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a�
]
a2 2 .t ae
Because the stereographic projection is conformal, 3 z(x) is (locally) a Brownian motion run with a new clock. 3 # 0 for t # 0�, so x cannot hit the south pole (0, 0,  1 ) for t # 0, and since G commutes with spherical rotations, it cannot hit the north pole (0, 0, 1) either. But this means that for x(O) # (0, 0, 1 ), the projection 3 is welldefined for t � 0, and the fact that x visits each spherical disk i.o. , as t j oo § is mirrored iri the fact that 3 hits each plane disk, i.o. , as t j oo. Consider th e Riemann surface R of w = lg z as a plane divided into horizontal strips of height 2n, with projection z = ew mapping it onto the punctured plane R 2  0 as in Fig. 2. R can be viewed as the universal =
•
FIG. 2.
t 0 � cp colatitude � 7T , 0 � 8 longitude < 27T. t See Problem 7, Section 2.9. § See Section 4.4. =
=
4.6
COVERING BROWNIAN MOTIONS
111
covering surface of R 2  0 ; as such, its covering group is identified with the fundamental group Z 1 t of R 2  0. Because the plane Brownian motion 3 never hits 0 if 3(0) =I= Ot, a Brownian path on R 2  0 can be lifted up to R. Levy tells us that this lifted path is a Brownian motion run with a new clock, and it follows that the lifted motion hits each disk of R, i .o. , as t j oo . Regarding R as the universal cover of R 2  0, it now follows that the 2dimensional Brownian path winds both clockwise and counterclockwise about 0, i.o., as t j oo and also unwinds itself, i.o., as t j oo , reflecting the fact that the covering motion makes unbounded vertical excursions but comes back to the strip 0 � b < 2n, i.o. , as
t i 00 .
Define R to be the open upper halfplane for the next application. Given k2 =1= 0, 1 , the inverse function of the elliptic integral
fo
dtf[(I  t 2 )(1  k 2 t 2 )] 1 / 2
is a Jacobi elliptic function, and Jacobi's modulus k 2 , expressed as a function of the ratio w E R of its fundamental periods, maps R onto the twicepunctured plane R 2  0  I . k 2 is a modular function of the group G of substitutions iw + j O i j modulo 2, i l  kj = + 1 = 0 w + 1 k kw + 1 , [modular group of second level ] . G maps R onto R, and dividing R into sheets in accordance with this action as in Fig. 3, k 2 maps each sheet 1 : 1 onto R 2  0  1 . R can be regarded as the universal covering surface of the twicepunctured plane R 'b  0  1 . G is both the covering group R and the fundamental group of R 2  0 1 ; as such, it is iso morphic to the free group on 2 generators. A plane Brownian path 3 cannot meet 0 or 1 if 3(0) =I= 0 or 1 , so such a path can be lifted up from the punctured plane to R and will perform on R a Brownian motion run with a new clock. R is a halfplane, so this covering motion tends to the line R 1 X 0 = o R as t i 00 . Regarding each sheet of R as labeled by an element of the fundamental group G, it follows that, unlike the
l ]  [1 1J

case of the oncepunctured plane, the winding of the Brownian path about the two punctures 0 and 1 becomes progressively more complicated as t j oo and never gets undone. t Z 1 denotes the rational integers 0, ± 1 , ± 2, etc.
t See Problem 7, Section 2.9.
1 12
4
STOCHASTIC INTEGRAL EQUATIONS
(d � 2) R
I
I 2
 
I 3

I 3
0
I 2
FIG . 3.
Weyl [1] defines the class surface of the twicepunctured plane to be the biggest surface R 1 between the modular figure R and R 2  0  1 still having a commutative covering group G1 . R 1 can be regarded as the Riemann surface of w = lg z + J 1 lg (z  1) ; as such, it can be depicted as an infinite number of copies of the Riemann surface w lg z connected by logarithmic ramifications at the points 2n ri x Z 1 . G1 is just the group G made commutative, i.e. , considered modulo its commutator subgroup. G1 can also be identified with the homology group Z 2 t of R 2  0  1 . Now lift up the Brownian path from R 2  0  1 to R 1 • Levy tells us that this lifting is a Brownian motion on R 1 run with a new clock. But such a Brownian motion visits each disk of R 1 , i.o. , as t i oo (the proof is explained below), so the winding of the plane Brownian path about the points 0 and 1 undoes itself, i.o., as t j oo from the point of view of homology with integral coe.fficients. =
t Z2 denotes the lattice of integral points of R2 under addition.
