the 1-equilibrium potential eA such that 0  e eA .x/  1; 8x 2 X .

372

7 Construction of symmetric Markov processes

e \ Bb .X/ such that e the smallest subfamily of F Denote by H e B0 ; ¹e (H.1) H eA ; A 2 O1 º, e/  H e ; t 2 QC ; R e1 .H e/  H e. (H.2) e p t .H e is an algebra over Q. (H.3) H e is then countable. To see this, consider the mappings from ¹F \ Bb .X/º2 H into F \ Bb .X/ defined by s1 W .u; v/ 7! u C v; s2;a W .u; v/ 7! au; s3 W e1 u. It then suffices to apply .u; v/ 7! uv; s4;t W .u; v/ 7! e p t u; s5 W .u; v/ 7! R Lemma 7.1.1 to G D B0 [ ¹e eA W A 2 O1 º and S D ¹s1 ; s2;a ; s3 ; s4;t ; s5 W a 2 Q, t 2 QC º. Lemma 7.1.3. There exists a regular nest ¹Fk º satisfying the following: If Y1 D S1 F kD1 k , then e  C1 .¹Fk º/; Fk  F 0 ; k D 1; 2; : : : ; (i) H k

(ii) e eA .x/ D 1; x 2 A \ Y1 ; A 2 O1 , e 1e e1 u.x/e tk R p tk u.x/ ! u.x/, .1=tk /.R p tk u.x// ! (iii) 9tk  QC , tk # 0, e e u.x/, x 2 Y1 , u 2 H , e, p s u.x/ D e p t Cs u.x/; x 2 Y1 ; t; s 2 QC ; u 2 H (iv) e pte e1 u.x/ D R e 1e e1 u.x/  R e1 u.x/; x 2 Y1 ; t 2 QC ; p t u.x/; e t e pt R (v) e pt R e C, u2H p te eA .x/  e eA .x/; x 2 Y1 ; t 2 QC ; A 2 O1 , (vi) e t e (vii) 0  e eA .x/  1; x 2 Y1 ; A 2 O1 , eB .x/; x 2 Y1 ; A; B 2 O1 ; A  B. (viii) e eA .x/  e Proof. By Lemma 2.1.4, e eA .x/ D 1 q.e. on A. Since O1 is countable, we can find a set N1 such that Cap.N1 / D 0 and e eA .x/ D 1;

8x 2 A n N1 ; 8A 2 O1 :

.7:1:10/

e admits a sequence tk 2 QC tk # 0 By Lemma 1.3.3 and Theorem 2.1.4, each u 2 H e such that (iii) holds for q.e. x 2 X. Since H is countable, ¹tk º can be selected e and (iii) holds for every u 2 H e and for every x 2 independently of u 2 H X n N2 ; N2 being a suitable set of zero capacity. By virtue of Theorem 2.1.2 (i), there exists a nest ¹Fk º satisfying the condiS tion (i) and X n 1 F kD1 k N1 [ N2 . Denote the m-regularization by ¹Fk º again. Obviously, (i), (ii), and (iii) are satisfied for this ¹Fk º. (iv) and (v) hold because they are valid m-a.e. (vi), (vii) and (viii) follow from Lemma 2.1.1 and Theorem 2.2.1.

7.2

Construction of a symmetric Hunt process

373

(III) A Markovian transition function p t ; t 2 QC Lemma 7.1.4. (i) There exists a Borel set Y2  Y1 with Cap.X n Y2 / D 0, e p t .x; X n Y2 / D 0; 8x 2 Y2 ; 8t 2 QC . (ii) Let

´ e p t .x; A/; p t .x; A/ D 0;

x 2 Y2 ; A 2 B.X/ x 2 X n Y2 ; A 2 B.X/;

.7:1:11/

Then ¹p t º t 2QC is a Markovian transition function on .X; B.X//; namely, each p t is a Markovian kernel on .X; B.X// and property .1:4:10/ is satisfied for every t; s 2 QC . p t .x; Proof. (i) T t 1X nY1 .x/ D 0 m-a.e. because X n Y1 is m-negligible. Since e X n Y1 / is a quasi continuous modification of T t 1X nY1 by Lemma 7.1.2 (ii), there .1/ .1/ exists a Borel sets Y1  Y1 such that Cap.X n Y1 / D 0; e p t .x; X n Y1 / D 0, .1/ 8x 2 Y1 ; 8t 2 QC . In this way, we can find a sequence of Borel sets Y1 .1/ .2/ .k/ .kC1/ Y1 Y1    such that e p t .x; X n Y1 / D 0; 8x 2 Y1 , 8t 2 QC , T .kC1/ .k/ 1 / D 0: Y2 D kD1 Y1 then possesses property (i). Cap.X n Y1 (ii) It suffices to show the semigroup property, p t ps u.x/ D p t Cs u.x/;

x 2 X; t; s 2 QC ; u 2 B0 :

.7:1:12/

Both sides vanish when x 2 X n Y2 . When x 2 Y2 , (i) implies p t ps u.x/ D pt . e p s u/.x/, which is also equal to e p t Cs u.x/ D p t Cs u.x/ by e p t .ps u/.x/ D e virtue of the preceding lemma.

7.2

Construction of a symmetric Hunt process

Let X D X [ ¹º be the one-point compactification of X. When X is already compact,  is adjoined as an isolated point. Extend the Markovian transition function p t on .X; B.X// of Lemma 7.1.4 to .X ; B.X // by 8