II P( IfA fA I IE > 0 I)
(ii) Suppose lim
A1
+
EP( IfA fA I IE < 0 I) < E( 2 II~ II
+ 1)
E E}.
Jn lftA(w) fA(w)l~ dP(w)
= 0
uniformly in A EA.
Then the first assertion in (ii) follows immediately from the Chebyshev inequality
P(lf~
fAIE
>E)<~ J lf~(w) fA(w)l~ E
n
dP(w)
which holds for any E > 0, t > 0, A E A (Theorem (1. 2.4)). To prove the second assertion in (ii), use the first assertion and a similar argument to that in (i). c We now have the following theorem. Theorem (3.1): For the stochastic FOE (I) suppose Hypotheses (M) are satisfied. Then the family {p~1:o < t 1 < t 2
67
(i)
with t1 t (ii) Pt o pt2 2
=
3
t pt1' 0 < t 1 < t 2 < t 3
In particular, for the autonomous stochastic FOE (IV), the family {pt = P~:O < t
Proof:

2
1
+
2
t 2 E [O,a].
By Theorem (11.3.1), the map Ttt 1:C(JJRn) ~ 2
t
2(n,C(JJRn);Ft ) is 2
Lipschitz for 0 < t 1 < t 2
'E
3
{P: 1rP: 2(,)J}(n) 2
3
=
3
2
J J g
=
'{T!2rT: 1(n)(w)](w')}dP(w')dP(w) 3
g
2
t1 '[Tt (n)(w)]dP(w),
J
3
g
(Cf. proof of Theorem (1.1)). In the autonomous case, Pt

2
= Pt0
=
2
t1 Pt +t , t 1,t2 E [O,a], t 1 + t 2 E [O,a], 1
2
because of timehomogeneity. Thus pt opt = p~ o·p!1+t = pt +t. D 1 2 1 1 2 1 2 The semigroup {Pt}t>o of the above theorem will be studied in some detail in the next chapter. Let M(C) be the complete topological vector space of all finite Borel measures on C : C(JJRn) given the weak * (or vague) topology (§1.2, Parthasarathy [66], Schwartz [71], Stroock and Varadhan [73]). Then we have a bilinear pairing 68
<·,·>:. cb
x
M(C)> :R, <4>,ll>
=
Ic4>(n)dJJ(n),
4> € Cb' ll € M(C). Following Dynkin [16], define the adjoint semigroup * {Pt}O
I
nEC
p(O,n,t,B)dlJ(n), B €. Borel C(J,:Rn), 0 < t
Then for any 4> € Cb' <4>,P;ll>
=
I
~€C
4>(~) d(P;ll)(~) =
f
J~EC
=
J
nEC
I
~EC
4>(~)
I
p(O,n,t,d~)dll(n)
n€C
4>(~)p(O,n,t,d~)dll(n)
Pt(cp)(n)dll(n) =
Since {Pt}O
2
1
2
t 1,t 2 € [O,a], t 1+t 2 € [O,a]. An invariant probabiLity measure for the stochastic FOE (IV) is a probability measure llo € M(C) such that P; llo = llo for all t € [O,a]. If a = ~. a probability measure llo € M(C) is invariant if and only if lim p(O,n,t,·) = llo for some n € C(JJRn). This t+<><>
follows easily from the fact that P; p(O,n,t',·) = p(O,n,t+t',·), t,t' > 0, n € C(JJRn). Observe also that the family of transition probabilities {p(O,n,t,.):n € C(JJRn), t > O} for (IV) is left invariant by the semigroup {p;} t>O when a = ~. It would be interesting to find generic conditions on the coefficients H, G of (IV) which guarantee the existence of a (unique) invariant probability measure. Some partial results in this connection may be found among the examples of Chapter VI (§VI.4). See also Ito and Nisio ([41]) and Scheutzow [69]).
69
IV The infinitesimal generator
§1o
Notation
For the present chapter we keep the notation, general setup and the standing assumptions of the last chapter (§III 1,2)o In particular, we focus our attention on the autonomous stochastic FOE: n(O)
+I:
H(xu(w))du+ (w)
I:
G(xu(o))dw(o)(u)
x(w)(t) = { n(t)
t >0 (I)
t E J = [r,O]
Very frequently, the solution x through n will be denoted by nx; and throughout the chapter we shall assume that the coefficients H:C(J,Rn)+ Rn, G:C(J,Rn) L(Rm ,Rn) are g'LobaZZ.y bounded and Lipschitzo The driving Brownian motion w is in Rm, generating a filtration (Ft)t>O on the probability space (O,F,P)o For brevity, symbolize the above stochastic functional equation by the differential notation: dx(t) = H(xt)dt + G(xt)dw(t)
(I)
x0 = Tl € C(J,Rn) Similarly for any t 1 > 0 we represent the equation
t
n(O)+t H(xu(w))du+ (w) G(xu( o))dw( o)(u).t > t 1 x(w)(t) = { t1 t1 t 1r < t < t 1 n( tt 1) by the stochasticdifferential notation dx(t)·= H(xt)dt + G(xt)dw(t)
t > t, > 0 }
}
(I)
1
I
xt = n € C(J,Rn) 1
Recall that at the end of the previous chapter, we constructed a contraction semigroup {pt}t>O associated with the stochastic FOE (I) and defined on the Banach space Cb of all bounded uniformly continuous functions ~:C(J,Rn) > Ro 70
Indeed
Now, for an ordinary stochastic differential equation (Stochastic ODE, r = 0), it is wellknown that the semigroup {Pt}t>O is strongly continuous on Cb with respect to the supremum norm, and its strong infinitesimal generator is a second order partial differential operator on the state space of the solution process ([16], [22]). Our first objective is to show that, when r 0, the semigroup {Pt}t>O is neve~ strongly continuous on the Banach space Cb. Furthermore, we shall derive an explicit formula for the (weak) infinitesimal generator of {Pt}t>O. §2.
Continuity of the Semigroup
For each n € C(JJRn) and t > 0, define ~:[r,=) +Rn by "'
n(t)
=
{~ n( O)
t>O
n(t)
t € J R"
R
0
Define the shift St:Cb
~
"' st<•>
Cb, t > 0, by setting €
C(JJRn ) , •
€
cb.
The next result then gives a canonical characterization for the strong continuity of {pt}t>O in terms of the shifts {St}t>O.
71
Theorem (2.1): The shifts {St}t>O form a contraction semigroup on Cb, such that, for each n € C(J,Rn), lim St(ct>)(n) = lim Pt(ct>)(n) = cl>(n) for all t+0+ t+0+ 4> € Cb. Furthermore lim Pt(cl>)(n) = cl>(n) uniformly in n € C(J,Rn) if and t+0+ only if lim St(cl>)(n) = cl>(n) uniformly in n € C(J,Rn). t+0+ Proof:
Let t 1,t2 > 0, n € C(J,Rn),
4>
€ Cb, s € J.
Then
where
"' )(0) (nt
~
"' )t (s) (nt 1 2
=
{
"'
=
n(O)
t
(nt }(t2 + s) 1
=
+
s
< 0
"' +t )(s). (nt 1
Hence
i.e.
r < t 2
st (St (ct>))(n) 1 2
=
2
"' +t ) ct>(nt 1 2
=
st +t (ct>)(n) 1 2
Since lim . "' nt = n , it is clear that lim St(cl>)(n) = ct>(n) for each t+0+ t+0+ n € C(J,Rn), 4> € Cb. Also by sample paths continuity of the trajectory {nxt:t > O} of (I) (Theorem (11.2.1)) together with the dominated convergence theorem, one obtains
72
for each ~ € Cb and n € C(JJRn). To prove the second part of the theorem, suppose K > 0 is such that IH(n) I < K and IIG(n) II< K for all n € C(JJRn). Then for each t > 0 and almost all w € Q we have t+s t+s J H(nx (w))du+(w) J G(nx (·))dw(·)(u) t+S>O 0
u
0
u
r < t
+
= {
0
s
<
0,
s € J
Thus, using the Martingale inequality for the stochastic integral (Theorem (1.8.5)), one gets "' 2 Jt+s 2 unxt(·) ntll 2 < 2E sup I H(nxu(·))dul £ (Q,C) S€[t,O] 0 + 2E sup S€[t,O]
IJ
t+s
G(nxu(·))dw(·)(u)l 2
0
< 2K 2t 2 + 2K 1 J: E IIG(nxu(·))ll 2 du < 2K2t 2 + 2K 1t, some K1
> 0.