4.6 Proof th at
a
1 13
COVERING BROWNIAN MOTIONS
Brow n ian motion o n R 1 vis its eac h d i sk, i . o. , as
t j oo
Define Q to be the commutator subgroup of G. As n i oo , the image of J=1 under the commutator
tends to jf l, and, for the first factor running over G (j even and I odd with no common divisors), such fractions are dense on the line o R. Q also maps R onto R, so it is a principalcircle group of the first k ind, socalled, and therefore by a theorem of Poincare,
L ( i 2 + j2 + k2 + 1 2)  1 = Q
oo . t
Now consider a standard Brownian motion w = w(O) + a + J  1 b = x + Fl l) on R run with the clock t 1 inverse to t(t) = : l)  2 •
J
Clearly t  1 is defined up to time J: lJ  2 , e being the exit time min (t : t) = 0) of w from R, and this integral is + oo since  oo = lg
tJ (e  )/tJ(O) = lim t je
rrlJ  1 d b  1 rlJ  2 ds] . 0
0
But then the projection 3 1 of w(t  1 ) onto R 1 is a Brownian motion run with a new clock defined for 0 � t < oo , and since the expected time spent by w(t  1 ) in a disk D R with indicator function f is c
J 00E[l)(t) 2, w(t) t < e] dt 00 = J dt t(2 t )  1 { exp (  lw  w(O) I 2 f 2 t) o =
E D,
0
n:
 exp (  lw*  w (O)I 2 /2t)}y  2 dx dy
1 =n
J
n
w*  w(O) Ig y  2 d d y ,+ w  w(O)
t Lehner [1 , pp. 1 791 83 ]. t The * means conjugate in this formula and the next.
X
+
1 14
4
STOCHASTIC INTEGRAL EQUATIONS
(d � 2)
it follows easily from the invariance of the volume element y  2 dx dy under G that the expected time 3 1 spends in the projected disk is
*  w (O)  2 d d " 1 J. w Ig y X y l...J O w w( ) Q n [� {] n � L constant Q
x
( i2 + j2
+ k2 + 12)  1
=
oo .
Because 3 1 is the same as the lifting of the plane Brownian motion from R 2  0  1 to R 1 up to a change of clock, it is now enough to verify that 3 1 visits each disk of R 1 , i.o. , as t i oo . But this follows easily from the divergence of the expected sojourn time of 3 1 in a disk and the fact that 3 1 begins afresh at its passage times. In fact, if D is a closed sub disk of the open disk B c R 1 , if t 0 � t1 � t 2 � etc. are the successive passage times of 3 1 to D via R 1 B, and if rx P[ t 0 < oo] and =

{3
=
E[measure (t : 3 1
E D,
t0 < t < t1 )]
are regarded as functions of 31 (0), then the total expected sojourn time of 3 1 in D is oo
=
=
00
I1 E [measure (t : 3 1 E D : tn _ 1 < t < tn)J
n=
00
L E[tn  1 < 00 , f3(31(tn  1))] n= 1
f
[ ]
sup rx � 1X(3 1 (0)) 1 n = oB
n l
sup p. oD
But for small B, {3 is bounded on o D as is clear upon lifting 3 1 back up to R. Besides, a is harmonic off D, t so the di vergence of the sum implies that rx 1 at some point of oB with the result that a = 1 ,t and now the proof is finished. =
t See Section 4.4. � See Step 3 of Section 4.4.
4. 7 4.7
115
BROWNIAN MOTIONS ON A LIE GROUP
BROWNIAN MOTIONS ON A LIE GROUP
A (connected) Lie group G is a manifold as defined in Section 4. 1 and also a group endowed with a smooth multiplication G x G � G Smooth means that for g1 (g 2 ) contained in a small patch U1 ( U2 ) , the product g = g1 g 2 is confined to a small patch and its local coordinates belong to ceoc ul X U2 ). Define ceo(l ) to be the class of germs of infinitely differentiable functions at the identity 1 of G. A derivation 1 D of ceo(l) is a map ceo( 1 ) � R with D(ft/2 ) = (D.ft)/2 (1) + /1(1)(D/2 ). Such a map can be expressed in terms of local coordinates x on a patch U about 1 as Df = a · gradf( l ) for some a E Rd , and this correspondence D � Rd is an additive isomorphism between Rd and the tangent space A of G at 1 , consisting of all derivations of ceo( 1 ). Define D( G) to be the class of all partial differential operators on G with coefficients from ceo( G) that commute with the left translations g : f � gf = f( g ) D E A can be viewed as a member of D(G) using the recipe Df(g ) = Dgf( l ) For members of A, it turns out that the commutator [D1 , D 2 ] = D1 D 2 D 2 D1 , computed in D(G) and then applied to ceo ( I), belongs again to A . A endowed with this commutator product is the Lie algebra of G. D (G) endowed with the usual product is the enveloping algebra of A , sonamed because, up to isomorphism, it is the smallest associative algebra containing A as a Lie subalgebra under the commutator product. A is provided with a mapping into G, the socalled exponential map defined in the neighborhood of 0 E A by the rule x(exp (tD)). = (Dx)( exp (tD)).t exp maps the ! dimensional subspaces of A onto the ! dimensional subgroups of G ; it is a local diffeomorphism. A simple example is provided by the group G = S0(3) of proper rotations of R 3 . S0(3) can be identified as 3 x 3 orthogonal matrices of determinant + 1 , A as 3 x 3 skewsymmetric matrices under the com mutator product, and exp as the usual exponential sum : exp (D) = I Dn/n ! . A is spanned by the three infinitessimal rotations : 0 0 0 0 1 0 0 1 0 D3 = 1 0 0 0 , D2 = D1 = 0 0  1 , 0 0 . .
.
.
.

0
1
0
1
0
0
[D 1 , D 2 ] = D 3 , [D 2 , D 3 ] = D1 , [D 3 , D1 ]
0
=
0
0
D2 ,
t The stands for differentiation with respect to t, and x for local coordinates on a patch about 1 . •
1 16
4
STOCHASTIC INTEGRAL EQUATIONS
(d � 2)
so that A is isomorphic to R 3 under the outer product. The exponential mapping sends into the rotation about the axis a E R 3 through the angle lal in the sense prescribed by the righthand screw rule. t Besides the above general facts, only Ado's theorem is needed here (see Step 1 of Section 4.8). Helgason [ 1 ] is recommended for proofs and general information. A (left) Brownian motion on G is a continuous movement 0 � t � 3(t) E G with 3(0) = 1 , beginning afresh at its stopping + times t in the sense that, conditional on t < oo , the future 3 (t) = 3(t)  t 3(t + t) : t � 0 is independent of the past 3(s) : s � t + (i.e., indepen dent of the usual field Bt + ) and identical in law to the original motion 3(t) : t � 0 starting at 1 . This is the analog of the differential property of the !dimensional Brownian motion. Yosida [ 1 ] proved that such a Brownian motion is governed by a (possibly singular) elliptic differential operator G E D( G) , expressible in terms of a basis D = ( D1 , . . . , Dd) of A as G = !D eD + f D = t L eii D i Di + L h D i i , j� d i�d with constant 0 � e = e * E Rd ® Rd and f E Rd . For nonsingular e , the statement means that the density p(t, g) of P[3(t) E dg] , relative to the volume element of G, is the elementary solution of oufot = G * u with pole at 1 . A formal proof consists of using the (left) differential property of 3 to check that, for the map G E D(G) defined by Gf(l) = lim t  1 [E[f(3)]  f( l )], t ..l 0
E[gf(3)] = exp (tG)f, and then using Problem 1 , Section 4. 1 , and the fact that [A, A] c A to reduce G to the desired form. Ito [4] proved that every such G arises from some (left) Brownian motion by constructing the associated sample paths as in Section 4.3. Y osida also proved this fact by constructing the elementary solution of ouf ot = G * u. A third method is to inject the differentials of a ddimensional (skew) Brownian motion Je b + ft from A (identified with Rd) t GelfandSapiro [1 ] can be consulted for detailed information about this group.
4.8
1 17
INJECTION
into G via the exponential map and then to put them back together as a socalled product integral:
n exp [Je d b + f ds J
s�t
= lim fl exp [J; [ b (k2 ")  b ((k  1 ) 2  ")] + f 2 "] . n t oo
k � 2 "t
This program is carried out in Section 4.8. t 4.8
INJECTION
Given constant 0 � e = e * and f, and a standard ddimensional Brownian motion b, regard a = Je b + ft as a (skew) Brownian motion in the Lie algebra A of G, i.e. , identify a E R d and a · D E A , and let us verify the following recipe : if 3n( t )
= =
( t = 0) ( t � 0, l = [2" t]) ,
1 3n ( l2  ") exp [a (t)  a ( l2  ") ]
then the socalled product integral: 3 oo (t )
=
n exp ( d a ) = lim 3n (t) n t oo
s�t
exists and is a left Brownian motion on G governed by G = DeD/2 + fD. Step 1
Ado's theoremt states that A can be faithfully represented as a Lie subalgebra of Rm ® Rm , under the customary commutator product, for some dimension m. The classical exponential sum exp (a) = L a"fn ! maps this faithful copy of A onto a Lie subgroup of the general linear group GL(m, R 1 ) and this subgroup is locally isomorphic to G but perhaps not globally so. But the injection recipe is local, so it is per missible for the proof to suppose that G is faithfully embedded as a Lie ,
t McKean [2] did this for the special case G S0(3) . See also Gangolli [1 ] for a more general application of the same idea, and Perrin [1 ] for a discussion of Brownian motion on S0(3) from a more classical standpoint. � See Bourbaki [1 ]. =
1 18
4
STOCHASTIC INTEGRAL EQUATIONS
(d � 2)
subgroup of GL(m, R 1 ). Rm ® Rm is provided with the norm Jsp a * a, not to be confused with the bound Ia I of a as an application of R'". The bounds
a l � Jsp a *a � m la l , lsp a c b l � constant
x
Jsp a * a Jsp b* b ,
sp (a + b)* ( a + b) � 2 sp a* a + 2 sp b * b will be of frequent use. Warning : * a stands for the transpose of a E A as an application of R'" ; this will make the formulas come out more neatly. Step 2
be
The product integral for 3oo suggests that in the enveloping algebra D(G) c Rm ® R'", d3oo = 3oo [exp (da)  1 ] = 3oo [da + !(da) 2 ] ; it is to proved(that this problem has just one nonanticipating solution 3 : t D( G) with 3(0) = 1 . +
Proof of u n i q uen ess
Consider the difference 1J of two nonanticipating solutions and let the Brownian stopping time t be the smaller of t � 0 and min (t : 11J I = n). An application of Ito's lemma to lJ * lJ implies lJ* t) (t) =
f t)[dj + *dj + dj *dj] * t) t
0
with dj = da + (da) 2 /2 = Je db + k dt. Using the bounds cited in Step 1 , it develops that the expected spur D of l) * l) (t ) is � (mn) 2 < oo and bounded as in
D
=
E [( sp tJ(k ds + *k ds + � db * Je db)* lJJ
:::;; constant
x
Jo D, t
permitting us to conclude that D = 0. The proof is completed by making n j oo .