Therefore, lim llnx  ~ II = 0 uniformly in n € C(JJRn). Using the t+0+ t t £2(Q,C) uniform continuity of ~ € Cb it is then not hard to see that lim {E~onxt(•)~(~t)} =0 unifom"ly inn € C(JJRn) (Cf. proof of Lemma (III. t+0+ 3.1 )). So 1 irn {Pt(~)(n)St(~)(n)} = 0 uniformly in n € C(JJRn). Finally, \'lriting t+0+
the second assertion of the theorem is now obvious. c Let C~
Cb be the set of all ~ € Cb such that lim Pt(~) = ~(= lim St(~)) t+0+ t+0+ in Cb. Then C~ is a closed linear subalgebra of Cb which is invariant under the semigroups {pt}t>O, {St}t>O. Both {pt}t>O and {St}t>O restrict to strongly continuous semigroups on C~ (Dynkin [16], pp. 2226). c
Theorem (2.2): The semigroup {pt}t>O is not strongly continuous on Cb with respect to the supremum norm. ~:
It is sufficient to find~
€
Cb(C(JJRn)JR) such that St(~) ~ ~as 73
t +0+ i.n Cb;. (but St(lll)(n) + lll(n) as t + 0+ for each n €. C(J,Rn).) Let B c: C(J,Rn) be the closed unit ball. Fi.x any r < s0 < 0 and define lii:C(J,Rn) ~ R by n(s 0) lll(n) = {
lin
II
< t
1 lin II >
n
t
Clearly 111 is continuous; indeed 111 is globally Lipschitz (and hence unifonmly continuous) on C(J,Rn). To prove this let n. n• E C(J,Rn) and consider the following cases (i)
n,n• e B.
( ii )
n, n•
t.
Then llll(n)  lll(n•)
i nt B• i.e.
II n II >
t,
= ln
II n• II >
1•
Write
I
n(s 0) n'(s 0) J llll(n)  lll(n•) I =   •  lin II lln'll n(s 0) n(s 0) J n(s 0)
< =
<
Iw~~~ I InI 0 llln• llnlllln•ll
L
lin• nil
lln•ll (iii) n e int B, n•
74
+
II 
n'(s 0)
~~~M~
lin II I
+
I
 1ln
lln'll
+
L
lin n•ll < 2lln n•ll
lln'll
t.
B;. i.e. lin II < 1,
lin•
II>
1.
Find n" e
as
II
(where 38 is the boundary of B) such that n" lies on the line segment joining nand n• i.e. find >.0 E [0,1].such that n" = (t>.0 )n.., >.0n• and lln"ll = 1. Define the function f:[0,1]> R by f(>.) = 11(1>.)n • >.n'll1, >. E [0,1]. Then f h clearly continuous. Also f(D) = lin II t < 0 and f(1) = llrlll1 >0. Hence by the IntermediateValue Theorem there exists >. 0 E (0,1) such that f(A 0 ) = o i.e. taken"= (1><0)n ... >. 0n• and lln''ll = t. Hence lljl(n)  ljl(n') l < lljl(n)  ljl(n")
I ...
lljl(n")  ljl(n')
II
R"
r
i.e.
0 nn(s) = { [s 0 • 1
*
s]n
so
1 +
n
s0 < s < s0 +
r < s < s 0
Note that llnnll = t for all n. such that s 0 • Consider for each fixed n
*
*
< 0; i.e. nnE 38 for all n.
75
St(~)(nn)
= ~<~{' = nn(t + s 0>. o < t
s 0•
< 
Therefore lim St(~)(nn) t+o+
= lim
nn(t + s 0)
t+o+
= nn(s 0) = ~(nn)
(*)
Now(*) is not unifo~ inn; for if (•) were uniform inn. given 0 < E < 1, there is a 0 < 6 < s0 (6 independent of n) such that lnn(t+s 0)nn(s 0) I
0
no
no
n (t 0 + s0)  n
(s 0) = lo 1 I = 1.
But, by choi.ce of 6, lnn°(t0 + s 0)  nno(s 0) I < tradicts the choice of E. Therefore ~ t cg.
E •
Hence 1 <
E,
which con
a
Remarks (i)
One could have chosen
~· €
Cb in the above proof such that
II n II <
1
~*(n) =
II n II
> 1
It can be shown that ~· t c~. However, note that, while ~· is only continuous. ~ is smooth in any small neighbourhood of 0. (ii) Later on, in §4, we construct a concrete class of smooth functions on C(JJRn) lying entirely in C~. These functions are called quasitame fUnctions. The class of quasitame functions is sufficiently rich to generate Borel C(JJRn). in exactly the same fashion as the tame functions do. (iii) Note that the domain of strong continuity C~ of {Pt}t>O is independent of the coefficients H. G and the Brownian motion w. As yet, a complete characterization of C~ is unknown to me. §3. The Weak Infinitesimal Generator In this section we obtain a general formula for the.weak infinitesimal generator of {Pt}t>O (Dynkin [16] Vol. 1 pp. 3643). Define a weak topology on Cb as follows: Let M(C(JJRn)) be the Banach 76
space of all finite regular measures on Borel C(JJRn) given the total variation norm. Then there is a continuous bilinear pairing <·,·>:CbxM(C(JJRn))~ defined by
<~.~>
=
fC(JJRn) $(n) d~(n),
$ € Cb'
~
€ M(C(JJRn))
Say a family {$t:t > O} in Cb conve~es ~eakLy to $ € Cb as t ~ 0+ if lim <~t.~> = <~.~> for all ~ € M(C(JJRn)). Write this as $ = wlim ~t• t~+
t~+
Proposition (3.1) (Dynkin [16], p. 50): For each t > 0 let ~t' ~ € Cb. Then $ = wlim ~t if and only if £11~tll: t > 0} is bounded and $t(n) ~ ~(n) t~+
as t ~ 0+ for each n € C(JJRn). Proof: For each n € C(JJRn) let on be the Dirac measure concentrated at n, defined by 1 ,
n € B
o, nt B Define ~t € [M(C(JJRn))]* by
for all B € Borel C(JJRn).
$t(~) = <~t'~>. ~ € M(C(JJRn)).
If$= wlim
t~+
~t'
then
~(n) =<~,on>
"'
= lim
t~+
<~t.on>
=lim
t~+
~t(n)
for all
n € C(JJRn), and the set {$t(~):t > O} i! bounded for each~ € M(C(JJRn)). By the uniform boundedness principle {II $til : t > O} is bounded. But ll~tll = ll~tll for each t > 0, so £11~tll : t > O} is bounded. Conversely suppose {II $til : t > O} is bounded and ~t(n) ;:. $(n) as t ~ 0+, for all n € C(JJRn). By the dominated convergence theorem
J
n C(JJR )
as t ~ 0+, for all ~ € M(C(JJRn)). The
~ak infinitesimaL generatoP
A($) = wlim t~+
~(n)d~(n)
Thus ~ = wlim $t.
= <~.~>
D
t~+
A:V(A)
c
Cb
~
Cb of {pt}t>O is defined by
Pt($)  $ t 77
where O(A) is the set of all ' for which the above weak limit exists. By continuity of the sample paths and the dominated convergence theorem it is easy to see that wlim Pt(') = 'for ev.ery ' € Cb. Moreover the t~+
following properties can be found in Dynkin ([16] Vol. 1 Chapter I §6, pp. 3643). Theorem
(3.1)~
(i) O(A)
=C~.
O(A) is weakly dense in Cb, Pt(O(A)) ~ O(A) for all t >
o.
(ii) If ' € O(A), then the weak derivative
exists and
~ Pt(') = A(Pt(,)) = Pt(A(,)), Pt(')  ' =
J:
Pu(A ('))du for all t > 0.
(iii) A is weakly closed i.e. if {'k};= 1 c O(A) is weakly convergent and {A(,k)};= 1 is also weakly convergent, then wlim 'k € O(A) and wlim A(,k) = A(wlim 'k). k~ k~
(iv)
k~
For
each~>
(~id

0,
A) 1 (~)
~idA
=
is a bijection of O(A) onto Cb.
J: e~tpt(~)dt
for all
~
Indeed
E cb.
The resolvent R~ = (~id A)1 is bounded linear and IIR~II< {for all ~ > 0. (v) wlim
~~
~R~(~)
= ~ for every
~
E Cb.
In order to deriv.e a general formula for the weak infinitesimal generator A, one needs to augment the state space C(JJRn) by adjoining a canonical ndimensional direction. The generator A will then be equal to the weak infinitesimal generator S of the shift semigroup {St}t>O of §2 plus a second order partial differnetial operator along this new direction. The construction is as follows. Let Fn be the vector space of all simple functions of the form vx{O}where vERn and x{O}:J ~ R is the characteristic function of {0}. Clearly 78
c(JJRn) n Fn = {0}. plete norm lin
+
Form the direct sum C(JJRn) • Fn and give it the com
vx{o}ll = sup ln(s) I s€J
+
lv I
• n e: C(JJRn). v € Rn
Indeed C(J,Rn) • Fn is the space of all functions ~:J +Rn which are continuous on [r,O) and possibly with a finite jump discontinuity at 0. The following lemma shows that the above construction admits weakly continuous extensions of linear and bilinear forms on C(J,Rn) to C(J,Rn) • Fn. Lemma (3.1): let a € C(J,Rn)*. Then a has a unique (continuous) linear extension a:C(J,Rn) • Fn ~ R satisfying the weak continuity property (w 1) if {~k};= 1 is a bounded sequence in C(J,Rn) such that ~k(s) ">· ~(s) as k +=for all s € J for some ~ € C(J,Rn ) • Fn• then a(~ k) . a(~) as The extension map e : C(J,Rn) *> [C(J,Rn) • FnJ*
is a linear isometry into. Proof: We prove the lemma first for n = 1. Suppose a € C(J,R)*. By the Riesz representation theorem (Dunford and Schwartz [15], §IV.6.3 p. 265) there is a (unique) regular finite measure ~= Borel J +R such that a(n) = Jt r
n(s)d~(s)
for all n € C(J,R).
Define a:C(J,R) e F1 ~ R by a(n
+
v1x{O}) = a(n)
+ v 1 ~{0},
n € C(J,R), v1 € R.
a is
clearly a continuous linear extension of a. k let {~k};= 1 be a bounded sequence in C(J,R) such that ~ (s) + n(s) + v 1 ~ 0 js) as k +=for all s € J where n € C(J,R), v1 € R. By the dominated convergence theorem,
By
79
The map e:ceJ,R) * ~ rceJ,R) e F1J* is clearly linear. a
•>
a
Higher dimensions n > 1 may be reduced to the 1dimensional situation as follows: Write a E ceJ,Rn)* in the form n . aen) = E a 1 en.) 1 i=1 where n = en,_ ••• ,nn) E ceJ,Rn), ni E ceJ,R), 1 < i < n, aien*) = aeo,o, ••• ,O,n*,O, ••• ,O) with n* E ceJ,R) occupying the ith place. Hence ai E ceJ,R)* for 1 < i < n. Write Fn=F~, where F1 = {v*x{O}:v* ER} by taking vX{O} = ev 1x{O}'"" ., vnx{O}) for each v = ev 1, ••• ,~n) ERn, vi E R 1 < i < n. Let al E [CeJ,R) e F1J* be the extension of a 1 described before and satisfying ew 1). I_t is easy to see that ceJ,Rn) e Fn = rceJ,R) e F1J
x ••• x
rceJ,R) e F1J en copies)
i.e. n + VX{O} = en 1 + v1x{O}'"""'nn + VnX{O}) Define a E [CeJ,Rn) e FnJ* by n
•
aen+ VX{O}) = i~ 1 a 1 eni + ViX{O}) when n = en 1 ,.~··nn), v = ev 1, ••• ,vn). Since ai is a continuous linear~ extension of a 1 , then a is a continuous linear extension of a. Let {~k}k= 1 be bounded in ceJ,Rn) such that ~kes) ~ ~es) as k + ~ , for all s E J, n k k k k where~ E ceJ,R) e Fn. Let~ = e~ 1 ••••• ~n), ~ = e~ 1 ••••• ~n), ~i E ceJ,R), ~i E ceJ,R) e F1, 1 < i < n. Hence {~~};= 1 is bounded in C(J,R) and 80
t~(s) ~ l;;i(s) as k k
+
Q), s
€
n
i
J, 1 < i < n. Therefore k
n
i
lim a(l;; ) = lim E a (l;;i) = E a (l;;i k.._, k.._, i=1 i=1 ·so
)

= a(l;;).
a satisfies
(w 1). To prove uniqueness let "' a € [C(JJRn) e Fn] * be any continuous linear extension of a satisfying (w 1). For any vx{O} € Fn choose a bounded sequence {t~};= 1 in C(JJR) such that l;;~(s) ) YX{O}(s) as k + Q), for all s € Ji e.g. take. (ks + 1)v, 
r1 < s < 0
= {
r < s < 
0
r1
r
Note that lll::~ll = lvl for all k > 1. Also by (w 1) one has

for all n € C(JJRn). Thus "' a = a. Since the extension map e is linear in the onedimensional case, it follows from the representation of a in terms of the a; that e:C(JJRn) * ~ a
;,
a 81
is also linear. But a is an extension of a, so llall > llall. Conversely, let ~ = n + vx{O} € C(J,Rn) e Fn and construct {~~};= 1 in C(J,Rn) as above. Then
a(~) = lim·a(n k+OO
+
~~).
But Ia( n
+
~~ > I < II a II II n .., ~~ II < II a II [ II n II = II a II
+
II ~ ~ II J
[ II n II .., IvI J = II a II II ~II for a11 k > 1•
Hence Ia(~) I= lim la(n +~~)I< llall II~ k+OO
II for every~
€
C(J,Rn) e Fn.
Thus llall < llall. So lliill = llall and e is an isometry into.
o
Lemma (3.2) :. Let B:C(J,Rn) x C(J,Rn) ~ R be a continuous bilinear map. Then B has a unique (continuous) bilinear extension S:[C(J,P.n) eFn] x [C(J,Rn) E&Fn] +F. satisfying the weak continuity property: (w2) if {~k};= 1 , {nk};= 1 are bounded sequences in C(J,Rn) such that ~k(s)!> ~*(s), nk(s)!> n*(s) as k + oo, for all s € J, for some ~*.n* € ·c(J,Rn) e Fn, then B(~k.nk)+ B(~*.n*) ask+ oo. Proof: Here we also deal first with the 1dimensional case: Write the continuous bilinear map B:C(J,R) x C(J,R) + R as a continuous linear map B:C(J,R) !> C(J,R)*. Since C(J,R) *is weakly complete (Dunford and Schwartz [15], §IV. 13.22, p. 341), B is weakly compact as a continuous linear map . C(J,R) ~ C(J,R) * (Theorem (1.4.2); Dunford and Schwartz [15], §VI. 7.6, p. 494). Hence there is a unqiue measure A:Borel J ~ C(J,R)* (of finite semivariation IIA II (J) < oo ) such that for all ~ € C(J,R)
B(~) = ~~r ~(s)
dA(s)
(Theorem (I.4.1)). Using a similar argument to that used in the proof of Lemma (3.1), the above integral representation of B implies the existence of a unique continuous 82
nnear extension a:C(J,R) C9 F1 ~ C(J,R)* satisfying (w1). To prove this, one needs the dominated convergence theorem for vectorvalued measures (Dunford and Schwartz [15] §IV.10.10, p. 328; Cf. Theorem (1.3.1)(iv)). Define a:C(J,R) e F1 > [C(J,R) e F1J* by B = e o 13 where e is the extension isometry of Lemma (3.1); i.e.
Clearly a gives a continuous bilinear extension of 13 to [C(J·,R) • F1] X [C(JJR) • F1]. To prove that a satisfies (w2) let kco kco k {~ }k= 1' {n }k= 1 be bounded sequences in C(J,R) such that~ (s) + ~*(s), nk(s) + n*(s) as k +co for all s € J, for some~·. n* € C(J,R) e F1• By ,.. ,.. * k (w 1) for 13 we get 13(~ ) = lim 13(~ ). Now for any k, k
,..
But by (w 1) for
13(~*)
we have
lim la(~*)(nk) a(t*)(n*)l = 0 kSince {llnkll };= 1 is bounded, it follows from the last inequality that lim 13(~k)(nk) = ~(~*)(n*). kWhen n > 1, we use coordinates as in Lemma (3.1) to reduce to the 1dimensional case. Indeed write any continuous bilinear map 13:C(J,R)n x. C(J,Rn) ~ R as the sum of continuous bilinear maps C(J,R) x. C(J,R) + R in the following way 13((~1•···•~n),(n1, ••• ,nn>> =
n
..
E 13,Jc~ .• n.) i,j=1 1 J
~~re (~ 1 ••••• ~n>• (n 1, ••• ,nn) € C(J,Rn), ti,ni € C(J,R), 1 < i < n,
13 1 J:C(J,R)
x
C(J,R) +R is the continuous bilinear map 83
aij(~'.n'> = a((o,o, ••• ,o.~·.o, ••• ,o), (o,o, ••• ,o.n·.o •••• ,o)), for ~·, n' € C(J,R) occupying the ith and jth places respectively, 1
where~·= (~1·····~~), n* = (n; ••••• n~) € C(J,Rn) • Fn, ~i· nj € C(J,R) • F1, 1 < i, j < n. It is then easy to see that is a continuous bilinear extension of a satisfying (w2). Finally we prove uniqueness. Let e:C(J,Rn) e Fn ~ [C(J,Rn) e Fn]* be any continuous bilinear extension of a satisfying (w2). Take~· = ~ + v1x{O}' n* = n + v2XV} € C(J,Rn) e Fn' where~. n € C(J,Rn), v1, v2 € Rn. Choose k k n) such th:t ~ k( s ) + v x{O} ( s ) , b~unded sequences {~O}k=t' {n 0}k=t 1n CkJ,R 1 0 n0(s)  > v2x{O}(s) as k~oo, s € J; ll~oll = 1v 1 1. llnoll = 1v 2 1 for all k > 1. koo koo . k k k k Let~ = ~ + ~ 0 • n = n + n0• Then{~ }k= 1' {n }k= 1 are bounded sequences 1n C(J,Rn) such that ~k(s) ~ ~*(s), nk(s) ~ n*(s) as k ~ oo, for all s € J. So by (w2) for "'a and a one gets
a
00
00
•
(
B(~*)(n*) =lim a(~k)(nk) = B(~*)(n*) k~
Thus ~ =
s.
c
For each n € C(J,R0 ) let nx € t 2(n,C([r,a],Rn)) be the solution of (I) through n. and ~t € C(J,Rn) be defined as in §2 for each t € [O,a]. Lemma (3.3):
There is a K > 0 (independent of t,n) such that
II{E(nxt  ~t) II C < K for all t Also if a
€
lim
t~+
Proof: 84
>
0 and n
€
C(J,Rn).