4. 8
1 19
INJECTION
Proof of existe n ce
Define 3 to be the sum of t) 0 (t) = 1 and lJ n ( t) = much as in Section 2.7. Then
Dn = E[sp tJ/tJ nJ � constant
x
J0 Dn  1
J� lJ n  1 dj for n ;;;: 1 ,
t
(n � 1 )
is proved just as the bound for D above, and using the martingale inequality and the first BorelCantelli lemma much as in Section 2. 7, the sum for 3 is found to converge geometrically fast to a solution of 3 = 1 + 3 dj.
J�
Step 3
Define 3 n(O) = 1 and 3 n (t) = 3 n(/2  n) exp [a ( t)  a(/2  n)] for I = [ 2"t] , t � 0, and n � 1 ; it is to be proved that
[
P max l3n (t) l � 2et", n j t� 1
oo
]
for any rx > 0.
=1
Proof
The norm of exp (a) is � exp (I Jel lbl + 1/l t). Because of E[exp (yb 1 )] exp (y 2 t/2), E[lexp (a) I P ] is bounded for t � 1 , for each f3 > 0 separately, so E[I31 P] is bounded too, and =
[
]
P max l3n (l2  ") l > 2et " � constant x 2n [ l  et P J �� 2" is the general term of a convergent sum for rx/3 > 1 . An application of the first BorelCantelli lemma, completes the proof. Step 4
[
1 P max 3n  1  J 3 n dj � 2  8" , n j 0 t� 1
oo
]
= 1
for any (} < 1.
Proof
Define A = [(k  1)2 " , k2  " ] and a(d) = a(k2  ")  a((k  1)2  ") for k � 2 ". Using Levy's modulus (Section 1 .6), its counterpart for
4
1 20
STOCHASTIC INTEGRAL EQUATIONS
(d � 2)
Brownian integrals (Section 2.5) , and the bound of Step 3, we find for n i oo and (} < ! , that up to errors of magnitude � constant x 26", 1 3n ( t )  1  3 n dj
J
=
0
[
L 3n (( k  1) 2") exp [a(L\) ]  1 
k�l
=
L 3 n (( k  1) 2" )h ( L\)
JA dj]
k�l
J,�,
with h{L\) = [Je (b(L\)]2  (J e db )1. Because the final sum ( = 1) 1 ) is a martingale, sp lJ L * lJ L is a submartingale, and using the bound E[ l3 n l 2 ] � constant (t � 1 ) from Step 3 and the independence of 3 n ((k  1)2  ") and h (L\), it develops that
[
]
P max sp 1) 1 *lJ z > 2  26 " l� " 2 � 226"E[sp l) 2 n * 1J 2 n] =
[
226"E L SP 3n CCk  1) 2")h(L\)*h(L\)* 3n CC k  1 ) 2" ) k � 2"
� 2 26n m 2 L E[ / 3n ((k  1) 2 n) j 2 ] E [ /h (L\)j 2 ]
]
k � 2"
� constant
x
226n + n  2 n .
But for (} < ! , this is the general term of a convergent sum, so an applica tion of the first BorelCantelli lemma does the rest. Step 5
[
P max l 3n  31 � 2  on , t� 1
n
j
oo
]
=
1
for any 0 < ! .
Proof
s:
s:
3 n  3 lJn + (3 n  3) dj with lJ n = 3 n  1  3 n dj. This last expression is of magnitude � 2  o n for t � 1 , n i oo, and (} < 1 in accord ance with Step 4. Bring in the Brownian stopping time t n defined either as the first time t � 1 such that 1 3 n  3 1 2cxn or ltJ n l = 2  6", or as t 1 if neither of these events occurs before. Because of Steps 24, tn = 1 for n n i oo . D = E [ sp ( 3 n  3) * (3n  3 )(t n ) J < m 2 2cx < oo can be bounded as in =
=
=
4. 8 D
121
INJECTION
� 2E [sp lJ n * lJ n (tn)] " sp (3.  3) [dj + * dj + dj * djJ *C3n  3) + 2E
r(
:::;; 2m 2 2  2 9" + constant
x
J D, t
1
0
with the result that D is bounded by a constant multiple of 2  2 0" for t � 1 , and now the usual martingale trick applied to the sub martingale tn t tn t sp (3 n  3) j e db (3n  3) j e db *
f
J\
J\
0
0
implies
f
P max (3.  3) je db � 2  0", t� 1 0 The analogous bound
n
j
oo
] = 1.