C(J,Rn)*, then 1 nxt "' tEa( nt) = a(H(n)x{O}) for each n
€
C(J,Rn).
Denote by E the expectation for Rn or C(J,Rn)valued random variables.
Let K > 0 be such that IH (n)l < K for all n € C(J,Rn). Now t+s Jt+s n ~ { J H(nxu(w))du + (w) G(nxu(·))dw(·)(u), t+S>O xt(w)(s)nt(s) = 0 0 0 t + s < 0, for each t E [O,a], a.a.w € o, s € J. Since evaluation at each s € J is a continuous linear map C(J,Rn) then it commutes with the expectation i.e.
+
Rn,
1n ~ 1n ~ [Eft< xt nt)}](s) =Eft£ xt(•)(s)  nt(s)]}, s € J n 1 Jt+s = { t 0 E(H( xu))du, 0
t +s>0
t + s <
,
o.
t > 0
using the Martingale property of the Ito integral. Thus lim t J: E(H(nxu))du lim [E{t(nxt ~t)}](s) = { t+0+ 0 r < s < 0 t+0+ =
s =0
H(n)x{O}(s), for all s € J.
We prove next that lit E(nxt  ~t) lie is bounded in t
>
0 and n. Clearly
Jt
~ 1 Jt+s IE(H(nxu))ldul
< K for all t
>
Therefore lit E(nxt  ~t) lie < K for all t If a
0 and n
>
C(J,Rn).
€
o and n
€
C(J,Rn).
C(J,Rn) *, then
€
~ ) 1 (nxt  nt tEa = t1 a ( E(nxt  ~nt)) = a[t1 E( nxt  ~nt)].
By the weak continuity property (w1), one gets .
1
n
~

11m tEa( xt  nt) = a(H(n)x{o}>· t+0+
a
85
Lemma (3.4): For each t
w~(w)(s)
>
0 and a.a. w €
n,
define wt(w)
1 [w(w)(t+s)  w(w)(O)] = {
€
C(JJRn) by
t < s < 0
If r < s < t
0
Let a be a continuous bilinear form on C(JJRn). Then 1 nxt  "' lim [tEa( nt, nxt  "' nt)  Ea(G(n) t+0+
wt, G(n)
o
o
wi)l =
o.
Proof: We prove first that (1)
Observe that 1 Jt+s H(nx )du + 1 Jt+s G(nx )dw(u) If 0 u If 0 u 1 Jt+s dw(u)], t  G(n)[ If 0
<
s<0
r < s < t
0
almost surely.
for t > 0, and some K1 > 0. Since Hand G are Lipschitz with Lipschitz constant L, then almost"surely IH(nxu) 12 < 2IH(n) 12 + 2L 2 11 nxu nil~, u and
86
IIG(nxu)  G(n) 11 2 < L2 llnxu nil~, u
>
0.
>
0;
Hence (3)
and (4)
Now by the main existence theorem (Theorem (11.2.1)), the map [O,a] ?u~o+ xuEi(n,C(J,.r.n)) is conttnuous;. so Hm Ellnxunii 2=0. Therefore the u...O~: last two inequalities (3) and (4) imply that {EIH(nxu>l 2: u E [O,aJ} is bounded and Hm Ell G(nxu)  G(n) 11 2 = 0. u+0+ Letting t+ 0+ in (2) yields (1). Since a is bilinear,
'  a(G(n) o wi• G(n) o wi> tt a( nxt  " nt•' n xt "  nt) a( __ 1 (nxt  ~t)  G(n) o wi• __ 1 (nxt  ~t)  G(n) o wi> If If 1 (nxt~t)G(n)owi•G(n)owi) + a(G(n)owi• l{nxt~t)G(n)owt). + a(If If
=
Thus, by continuity of a and H6lder•s inequality, one gets It Ea(nxt ~t• nxt ~t) Ea(G(n) o wt• G(n) o wi>l < llall
Ell~ (nxt ~t)  G(n) o wi11 2
1 + 211a II [EIIIf for all t > 0.
(nxt~t)  G(n)owi11 2J 112 [E IIG(n)owi11 2J 112 •
(5)
But E IIG(n) o wi11 2 < E sup I w(t+s~w(O)I 2 IIG(n) 11 2 sE[t,O] = =
t E SlAP lw(t+s)w(O) 12 IIG(n) 11 2< t t IIG(n) 11 2 sE[t,OJ IIG(n) 11 2 • for all t > 0. (6) 87
Comb;n;ng (6) and (5) and lett;ng t + 0+ g;ves the requ;red result.
a
Lemma (3.5): Let ;n~n + Fn be the ;somorph;sm ;n(v) = vx{O}' v € Rn, and G(n) x G(n) denote the l;near map
(v,.v 2)
~
(G(n)(v 1), G(n)(v 2)).
Then for any cont;nuous b;l;near form Bon C(J,Rn) Hm t+O+
t EB(nxt~t•nxt~t) = trace [B
o
(;n x in)
o
(G(n)
x
G(n))]
for each n € C(J,Rn), where B ;s the cont;nuous b;l;near extens;on of B to C(J,Rn) • Fn (Lemma (3.2)). Indeed ;f {ej}J= 1 ;s any bas;s for Rm, then . "' nxtnt) "' = .Em B(G(n)(ej)X{o}•G(n)(ej)X{o}>· 11m t1 EB( nxtnt• t+O+ J=1
Proof:
In v;ew of Lemma (3.4) ;t ;s suff;c;ent to prove that m
1;m EB(A t+O+
o
wt• A o wt) = _E B(A(eJ.)X{O}' A(eJ.)X{O}) J=1
(7)
for any A € L(Rm ,Rn). We deal f;rst w;th the case m = n = 1, v;z. we show that 1;m EB(wi• wi> = B(x{o}•X{o}> t+O+
(7)'
for onedimensional Brown;an mot;on w. If t, n € C = C(J,R), lett 8 n stand for the funct;on J x J +R def;ned by (t 8 n)(s,s') = t(s)n(s')for all s, s' € J. The projective tensor product C 8~ C is the vector space of all functions of the form E~= 1 ti 8 ni where ti,ni € C, i = 1,2, ••• ,N. It carries· the norm N N llhll 8 = inf { E llt11 lin; II :h = E ti 8 ni,ti,ni €C, i=1,2, ••• ,Nl. i =1 ~ i =1 1 The infimum is taken over all possible finite representations of h € C 8~ C. Denote by C i~ C the completion of C 8~ C under the above norm. It is well known that C ~~ C is continuously and densely embedded in C(J x J,R), the Banach space of all continuous functions J x J +R under the supremum norm 88
(Treves [75], pp. 403410; §1.4). Since C is a separable Banach space, so is C in C. . countable dense subset of C. Then the countable set N
Y8 Y = { t
~
i=1
1
For let Y c C be a
8 n. : ~ 1.,n 1• E Y, i = 1, ••• ,N, N = 1,2, ••• } 1
is dense in C en C and hence in C in C. The continuous bilinear form Bon C corresponds to a continuous ZineaP ,.. C] * (Treves [75] pp. 434445; Cf. Theorem 1.43). functional "'BE [C ew Now let ~,.~2 e t2(g,c). The map
cec+ cenc (~.n)
~~en
is clearly continuous bilinear. Thus ~, ( •)
e ~2 (.)
:
n  > c in c w1
is Borel measurable.
for almost all wE
>
~ 1 (w) 8 ~ 2 (w)
But
n;
hence by HOlder's inequality the integral
fn 11~ 1 (w) 8 ~2 (w) 11 8
dP(w) exists and
n
From the separability of C •n C the Bochner integral (§1.2) E
~ 1 (.) 8 ~2 (·)
=
J0 ~ 1 (w) 8 ~2 (w)dP(w)
exists in C•n C. Furthermore, it commutes with the continuous 1inear functional "'B; viz.
89
Fix 0 < t < r and cons;der
t E[w(·)(t+s)  w(•)(O)][w(·)(t+s')  w(•)(O)],
s,s• € [t,O]
= {
s
0 1 +
[r,t) or s•
€
t m;n (s,s•)
s,s•
€
€
[r,t).
[t,O]
= {
s
0
[r,t) or s•
€
€
[r,t) (9)
Def;ne Kt:J x J +R by lett;ng Kt(s,s•) = [1 +
t m;n (s,s')]X[t,O](s)x[t,O](s•)
s,s• € J
(10)
;.e. Kt
=
E[wt(•)
€ £ 2(n,C),
S;nce wt
8
wt(•)].
;t ;s clear from (8) that Kt
ES(wt(•), wt(·))
=
"'B(Kt)
€
C e~ C and
•
( 11)
In order to calculate 1;m ~(Kt)• we shall obta;n a ser;es expans;on of Kt. t+O+ We appeal to the follow;ng class;cal techn;que. Note that Kt ;s cont;nuous; so we can cons;der the e;genvalue problem
t
r
Kt(s,s•)~(s')ds'
=
'~(s)
s
€
J
(12)
s;nce the kernel Kt ;s symmetr;c, all e;genvalues 'of (12) are real. (10) rewr;te (12) ;n the form t
J0t ~(s')ds' + Jt0 m;n(s,s•)~(s')ds' '~(s) =
Therefore 90
0
= 't~(s)
s € [t,O] (;)
s € [r,t)
(;;)
} (13)
t J0 E;(s')ds'.., Js s'E;(s')ds' t t
+
s J0 E;(s')ds' = UE;(s) s
sE[t,O] ( 14)
Differentiate ( 14) with respect to s, keepi.ng t fixed, to obtain
t
( 15)
E;(s')ds' = U E;'(s), s E (t,O]
s
Differentiating once more, E;(s) = ).t E;"(s) ,
s
E
(t,O]
(16)
When).= 0, choose E;~~J +R to be any continuous function such that ~~(s) = 0 for all s E [t,O] and normalized by
frt t;0(s) 2 ds = t. t
Suppose)."/ 0. Then (13)(ii) implies that E;(s) = 0 for all s E [r,t)
( 17)
In (14) put s = t to get ~(t)
(18)
= 0·
In (15) put s = 0 to obtain ~·co> =
o
( 19)
Hence for). I 0, (12) is equivalent to the differential equation (16) coupled with the conditions (17), (18) and (19). Now solutions of this are given by
A,eth~is
+
A2eth~is • s E [t,O]
E;(s) = {
(20) s E [r,t), i = rT
0
Condition (19) implies immediately that A1 = A2 = 1, say. e
t~A~it
+
e
t~A~it
From (18) one gets
= 0.
Since the real exponential function has no zeros, it follows that ).~ cannot 91
he
imaginary i.e. A > 0.
Being a covariance function, each kernel Kt is non
negative definite in the sense that
tt
Kt(s,s')t(sH;(s')ds ds' > 0 for all t
€
c.
Using (18), we get the eigenvalues of (12) as solutions of the equation t(t) = 2 cos [  1 (t)] = 0 •
liT
Therefore the eigenvalues of (12) are given by k = 0,1,2,3, •••
(21)
and the corresponding eigenfunctions by t (2)i (2k+1)~s tk(s)= t x[t,OJ(s) cos [ 2 t J,
s € J, k = 0,1,2, •••
(22)
after being normalized through the condition t 2ds = 1, k = 0,1,2, ••• JJ tk(s)
Now, by Mercer's theorem (Courant and Hilbert [10] p. 138, Riesz and SzNagy [68] p. 245), the continuous nonnegative definite kernel Kt can be expanded as a uniformly and absolutely convergent series 00
Kt(s,s') = E Ak t~(s) t~(s'), k=O 00
E
k=O = {
0
8
~2(2k+1)2
(23)
s,s' € J
cos [(2k+1)~s] cos [(2k+1)~s·J 2t 2t s,s' € [t,O]
(24)
s € [r,t) or s' € [r,t)
But from the definition of Kt' one has Kt(O,O) = 1 for every t putting s = s' = 0 in (24) we obtain
>
0. Thus (25)
From the absolute and uniform convergence of (24), it is easy to see that Kt can be expressed in the form 92
(26) where ~
~k(s) =cos [
(2k+1)ns zt Jx[t,O](s),
s € J.
Note that the series (26) converges (absolutely) in the projective tensor ,.. product norm on C en C. Hence we can apply "'B to (26) getting from (11) the equality EB(wi(·),wt(.)) =
=
"" 8 "' '!1: E 2 2 B(~k k=O n (2k+1)
'!1:
8 ~k)
""E
8 B(~ ~t) k' k k=O n2(2k+1)2
(27)
But II~ II< 1 for all k > 0 and all 0 < t < r; so the series..,(27) is uniformly convergent in t, when compared with the convergent series E 2 8 2• ~ k=O n (2k+1) Moreover, for each s € J, ~k(s) ~ x{O}(s) as t + 0+, k = 0,1,2, •••• Thus if we lett+ 0+ in (27), we obtain
""
= E 2 8 2 S(x{O}' X{O}) k=O n (2k+1)
using (25) and Lemma 2. This proves (7)'. For dimensions n > 1, write e:C(JJRn) X C(JJRn) +R in the form n
..
.
.
e<~1.~2> = . ~ e,J<~~.~~> 1 ,J=1  ( 1 n) lJ. ( ) ( ) where ~ 1  ( ~ 11 ••••• ~ n) 1 , ~ 2  ~ 2 •••• ,~; 2 and each B .C JJR x C JJR ~ R is continuous bilinear. Let A € L(Rm, Rn) and {ek}~= 1 ' {ei}~= 1 be the canonical bases for Rm and Rn, respectively; i.e.
93
Write mdimensi.onal Brownian motion w in the form w = (w1,w2,. ••• ~) where wk (t) = ~(t),ek>' k = 1, ••• ,m, are independent onedimensional Brownian motions. Then
Letting t
=
n 1: i ,j=1
=
n 1: i ,j=1
+
0+ and using (7)' gives
lim Ea(Aowt,Aowt) t+O+
=
m 1: k=1
To obtain the final statement of the lemma, take A = G(n) and note that the last trace term is independent of the choice of basis in Rm. c Let V(S) c C~ be the domain of the weak generator S for the shift semigroup {St}t>O of §2. We can now state our main theorem which basically says that if' € V(S) is sufficiently smooth, then it is automatically in V(A). Furthermore, A is equal to S plus a second order partia~ differential operator on C(J,Rn) taken along the canonical direction Fn. The following conditions on a function ':C(J,Rn) + R are needed. Conditions (OA): (i )
' € 1)( s)t
( i i) ' is c2; (iii) 0,, o2' are globally bounded; (iv) o2, is globally Lipschitz on C(J,Rn). Theorem (3.2): Suppose ':C(J,Rn) ~ R satisfies Conditions (OA). Then 94
• €
O(A) and for each n
C(JJR")
€
A( 4> )(n) = S( l!>)(n)+( Dl!>(n )o in)(H(n))+Hrace[D24>(n) o( in Xin) o(G(n) XG(n))]
where~. D24>(n) denote the canonical weakly continuous extensions of o•(n) and D24>(n) to C(JJR"> e Fn' and in~" ~ Fn is the natural identification v t> vx{O} • Indeed if {ej}J= 1 is any basis for R", then n :2:::'

A(l!>)(n) = S(l!>)(n)+DII>(n)(H(n)x{O})+i j: 1o l!>(n)(G(n)(ej)X{o}•G(n)(ej)X{o}>· ~: Fix n € C(JJR") and let nx be the solution of the SRFDE (I) through n. Suppose 4> satisfies (DA). Since 4> is c2, then by Taylor•s Theorem (Lang [47]) we have
where (28)
Taking expectations, we obtain
Since 4>
€
O(S), then (30)
In order to calculate lim following two limits
t~+
t E[ll>(nxt)  l!>(n)], one needs to work out the
"' n ~+ t1 E04>(nt)( xt  "' nt)
lim
t~O+
t ER2(t)
We start by considering (31). that
(31) (32)
From Lemma (3.3), there exists a K > 0 such
95
Hence
Let t
+
0+ and use the cont;nu;ty of 0' at n to obta;n
"n ' = 1;m t1 EO,(n)( n " nt) ' 1;m t1 EO,(nt)( xt "  nt) xt
t~+
t~+
(33) by Lenvna (3.3). Secondly, we look at the l;m;t (32). Observe that ;f K ;s a bound for H and G on C(JJRn) and 0 < t < r, then
< 8K4t 4 + 8K2t
J:
Ell G(nxu) 11 4 du
< "'K(t4 + t 2 ), some K2, "'K > 0,
(34)
where we have used Theorem (1.8.5) for the Ito ;ntegral. Furthermore, ;f u € [0,1] and 0 < t < r, then t 2'(nt+u( "' nxtnt))( "' nxtnt, "' nxtnt) "' Il£0
<
t 2'(n)( nxtnt, "' nxtnt) "' 1  l£0
t E llo2'(~t + u(nxt~t))  o2,(n) II
II nxt~tll2
< [E llo2,(?) + u(nx ~ ))o2'(n) 112J 1/2 [ 1 Ell nx ~ 114J 1/2 tt t2" tt t < ~1/2(t2+1)1/2[E llo2'<~t+u(nxt~t)) o2'(n)112J1/2. 96
(35)
But o2$ is globally Lipschitz., ~i.th Li.pschi.tz..constant L say;. so E IID 2 $(~t""u(nxC~t))  D2$(n) 11 2 < L2E( II ~tnll
+
II nxt~tll >2
< 2L 2 11~t  n11 2 ... 2L 2 [~1nx t  ~t 11 4 J 112 < 2L 2 ll~t  n II 2 ... 2L 2~ 1 1 2 t(t 2 because of the inequality (34). . tED 1 2$(nt "' 11m
+
t+O+
+ 1) 112
Letting t
+
(36)
0+ in (35) and (36), we obtain
u( nxt "' nt))( nxt "' nt• nxt "' nt) (37)
uniformty in u E [0,1].