P max (3.  3)k ds � 2  0" , t� 1 0 is even easier to prove, and the result follows.
n
j
oo
]
r
f
t
{
r
= 1
Step 6
3 oo = limn t oo 3 n
exists, and for nonsingular e, it is the left Brownian motion on G governed by G = DeD/2 + fD. Proof
Step 5 leads at once to the existence of the product integral 3oo = 3 for t � 1 , and the reader will easily check that this propagates for t � 1 . It is also plain that 3 oo is a left Brownian motion, and so it suffices to prove the last statement. But for compact u E C 00 (G), n i oo , t � 1 , I = [ 2" t ] , and () < i , it is easy to see that up to errors of magnitude � constant x 2  0" , u (3 00 )  u ( 1 ) = u(3n)  u ( 1 )
I { u [3n ( k2  ") J  U [3 n (( k  1 )2  n) ] } k�l = I I ai (L\) Di u + � I ai (L\ ) a j(L\ )Di Dj u i, j � d k� 1 i�d evaluated at 3n (( k  1)2  n), =
r
]
4
1 22
STOCHASTIC INTEGRAL EQUATIONS
and it is easy to see that as
n
(d � 2)
i oo , this expression tends to
But this means that on a patch U with local coordinates x, x = x(3 ocJ is a solution of dx = Je (x) db + f(x) dt, e and f being (just for the moment) the local coefficients of G. This permits us to identify 3oo as the left Brownian motion governed by G and completes the proof. A simple but amusing example of injection is provided by the motion of a 3dimensional unit ball rolling without slipping on the plane 3 R 2 x  1 c R while its center performs a standard 2dimensional 3 Brownian motion b = (b 1 , b 2 ) on the plane R 2 x 0 c R .t G = S0(3), the infinitessimal rotations
0 0 0 D1 = 0 0  1 0 1 0
0 0 1 0 0 0 , D2 = 1 0 0
'
1 0
0 0 0 0
span A, and the exponential maps a1 D 1 + a 2 D2 + a3 D3 E A into the righthanded counterclockwise rotation through the angle lal = 3 1 2 2 2 2 1 (a1 + a 2 + a3 ) about the axis a = (a1 , a 2 , a3) E R , as noted in Section 4.8. As the Brownian particle moves from b((k  1)2  ") [point 1 of Fig. 4] to b(k2  n) [point 2 of Fi g . 4] , the ball suffers the approxi mate rotation exp [e3 x b( �) D] = exp [  b 2 ( �)D 1 + b 1 ( �)D 2 ]t ·
of angle b(�), counterclockwise about the axis e3 x b( �), as in Fig. 4, so the total rotation suffered up to time t � 0 is just the corresponding product integral : namely, the (left) Brownian motion on S0(3) governed by G = !(Dt 2 + Dz 2 ) .
FIG. 4.
t McKean [2] ; see also Gorman [1 ]. t e3 (0, 0, 1). The x = the outer product. =
4.9
BROWNIAN MOTION OF SYMMETRIC MATRICES
1 23
Pro b l e m 1
Prove that the induced motion 3oo e3 of the north pole on the surface of the rolling ball is the spherical diffusion governed by a2 1 G + = 1 (sin cp )  a sin cp a + cot 2 cp a e 2
0�
qJ =
[
:
:
]
0 � (} = longitude < 2n.
colatitude � n,
Sol u tion
[ D 3 , G] = 0, so G commutes with the subgroup S0(2) of rotations about the north pole e 3 • Because of this, 3oo e3 is a diffusion on the spheri cal surface M = S0(3)/S0(2), and for the rest, it suffices to compute the 2 2 action G + of G = ! (D1 + Dz ) on u E C00(M) regarding u as a member oo of c ( G) depending only on cosets gS0(2). 4.9
BROWNIAN MOTION OF SYMMETRIC MATRICES
Regard R4 [d = n(n + 1 )/2] as the space of n x n symmetric matrices with coordinates x ii (i � j � d) and define M c R4 to be the submanifold of symmetric matrices with simple eigenvalues. O(n) acts on M by con jugation [x + o * xo] . M/O(n) can be identified with the submanifold R of diagonal matrices y with entries y 1 < · · · < Y n down the diagonal, and since the stability group of x E M is the (finite) subgroup D of diagonal rotations ( + 1 down the diagonal), M can be identified with R x O(n) considered modulo D, via the diffeomorphism (y, o) + o *yo . G = O(n) x ( + 1) x R4 acts as a motion group on R4 by conjugation [x + o * xo] , reflection [x +  x] , and translation [x + x + y] , and up to constant multiples, the only elliptic operator on C00(R4) commuting with the action of G is 2 G = t L o /ox� + ! I o 2 / oxfj · i�n
i <j
G governs a Brownian motion I lj ·
or
·
I
 I lJ (0) ·
·
on R4 expressible as =
b l} ·
·
(i = j) (i < j),
b ii (i � j) being independent standard ! dimensional Brownian motions.
4
124
STOCHASTIC INTEGRAL EQUATIONS
(d � 2)
An easy computation shows that
P[I(t 2 )  I(t 1 ) E dx I I(s) : s � t 1 ] ( 2nt)  nf 2 (n t)  n( n  l ) / 2 exp [  sp ( x) 2 /2t] dx for t = t2  t1 > 0, dx being the volume element ni � i dx ii . Using the =
in variance of this formula under the action of G, it is easy to see that the eigenvalues of I begin afresh at stopping times and perform on R the diffusion governed by the action G + of G on C 00 (R) : G + = 1 iLn o 2 f oy i 2  1 Ii (yj  Yi)  1 a ;ayi , j= � up to the exit time e of I from M.t A more picturesque statement is that as I performs the Brownian motion governed by G on M, its eigenvalues perform a standard ddimensional Brownian motion on R subject to mutual repulsions arising from the potential U : e  2 = n ( yj  Y i) . t j>i Because of this repulsion, it is natural to conjecture that the exit time e is infinite if I(O) E M, as will now be proved.
u
Step 1
Rd
=
M u o M, oM being the sum of d  1 submanifolds like
Y 2 < · · · < Yn J ' d  2 submanifolds like M3 = [x : = y 1 y 2 = y3 < · · · < Y nJ , and so on, plus the single submanifold Mn = [x : y1 = y 2 = · · · = Y n] · It is to be proved in this step that codim o M = 2.