1 =~
n
:r.;:
.r o $(n)(G(n)(eJ.)X{o}•G(n)(eJ.)X{o}>·
J=1
(38)
Since $ E V(S) and has its first and second derivatives globally bounded on C(JJRn). it is easy to see that all three terms on the right hand side of (29) are bounded in t and n. The statement of the theorem now follows by letting t + o... in (29) and putting together the results of (30), (33) and (38).
c
It will become evident in the sequel that the set of all functions satisfying Condition (DA) is weakly dense in Cb. Indeed within the next section we exhibit a concrete weakly dense class of functions in Cb satisfying (DA) and upon which the generator A assumes a definite form. §4. Action Of the Generator on Quasitame Functions The reader may recall that in the previous section, we gave the algebra Cb of all bounded uniformly continuous functions on C(JJRn) the weak topology induced by the bilinear pairing ($,lJ). ~ J n $(n)dlJ(n), where C(JJR ) ~ € Cb and lJ runs through all finite regular Borel measures on C(JJRn). Moreover, the domain of strong continuity C~ of {pt}t>O is a weakly dense proper 97
subalgebra of Cb. Our aim here is to construct a concrete class Tq of smooth functions on C(JJRn), viz. the quasitame fUnctions, with the following properties: (i)
Tq is a subalgebra of C~ which is weakly dense in Cb;
(ii)
Tq generates Borel C(JJRn);
(iii) Tq c V(A), the domain of the weak generator A of {pt}t>O; (iv)
for each ~ € Tq and n € C(JJRn), A(~)(n) is a secondorder partial differential expression with coefficients depending on n.
Before doing so, let us first formulate what we mean by a tame function. A mapping between two Banach spaces is said to be cPbounded (1 < p < m) if it is bounded, cP and all its derivatives up to order p are globally bounded; e.g. Condition (DA) implies c2boundedness; and c3boundedness implies (DA)(ii), (iii), (iv). Definition (4.1)
(Tame FUnction):
A function ~:C(JJRn) +R is said to be tame if there is a finite set {s 1,s 2, ••• ,sk} c Janda embounded function f:(Rn)k +~such that ~(n) = f(n(s 1), •••• n(sk)) for all n € C(JJRn).
(*)
The above representation of ~ is called minimal if for any projection p:(Rn)k + (Rn)k 1 there is no function g:(~n)k 1 +R with f =gop; in other words, no partial derivative Djf:(Rn)k + L(RnJR), j = 1, ••• ,k, off vanishes identically. Note that each tame function admits a unique minimal representation. Although the set T of all tame functions on C(JJRn) is weakly dense in Cb and generates Borel C(JJRn), it is still not good enough for our purposes. due to the fact that •most• tame functions tend to lie outside c~ (and hence are automatically not in V(A)). In fact we have Theorem (4.1): (i) The set T of all tame functions on C(J,Rn) is a weakly dense subalgebra of Cb' invariant under the shift semigroup {St}t>O and generating Botel C(JJRn). 98
(ii)
Let
~
E T have a minimal representation
~(n) = f(n(s 1), ••• ,n(sk))
where k > 2.

Proof:
Then
~ ~
n E C(JJRn)
0
Cb.
For simplicity we deal with the case n = 1 throughout.
lt is easy to see that T is closed under linear operations. We prove the closure ofT under multiplication. Let ~,.~ 2 E T be represented by $1(n) = f1(n(s 1), ••• ,n(sk)), $2(n) = f 2 (n(s1)~ ••• ,n(s~ )), for all n E C(JJR), where f 1 ~k +R, f 2 :Rm +Rare C bounded functions and s 1, ••• ,sk' s,, ••• ,s~ E J. Define f 12 :Rk+m +R by (i)
f 12 cx 1, ••• ,xk, x,, ••• ,x~) = f 1(x 1, ••• ,xk)f 2 (x1•····x~) for all x1, •.• ,xk, x,, ••• ,x~ E R.
00
Clearly f 12 is C bounded and
Thus ~,~~ E T, and T is a subalgebra of Cb. It is immediately obvious from the definition of St that if ~ E T factors through evaluations at s 1, ••• ,sk E J, then St($) will factor through evaluations at t + sj < 0. So T is invariant under St for each t > 0. Next we prove the weak density of T in Cb. Let T0 be the subalgebra of Cb consisting of all functions $:C(JJR) +R of the form ~(n) =
f(n(s 1), ••• ,n(sk)), n E C{JJR)
( 1)
when f:Rk +R is bounded and uniformly continuous, s 1, ••• ,sk E J. Observe first that T is (strongly) dense in T0 with respect to the supremum norm on Cb. To see this, it is sufficient to prove that if c > 0 is given and f:Rk +R is any bounded uniformly continuous function on Rk, then there is a k C bounded function g:R k +R such that lf(x)  g(x)l < c for all x E R. We Prov.e this using a standard smoothing argument via convolution with a C bump function (Hirsch [32], pp. 4547). By uniform continuity of f there is a 0 > 0 such that lf(x 1) f(x 2 )1
00
R
99
Define g~k +R by
the integral being a Lebesgue one on Rk. g(x) = J, k h(y)f(xy)dy R
=I
8(0,15)
h(y)f(xy)dy, x ERk.
(2)
By choice of 6 and property of h, it follows that lf(x)  g(x) I
= I JRk
h(y)f(x)dy  IRk h(y)f(xy)dyl
E
J, k h(y)dy R
= E
for all x € Rk. To prove the smoothness of g, use a change of variable y• = xy in order to rewrite (2) in the form g(x)
=
J
B(x,IS)
h(xy')f(y')dy'
Now fix x0 € Rk and note that B(x,IS) Therefore g(x)
= 
I
B(x 0 ,2&)
c
for all x
€
Rk.
(3)
B(x 0,2&) whenever x € B(x 0,&).
h(xy')f(y')dy'
for all x € B(x 0,&).
(4)
Since f is continuous and the map x ~ h(xy•) is smooth, it follows from (4) that g is em on B(x 0 ,&) and hence on the whoLe of Rk, because x0 was chosen arbitrarily. Indeed g and all its derivatives are globally bounded on Rk, for II oPg(x) II <
fB(x ,2&) II oPh(xy• >II
lf(y• >ldy'
0
< V.
supk IIDph(z) 11. supk lf(z) I
z~
z~
=
N, say,
for all x € Rk, where N is independent of x0 and V = J dy' is the 8(0,26) . volume of a ball of radius 215 in Rk. Secondly we note thatTOis weakly dense in Cb. Let ITk:r = s 1 < s 2 < s 3 < ••• < sk = 0, k = 1,2, ••• , be a sequence of partitions of J such that mesh nk + 0 as k + m. Define the continuous linear embedding Ik:Rk + C(JJR) by letting lk(v 1,v 2, ••• ,vk) be the piecewise 1inear path = (ssj_ 1) v. + (sjs) lk(v 1, ••• ,vk)(s) vJ._ 1 , s € [sJ._ 1,sJ.] sjsj_ 1 J sj 5 jt 100
joining the points
\L,. ... ,'lk E R,
j
= 2~ ••• ,k.
R
rs,
Denote by !k the ktuple (st•···•sk) E Jk, and by.p!k the map e(J,R) ~ Rk n
~
(n(st), ••• ,n(sk)).
Employtng the unifo.rm. continuity of .each n E. e(J,R) on the compact interval J, the reader may easily check that lim (Ik o.p k)(n)
!
k.....
=n
in e(JJR).
(5)
Now if' E eb' define 'k~e(J,R) +R by 'k = 'olk 0 Psk• k = 1,2, •••• Since' is bounded and uniformly continuous, so is ' o lk~ Rk + R. Thus each 'k E T0 and lim 'k(n) = '(n) for all n E e(J,R), because of (5). Finally k.....
note that II 'k II e < b
II' lie for a11 k > 1. Therefore ' = wHm 'k and b
k
T0
is
weakly dense in eb. From the weak density of T in T0 and of T0 in eb' one concludes that T is weakly dense in eb. Borel e(J,R) is generated by the class {, 1 (u)~U c:R open, 'E T}. For any finite collection !k = (s 1, ••• ,sk) E Jk let Psk ~ e(JJR) +Rk be as befor Write each' E Tin the form'= f o P5 k for someembounded f~k +R.
It is 101
a: 8
,}~~~r·,~~,0 I I I
. r
1
I I
.a
:.
~~
.e....
+~ ~
.a + i
• .6

~
.aI
~
102
wellknown that Borel C{JJR) is generated by the cylinder sets {psk 1 (U 1 x ••• xUk):Ui =:R open, i = 1, ... ,k,!k=(s 1, ... ,sk) e:J k, k=1,2, ... } (Parthasarathy [66] pp. 212213). Moreover, it is quite easy to see that Borel Rk is generated by the class {f 1(U) : U=:R open, f:Rk ..... R C00bounded} (e.g. from the existence of smooth bump functions on Rk). But for each open set U =R, $ 1(U) = P;~ [f 1{U)]. Therefore it follows directly from the above that Borel C(JjR) is generated by'· (ii) Let $ e: < have a minimal representation $= f o Psk where !k = (s 1, ••• ,sk) e: Jk is such that r < s 1 < s 2 < ••• < sk <0, f:Rk ..... R is Ceobounded and k > 2. Take 1 < jo < k so that r < Sj 0 < 0. Since the representation of $is minimal, there is a ktuple (x 1, ••• ,xk) e: Rk and a neighbourhood [xj 0E0 ,xj 0 + E0 J of xj 0 in R such that oj 0f(x 1, ••• ,xj 0_1,x,xj 0+1 , ••• ,xk) '0 for all x e: [x.
 E0 , x. + E0], with EO> 0. Jo Jo + E0 ] ..... R by
Define the function
g:[x. E 0 , x. Jo Jo g(x) = f(x 1 , ••• ,x. _1,x,x. 1 , ••• ,xk) for all x e: [x. E 0 ,x. +E 0]. Jo Jo+ Jo Jo co
Hence Dg(x) ' 0 for all x e: [x. E 0 , x. + E0] and g is a C diffeomorphism Jo Jo onto its range. Therefore, there is a A > 0 such that lx'  x"l
< Alg(x') g(x")l for all x', x"e:[x. E 0 ,x. +E 0] Jo Jo
Pick o0 > 0 so that o0
(6)
(sj2o 0 , sj+2o 0 ), j =1,2, ••• ,k, of the argument we may assume, n are such that~< o0 • Conthe picture opposite
103
viz.
xj
sjoO < s
 0) 010 x.(ss.o J J
xJ.
sj+o 0 < s < sj+2o 0• j
~
~ x.(ss.+o0) + xJ.
sj2o 0 < s < sjo 0, j
 01o x.Jo (ss.Jo o 0)+x.Jo
s. +o0 < s < s. + 2o0 Jo Jo
0
nn(s) =
+
J
J
j0 ~
jo
(7)
0
Suppose, if possible, that ' € Cb. Then lim St(,)(nn) = lim f(nn(t+s 1), ••• ,nn(t+sk))=f(nn(s 1h···•nn(sk)) t+0+ t~+ unifo~ty in n.
But, for j ~ jo and 0 < t < o0, nn(t+sj) = xj fo~ att n.
Therefore
unifo~ty
inn; i.e. for any E > 0, there is a 0 < o < o0 such that
lg(nn(t+sJ. ))  g(nn(sJ. ))I < E for all n, all 0 < t < o. . 0 0
(8)
Now suppose 0 < t < o0• If*< t < o0, nn(t+sj 0)  xj 0 = o. If o < t < *· then lnn(.t+s.) x. I= lnE 0t + x. + .E0  x. I = E. 0 (1nt)
Note that 6 is independent of n. ln the above inequality, fix t < 6 and choose n0 large enough such that~< t. Then s. * < t + s. < s. + 60 , n
and n °(t
+
s. )
Jo
lnn°(t
+
=
x. ~so
i0
Jo
0
Jo
Jo
Jo
s. )  nn°(s. ) I
Jo
Jo
= lx.
Jo
(x.
Jo
+ £ 0)
which clearly contradicts the arbitrary choice of
I = £0 < l£
£.
Therefore
<1>
t
C~.
a
Definition (4.2) (Quasitame FUnctions)~ A function ~C(JJRn) +R is quasitame if there is an integer k > 0, C~bounded maps fj~n +Rn, 1 < j < k1, h:(Rn)k +Rand piecewise c1 functions gj:J +R, 1 < j < k1, such that
=
h( J:r f 1(n(s))g 1(s)ds, ••• ,
J~r
fk_ 1(n(s))gk_ 1(s)ds,n(O))
for all n € C(J,Rn). Each derivative gj is assumed to be absolutely integrable over J. Denote by Tq the set of all quasitame functions on C(J,Rn). Theorem (i)
(4.2)~
Tq
c
V(s)
m(n)
=
c~.
c
r
Indeed if
=
hom € Tq where
f 1 (n(s))g 1(s)ds, ••• ,
t
r
fk_ 1(n(s))gk_ 1(s)ds,n(O))
1, ••• ,k1, are as in Definition (4.2), then k1 S(
and h.,fj ,gj, j
=
 J0 fj(n(s))gj(s)ds} r
for all n € C(J,Rn). The function h:(Rn)k +R is considered as one of k n~dimensional variables and Djh are the corresponding partial derivatives (J = 1, ••• ,k).
(ii) Tq is invariant under the shift semigroup {Sf t>O" 105
(iii)
Tq is a weakly dense subalgebra of Cb generating Borel C(JJRn).
Proof: In proving statements (i) and (ii) of the theorem, we shall assume for simplicity that $ = hom where h:(Rn) 2 +R is C~bounded and m(n) = cf0 f(n(s))g(s)ds, n(O»for some C~bounded map f~n +Rn and a r
(piecewise) c1 function g:J +R. Let 0 < t < r and consider the expression { [$(~t)  $(n)J
={ =
[h(m(~t))  h(m(n))]
J: Dh(z~){¥m
using the MeanValue Theo~em for h, with z~ But {rm
 m(n)J}du =
(10)
(1u)m(n) + um(nt), 0 < u < 1.
frt f(n(t+s))g(s)ds + i Jot f(n(O))g(s)ds 0
{ J
f(n(s))g(s)ds,O)
r
=
cf 0
tr
f(n(s)) t{g(st)g(s)}ds + f(n(O)) t
Jt0 g(s)ds
tr  { Jr f(n(s))g(s)ds,O).
Now f is bounded and g is (piecewise) c1, so all three terms in the above expression are bounded in t and n. Moreover letting t + 0+ we obtain lim t+O+
¥m<~t)m(n)J
=
<J 0 f(n(s))g'(s)ds+f(n(O))g(O)f(n(r))g(r),O) r ( 11)
As Z~ is bounded inn and continuous in (t,u), it follows from (10), (11) and the dominated convergence theorem that $ € V(S) and S($)(n)
106
=
Dh(m(n)){lim ¥m<~t)  m(n)J} t+O+
=
D1h(m(n)){f(n(O))g(O)f(n(r))g(r) 
J~rf(n(s))g'(s)ds}.
For •q to be invari.ant under {St}t>O it i.s suffi.cient to prove that it is invariant under St• 0 < t < r, due to the semigroup property. So let $ be as above and t E [O,r]. Then St($)(n) = h(J0 f(n(s))g(st)ds tr
+
f(n(O)) J0 g(s)ds,n(O)) t
for all n E C(J,R'n). Define "' g:.J +:R by "' { g(st) tr < s < 0 g(s) = 0 r < s < tr Then clearly ~is piecewise c1 with g' absolutely integrable over J. also ~:C(J,Rn) + (:Rn) 2• F:(:Rn) 2 + (:Rn) 2 and ~:.(:Rn) 2 + :R by
~(n) =
F(x,y) = f(y)
tr
Define
f(n(s))g(st)ds,n(O)),
Jt0 g(s)ds, h(x,y) = h[F{x,y)
+
(x,y)],
for all x,y ERn, n E C(J,Rn). Since f and hare c=bounded, so is F and hence "' h. Therefore St($) = "'"' hom E •q· We show that •q is a weakly dense subalgebra of Cb. It is an easy matter checking that •q is closed under addition and multiplication of functions in Cb. To prove weak density of •q in Cb" it is sufficient to show that •q is weakly dense in • u •q• because the set of tame functions • is already weakly dense in Cb (Theorem (4.1)(i)). Let$ E • have the representation $(n) = f(n(s 1) ••••• n(sk))
n E C(J,P.n)
where f:(:Rn)k +:R is c=bounded and s 1•••• ,sk E J. For each 1 < j < k, construct a sequence {gj}== 1 of piecewise linear functions on J with the property that gj(s)+ X{s.}(s) as m+ =and lgj(s)l < 1 for all s E J, for all m> 1; e.g.
J
107

1
0
r
Also choose a sequence {6m}== 1 of C~ bump funct;ons on Rn such that lxl < m lxl >m+1
0
and l6m(x)l < 1 for all x € Rn. Def;ne the sequence
(12)
Therefore '(n) = 1;m f(J 0 JII'+OO

r
tm(n(s))g~(s)ds, ••••
J0 r
tm(n(s))g~(s)ds)
(13)
Denot;ng the expression under the l;m;t;ng operat;on ;n (13) by 'm(n), ;t ;s clear that·'m € Tq for each m and l'm(n)l ^{
p!k:C ( JJRn)
+ ( Rn)k ,
! k = (s 1, ••• ,sk ) , sj E J, J. = 1, ••• ,k, k=1,2, •••
=
. aut this is agai.n generated by sets of the form {n:.n(sj) E C} where C Rn is closed. Now each closed set in Rn is a countable intersection of complements of open balls, so we need only prove that if BeRn is any open ball and B' is its complement, then {n:.n(sj) E B'} E a(Tq) for any sj E J. Suppose B has radius b > 1 and for each integer p > 0 let Up be the complement of the concentric closed ball of radius b  ~ Let the sequences {~}m= 1 and {gJ}m= 1 be as before, so that (12) is satisfied. Let ~mE Tq be given by
~m(n)
Jr ~(n(s))gj(s)ds,
= 0
We contend that
m> 1, n E C(JJRn).