M2
=
[X : Y 1
=
Proof
codim M2 is just 1 plus the dimension of the subgroup of O(d) com muting with the diagonal matrices belonging to M2 • But this subgroup is the product of a copy of 0(2) and the diagonal subgroup of O(d  2), so the codimension is 2. A similar proof shows that codim M3 = 2 + dim 0(3) = 5, and so on. t Section 1. 7 contains the prototype of the proof. t See Dyson [1 ].
4.9
BROWNIAN MOTION OF SYMMETRIC MATRICES
1 25
Step 2
I cannot hit a submanifold Z of R4 of codimension
2 for t =1= 0.
Proof
Define
2 2 p (x ) = J sp (x  y) ]  df + t d o , /
do
being the product of the volume element of Z and a positive function belonging to C00(Z) such that p < 00 off z. As X approaches a point of Z from the outside, p is bounded below by a positive multiple of f
rr /2
o
rr/2 (sin 0) 4 3 d O (sin 8)4 3 d O = 00 ' f 2 2 1 41 1 ] ] i t z dt 0 b [2 ( [2 ( 1 + £5) ( 1  cos 0 ) + cos 0)
b being the distance of x from Z. Now suppose I(O) E Z and define e to be the passage time of I to Z. e < oo implies lim t t e p(I) = oo , while for t < e , dp(I) = grad p di + Gp(I) dt is a pure Brownian differential since G[sp (x  y)2]  412 + 1 = 0 (x =1= y) . But this means that up to the passage time e , p(I) is a ! dimensional Brownian motion run with some clock,t and this leads to a contradiction as in the solution of Problem 7, Section 2.9, or Problem 5, Section 4 . 5 . ·
Pro b lem 1
Prove that the eigenvalues of I perform the diffusion governed by G + for n = 2 by direct stochastic differentiation of Y2 = � ( b 1 1 + b22) + Q = b i2/ 2 + ( b 1 1  b22)2/4
Y1 = } ( b 1 1 + b22) 
j Q,
j Q,
So l ution
in which
b1 1 
[2 1 + (  )i I[ 1  (  )i +
da i = �
hzzl Y2  Yt J
2
t See Problem 1, Section 2. 9.
db 1 1 + (  ) i
b11  b22 db22 Y2 Y t
J
b 12
Y2  Y1
db12
(j = 1 ' 2).
4
1 26
STOCHASTIC INTEGRAL EQUATIONS
(d ";3::. 2)
Now use Problem 2, Section 2.9, to prove that a1 and a 2 are independent ! dimensional Brownian motions. Pro b l em 2
Prove that for n 2, the determinant y 1 y 2 can be expressed as 1 (b 2  r 2 ), b being ! dimensional Brownian motion and r an independent 2dimensional Bessel process. =
So l ution
and d ( y 2  Y t )fJ2 = db 2 + ! [ ( Y 2  Yt )fJ2] 1 with new independent ! dimensional Brownian motions b 1 and b 2 Now use Section 3. 1 1 c to identify r = (y 2  y 1 )/ J2 as a Bessel process and express the determinant as i (b 1 2  r 2 ). •
Prob lem 3
Use the method of Step 2 to prove the topological fact that, for d '?:: 2 , R 4 minus a submanifold of codimension '?:: 2 is still connected. t So l ution
Denote the submanifold by Z, take x and y E R 4  Z, and draw about y a small ball A not meeting Z. 0 < P[x + x( l ) E A ] , and since, as in Step 2, x + x(t ) : t � 1 cannot meet Z, it is possible to find a continuous path joining x to y in R 4  Z by going from x to A via a Brownian path x + x(t) : t � 1 and then joining x + x( l) to y by a line segment. 4.10
BROWN IAN MOTION WITH OBLIQUE REFl ECTION
A nice example of a diffusion on a manifold with boundary is the Brownian motion with oblique reflection on the closed unit disk of R 2 • Consider the open unit disk M: l z l < 1 , assign to the point 0 � () < 2n of oM a unit direction I making an angle  n � qJ < n with the outwardpointing normal in such a way that exp (J  1 (/)) E C 00 (aM), and t See Helgason [ 1 ] .
4. 1 0
BROWNIAN MOTION WITH OBLIQUE REFLECTION
1 27
suppose that l cp l =1= n/2 except at a finite number of singular points at which cp' =1= 0. Denote this singular set by Z, and call a singular point attractive if cp' < 0, repulsive if cp' > 0. Brownian motion with oblique reflection along I is the diffusion on M  Z governed by G = A /2, subject to
oufo l = cos
qJ
au;an + sin qJ au;ae = 0
on
oM  Z.t
Dynkin [ 2] and Maliutov [ 1 ] have made a very complete study of this motion. For general information about diffusions on manifolds with b0undary, see Ikeda [ 1 ] , Motoo [2] , and SatoUeno [ 1 ] . Co nstructi on fo r cp
=
0 (sta n d a rd reflect i ng B rown i a n motio n )
Using Section 2.8, it is easy to deduce from Problem 9, Section 2.9, that the plane Brownian motion starting at 3(0) = r(O) exp (J  1 8) =1= 0 can be expressed as
r being a Bessel process starting at r(O) and a an independent ! dimen sional Brownian motion.t Replace r by the reflecting Bessel process on (0, 1 ] governed by A + /2 = i [o 2 /or 2 + r  1 ajar] subject to u  (1) = 0. This motion can be obtained as in Section 3. 10 from a ! dimensional Brownian motion b by solving dr = db + dtf2r  df for the path 0 < r � 1 and the local time f = lim ( 2a) 1 measure(s � t : r(s) > 1  a )
.
£tO
Using this modification of 3 , Ito ' s lemma gives
0 = E (j( t , 3 ) i�J = E
l(' (OjOt + A/2)j(t, 3) dt  (' (Oj/On) (t, 3) dfJ
for compact j E C 00 [ (0, 00) X MJ . Weyl's lemma now implies that the density p of the distribution of 3(t) belongs to C 00 [(0 , 00) X M] and t ojon denotes differentiation along the outwardpointing normal.
t &(0)
F
0 is assumed only to permit us to use this expression for
3.
4
1 28
STOCHASTIC INTEGRAL EQUATIONS
(d � 2)
solves opjot = Ap/2 inside M. Using Green ' s formula to transform . gives
0
=
Joo ! dt 0
]
[ I
oj op d (} p  j on on E oM
[Joo oj dfJ 0
on
'
granted that p belongs to C00 [(0, oo) X MJ .t But for j = it (t)j2 (r)j3 (0) with COmpact it E C00(0, 00 ) , COmpact J2 E C00(0, 1 ] , }2 (1) = 1 , j2  {1) = 0, and }3 E C00(oM), this gives
0= so opjon
=

Jooi t d t o
op j3 ; d (} , un oM
I
0 on oM, and the identification of the motion follows.
Construction fo r qJ # n/2
Using the reflecting Bessel process r, its local time f, and the inde pendent ! dimensional Brownian motion a, solve
1/J ( t ) =
1 J J O 1/J( ) + r da  tan
0
t
0
This is easily done, since (tan qJ )' is bounded. Define 3 Ito's lemma gives
0 = E[j(t, 3)!0]
=
=
r exp ( J  1 t/1 ) .
[(' (0/0t + A/2)j(t, 3) d t  (' ( OJ On + tan OjOfJ)j(t, 3) d fJ
E
cp
for compact j E C00 [(0, 00) X M. ] As before, it develops that p belongs to C00 [(0, oo) X M] and solves opjot = Ap/2 inside M, while opfon  o(tan
BROWNIAN MOTION WITH OBLIQUE REFLECTION
4. 10
1 29
,
Co n st r u ction i n t h e p rese n ce of re p u lsive s i ng u laritiecs o n ly
Figure 5 depicts the drift coefficient  tan
OJ ; this drift acts to push 3 away from the singular point. Because the standard reflecting Brownian motion [

FIG. 5.
 ta n
cp
e
e
FIG. 6.
Co n struction i n t h e p rese n ce of att ractive s i n g u larities
Figure 6 depicts the drift coefficient at an attractive singularity [
arg 3(e  ), but not both, (e) the density of the distribution of 3( t ) is the smallest elementary solution of oufot = Au/2 with pole at 3(0) subject to oujol = 0 on oM  Z.
The proof is carried out only in the simplest case : l = the horizontal direction [
1 30
4
Ste p
STOCHASTIC INTEGRAL EQUATIONS
(d � 2)
1
Z comprises just the two attractive singularities + J=1 , and for 3(0) ¢ Z, 3 can easily be defined up to the explosion time e = lim min ( t : I 3 n i oo
J  l l or I 3 + J  l l =
l /n )
so that (a) holds. Ste p 2
By Ito's lemma and Problem 1 , Section 2.9, du (3) = i Au d t +
au or
db +
= 1 Au d t + c(t)
au of)
[ au au ] + tan r  1 d a  df or
q>
ae
au
 df sec q> af
for u E C 00 (M  Z) and t < e, c being a ! dimensional Brownian motion and t the clock
Jo I grad u(3W . t
Define 3 = I + J  i 1) and put u = y. Because Au = 0 and oujol = ouj ox = 0 on oM, it follows that 1) is a ! dimensional Brownian motion run with the clock I grad u(3) l 2 = t up to the explosion time. Because 1) is bounded, e < oo , proving (b), t and the existence of 1)( e  ) is also evident. But then 1)( e  ) = + 1 by the definition of e, and this forces the existence of I( e  ), proving (c).