... n
p=t
lim inf
m
{n:~m(n)
E UP}
(14)
To see this, let n(sj) E B'. Then n(sj) E UP for all p > 1. Since n(sj) = lim ~m(n), then for each p there is an m0 > 0 such that ~m(n) E UP for all
m
m> m0• Hence, for each p > 1, belongs to the set on the right lim inf {n:~m(n) E UP} for each m m0 > 1 such that ~m(n) E Up for
n belongs to lim inf {n:~m(n) E UP} i.e. n m hand side of (14). Conversely, let n be in p > 1. Then for every p > 1, there is an
all m > m0• Taking m + '"' gives n(sj) E Up for all p > 1 i.e. n(sj) E Up. But B' = Up, so n(sj) E B'. Thus p=1 p=1 our contention is proved. As the sets {n:~m(n) E UP} are clearly in a(Tq), it follows directly from (14) that {n:n(sj) E B'} E a(Tq). Therefore a(Tq) =Borel C(J,Rn). a
n
n
The final result in this chapter asserts that every quasitame function is in the domain O(A) of the weak generator A of {Pt}t>O. Theorem (4.3): Every quasitame function on C(J,Rn) is in the domain of the weak generator A of {pt}t>O. Indeed if ' E Tq is of the form '(n) = h(m(n)), n E C(J,Rn), where
109
then A{~){n)
k1 = j;1 Djh{m{n)){fj{n{O))gj{O)fj{n{r))gj{r)
 J0
r
+
Dkh{m{n)){H{n))
+
fj{n{s))gj{s)ds}
t trace [D~h{m{n))o(G{n)
x
G{n))]
for all n E C{JJRn). Here again D.h denote the partial derivatives of h:{Rn)k +R considered as a functi~n of k ndimensional variables. Proof: To prove that Tq c V{A), we shall show that each ~=homE Tq satisfies®Conditions {DA) of §3. First, it is not hard to see that each ~ E Tq is C • Also by applying the Chain Rule and differentiating under the integral sign one gets

D~{n){~)
+
= Dh{m{n)){J 0 r
o
Dh{m{n)){ J
r
Df 1 {n{s)){~{s))g 1 {s)ds, ••• ,
2
D f 1 {n{s)){~ 1 {s).~ 2 {s))g 1 {s)ds, .•• ,
for all n, ~. ~,.. ~ 2 E C{JJRn). Since all derivatives of h, f., 1 < j < k1, are bounded, it is easy to see from the above formulae thatJD~ and o 2 ~ are bounded on C{JJRn). By induction it follows that~ is C bounded. Hence Codnitions {DA){ii), {iii), {iv) are automatically satisfied. Condition {DA){i) is fulfilled by virtue of Theorem {4.2){i). From the above two formulae we see easily that the unique weakly continuous extensions D~(n), o 2 ~{n) of D~{n) and o 2 ~{n) to ®
110
c(J,Rn) a Fn are given by D$(n)(vx{O}) = Dh(m(n))(O, ••• O,v) = Dkh(m(n))(v), ~
D $(n)(v 1x{o}•v 2x{o}> = o2h(m(n))((O, ••• ,o,v 1), (O, •••• o,v 2)) 2 = Dkh(m(n))(v 1,v 2),
for all v, v1,v 2 ERn. The given formula for A($)(n) now follows directly from Theor~~ (3.2) and Theorem (4.2). a Definition (4.3) (Dynkin [16]): Say n° E C(J,Rn) is an absoPbing state for the trajectory fietd {nxt:t > 0, n € C(J,Rn)} of the stochastic FDE(I) if p{w:w En, noxt(w) = n°} = 1 for all t > 0, where we take a =~here, i.e. p(O,n°,t, {n°}) = 1 for all t > 0. The following corollary of Theorem (4.3) gives a necessary condition for n° E C(J,Rn) to be an absorbing state for the trajectory field of the stochastic FDE (I). Corollary: Let n° E C(J,Rn) be an absorbing state for the trajectory field of the stochastic FOE (1). Then (i) n°(s) = n°(0) for all s E J i.e. n° is constant; (ii) H(n°) = 0 and G(n°) = 0. ~:
Let n° E C(J,Rn) be an absorbing state for {nxt:t > 0, n E C(J,Rn)}. For each t > 0 and s E J, define the Ftmeasurable sets 0
0
nt = {w:w E n, n xt(w) = n } 0
Qt(s) = {w:w En, n xt(w)(s) = n°(s)}. Then nt c Qt(s) for all t > 0, s E J and since P(nt) = 1, it follows that P[Qt(s)] = 1 for t > 0, s € J. Suppose if possible that there exist s 1,s 2 E J such that n°(s 1) ~ n°(s 2). 0 Without loss of generality take r < s 1 < s 2 < o. For each wEn, n x5 _ 5 (w)(s 1) = n°(s 2) ~ n°(s 1) and so ns _5 (s 1) = 2
1
~.
2
1
Hence P[Q5 s (s 1)J = 0, which contradicts P[Qt(s)] = 1, 2
1
111
t > 0, s € J. So n° must be a constant path. Th;s proves (;). To prove that n° sat;sf;es (;;), note that the absorb;ng state n° must sat;sfy A(~)(n°) = 0 for all ~ € V(A) (Dynk;n [16], Lemma (5.3), p. 137). In part;cular A(~)(n°) = 0 for every quas;tame ~:C(J,Rn) +R. Note f;rst that s;nce n° ;s constant then so ;s the map t + ~ and so S(~)(n°) = 0 for every~ € V(S). Take any ~:Rn +R C~bounded and def;ne the (quas;)tame funct;on ~:C(J,Rn) +R by ~(n) = ~(n(O)) for all n € C(JJRn). Then by Theorem (4.3), ~ € V(A) and
A(~)(n°) = D~(n°(0) )(H(n°))
+
i trace [D ~Cn°(0) ) (G(n°) 2
0
x
G(n°) )] = 0
The last ;dent;ty holds for every C~bounded ~:Rn +R. Choose such a~ w;th the property D~(n°(0)) = 0 and o 2 ~(n°(0)) = <·,·>, the Eucl;dean ;nner product on Rn e.g. take "'~ to be of the form "'~(v) = lv  no(0) I2 ;n some ne;ghbourhood of n°(0) ;n Rn. Then
for any basis {eJ. }mJ·= 1 of Rm. Therefore G(n°) = o. Thus D~(n°(0)) (H(n°)) = 0 ~ n ~ 1\1 ~~~~ for every C bounded ~:R +R. Now p;ck any C bounded A such that ~(v) = for all v ;n some ne;ghbourhood of n°(0) ;n Rn. Then DA(n°(0))(H(n°)) = IH(n°)1 2 = 0; so H(n°) = 0 and(;;) ;s proved. c Remarks (;) We conjecture that cond;t;ons (;) and (;;) of the corollary are also suff;c;ent for n° to be an absorb;ng state for the trajectory f;eld {nxt:t > 0, n € C(J,Rn)}. It ;s perhaps enough to check Lemma (5.3) of Dynk;n ([16], p. 137) on the weakly dense set of quas;tame funct;ons ;n V(A). (;;) An absorb;ng state n° corresponds to the o;rac measure onO be;ng ;nvar;ant under the adjo;nt sem;group {p~}t>O assoc;ated w;th the stochast;c FOE (I). Thus a necessary cond;t;on for the existence of ;nvar;ant o;·rac measures ;s that the coeff;c;ents H, G should have a common zero (cf. §III.3).
112
V Regularity of the trajectory field
§1.
Introduction
Given a filtered probability space (n,F,(Ft)G
=
H(xt)dt + G(xt)dw(t)
x0
=
n Ec
= C(JJRn)
0 < t < a
}
(I)
with coefficients H:C ~Rn, G:C ~ L(RmJRn) and mdimensional Brownian motion w. A version of the trajectory field of (I) is a measurable process x:n x [O,a] x C ~ C such that X(·,t,n) = nxt a.s. for all t € [O,a] and n € c. In order to deal with the question of the existence of versions of the trajectory field with reasonable sample function behaviour one needs to isolate two qualitatively different examples of stochastic FOE's. The first of these is when the diffusion coefficient G is independent of the past history, in which case we show that the trajectory field has a version whose sample functions are almost all compactifying (See Theorem (2.1)). In the second example, where G incorporates explicit dependence on the past (e.g. as in stochastic delayequations), we shall see that all versions of the trajectory field are almost surely highly irregular (See §3). Such erratic behaviour is essentially due to the Gaussiantype nature of delayed diffusions coupled with infinitedimensionality of the state space C. Under these considerations we are lead naturally to investigate regularity in probability of the trajectory field for the general system (1). This is done in §4 where the compactifying nature of the trajectory field is still shown to persist, though in a distributional sense, as in Theorem (4.6). It is perhaps worth noting here that for ordinary stochastic DE's (r = 0) satisfactory results on the sample function behaviour of the trajectory field are known. See Remark (i) of §3.
113
§2. Stochastic FOE's with Ordinary Diffusion Coefficients Consider the stochastic FOE dx(t) = H(xt)dt
+
g(x(t))dw(t)
x0 = n € c = C(J,Rn),
0 < t }