J�
Ste p 3
Consider the angles a and f3 depicted in Fig. 7 and define u = a  [3. u E C 00 (M  Z), Au = 0 inside M, and oujol = oujox = 0 on oM  Z, so u(3) can be expressed as a ! dimensional Brownian motion run with the clock t(t) = lgrad u(3) l 2 up to time e. Because u is bounded, t( e  ) < oo, t lim t t e u(3) exists, and it follows that as t j e, 3 approaches
J�
t See Problem 7, Section 2.9, or Problem 5, Section 4.5.
4
. 10
BROWNIAN MOTION WITH OBLIQUE REFLECTION
131
3( e  ) E Z at a definite angle. But as stated before, the standard reflect ing Brownian motion [ cp = OJ does not hit a point of oM named in advance, so r(t) = 1 , i.o. , as t i e, and it is immediate from the picture that as 3 approaches J=1 , say, a tends to + n /2. This proves (d).
FIG. 7
J1 Ste p 4
Using (a)(d), it is easy to prove (e) as in the nonsingular case, granting that p E C 00 [(0, oo) X M] . Problem
1
Prove that the standard reflecting Brownian motion [ cp = OJ does not hit a point of oM named in advance. So l ution
Define u = 2 lg lz  I I . Au = 0 inside M and oufon = 1 on oM, so, using Ito's lemma and Problem 1 , Section 2.9, as before, we find that u ( 3) is the sum of the local time  f and a ! dimensional Brownian motion run with the appropriate clock. As such, it cannot tend to  oo at a finite time, so 3 cannot hit 1 .
4
1 32
(d � 2)
STOCHASTIC INTEGRAL EQUATIONS
Problem 2
Prove that for the Brownian motion associated with the boundary condition oufox = 0 starting at z = x + � y,
[
P 3(e  )
]
1
= J  1 , lim a(3) = n /2 = l( l + y) + ( + x /2) . n t fe
rx
Sol ution
1) is a ! dimensional Brownian motion up to the explosion time, so
y = E [l) ( e  )] = 2 P[3(e  ) = J  1 ]  1 , showing that P[3(e  )
= J=1] = 1 (1 + y).
Define u = a  x/2. Au = 0 inside M and o uj o l = 0 on oM  Z, so
[ fe J
[ fe ]
n n x a  = E lim u(3) = E lim a(3) = Q  [ 1 ( 1 + y)  Q] , 2 t
t
2
2
Q being the desired probability. Now solve for Q. Pro b l e m 3
Prove that the functions 1 , y, a  x/2 , and f3  x/2 span the solutions of Au = 0 subject to the conditions : (a) u E C 00 (M  Z) , (b) oufox = 0 on oM  Z, (c) u approaches a finite value as point ofZ.
z
EM 
Z tends tangentially to a
Sol ution
Use Ito ' s lemma and Problem 1 , Section 2.9, as before to prove that any such u can be expressed as u = E[lim, t e u(3)] .
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Ann. Math . Berkeley Symp. Math. Statist. and Prob. DAN Dokl. Akad. Nauk SSSR IJM Illinois J. Math. JMP J. Math. Phys. MA Math . Ann.
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Math. 2.
SUBJECT INDEX
B
Levy's modulus, 1 8 on Lie group, 1 1 5 skew, injected, 1 1 7 of symmetric matrices, 1 23 winding of 2dimensional, 1 10 with oblique reflection, 1 27
Backward equation for d = 1 , 63 for d � 2, 98 Bernstein's theorem, 1 10 Bessel process, 1 8 Brownian motion !dimensional construction of, 5 differential property, 9 distribution of maximum, 27 law of iterated logarithm, 1 3 Levy's modulus, 1 4 local times, 68 nowhere differentiable, 9 passage time distribution, 27 scaling, 9 stopping times, 1 0 severaldimensional, 1 7 covering, 1 08 law of iterated logarithm, 1 8
c
CameronMartin's formula for d = 1 , 67 for d � 2, 97 D
Differential, stochastic, definition, 32 Ito's lemma for d = 1 , 32 for d � 2, 44 1 39
see also
Integral
1 40
SUBJECT INDEX
for severaldimensional Brownian motion, 43 under time substitution, 41 Diffusion 1 dimensional, 50 backward equation, 63 CameronMartin's formula, 67 explosion 0f, Feller's test, 65 forward equation, 60 generator, 50 reflecting, 71 stochastic integral and differential equations for, 52 on severaldimensional manifold, 90 backward equation, 98 CameronMartin formula, 97 exploeions of harmonic functions and, 97 Hasminskii's test, 1 02 forward equation, 91
iterated, and Hermite polynomials, 37 Ito's definition for d = 1 , 21 for d � 2, 43 simplest properties, 24 under time substitution, 29 Wiener's definition, 20 Integral equation, stochastic general idea, 52 general solution for d = 1 , 52 Lamperti's method, 60 on patch of a manifold, 90 singular examples, 77 solution of simplest, 35 K
Kolmogorov's lemma, 1 6
L
F
Lie algebras and groups, 1 1 5
Feller's test, 65 Forward equation for d = 1 , 61 for d � 2, 91
M
G
Manifolds, 82 Martingales, 1 1
Gaussian families, 3 H
T
Time substitutions, 29, 41
Hasminskii's test, 1 02 I
Integral , stochastic, see also Differential backward, 35 computation of simplest, 28
w
Weyl's lemma application for d = 1 , 61 for d � 2, 95 proof, 85