Aronson's estimates and conditional diffusion processes

ACTA MATHEMATICAE APPLICATAE SINICA Vol.10 No.2 Apr., 1994 ARONSON'S ESTIMATES AND CONDITIONAL DIFFUSION PROCESSES * QIAN ZHONGMIN (~,~ ~) WEI GUOQIANG (~ ~ ~) (Dep~nen$ of Mathemoi£c~ Statics, East 6vhm Nc.,vn~ Uni~z~'~ty,Shan@hai 20006~, China) Abstract Using time reversal for diffusions and A r o ~ n ' s e s t ~ A ~ s , we obtain several results on the compact properties of a conditional diffusion process in a small t i m e interval. In particular, we establish the large deviation property for a conditional ~ i o n process. Key words: Diffusion process, Aronson's estimate, time reversed process, large deviation §I. I n t r o d u c t i o n Suppose (M, O) is a complete Riemannian manifold, and (Xt) is the Riemannian Brownian motion. For each ~ E ( 0, 1], let Xe(t) denote the process X(et), P~,~ denote the law of the process Xe under the condition that X,(0) = y and X~(1) = z, that is, P~,x is the law of the Riemannlan Brownian bridge with life time 1. Then Pei Hsu [2] proved that P~,z has the large deviation property as ~ --~ 0, with the rate function 1 a Jy,~(w) = 2f0 [~(t)[dt- d2(y,z). (1.1) Using a quite differentmethod, Z. Qian and G. Wei[s| proved P~v,=also has the large deviation property if Xt is the s~metric process associated with the Dirichlet space (E,HI(Ra)) where E (f,g) = ~ , (V f, a, Vg) (z) dz, (1.2) provided that a(x) is continuous and satisfies )JR, < a(.) < A-XR, (1.3) for some constant A E (0,1]. In [2] and [5],the fact that X~ is symmetric plays a key role. In fact, Z. Qian and G. Wei [8 and Pei Hsu[2] used basic fact that the s ~ m e t r y of Xt implies the time reversibility. In this paper, we study a non-symmetric diffusionprocess Xt conditioned on X0 = y and )/i = z, Using the time reversed processes associated with the diffusion process Xt, Received September 17, 1990. Revised May 7, 1991. *This project is supported in part b y t h e National Natural Science Foundation of China. No.2 A R O N S O N ' S ESTIMATES AND CONDITIONAL DIFFUSION PROCESSES 149 we obtain several results on the compact properties of the conditional diffusion process. In particular, we establish the large deviation principle for the conditional diffusion process in a small time interval under mild regular conditions. W e malnly use Aronson's estimates and the time reversed process of a diffusion process to obtain the above result. §2. T i m e R e v e r s e d Processes Throughout this paper, assume a(z) = (a~j(z)) : R d ~ Rd@];~ d (d ~_ 3) is a symmetric, bounded measurable and matrix-valued function on R d, satisfying (1.3), and b(z) = (b~(z)) : R d --# R d is a bounded measurable function. Let 1 o o L : ~ .~. ~-~ma,j(-)~-mj + )a__ (2.1) . b,(- Ore," By using Aroson's estimates, we know that there is a unique strongly Feller continuous, Markov process associated with the infinitesimal generator (2.1), with a continuous transition density function p(t,y, z) with respect to the Lebesgue measure on R d. Moreover, there is a constant M depending only on A, d and Ibloo,such that 1 ~/t~ exp (. M[Ytz[2 ) <_p(t,y,z)<_ t~exp (. 'YM{ '2) (2.2) for all (t,y, z) E (0, 1] x R d × R d (see D.W. Stroock [6]). (2.2) is called Aronson's estimate or Moser's inequality. N o w we further assume that°a~j and bi are C~. Then p(t,y, z) is C 2 in y and z. Denote by L* the formal adjoint operator of L, that is, 1 d a a, 0 - ~ bdi ( L" = 2 .~. ~xi "J(')~xj 0 ")~mi - divb' (2.3) t~3 where div b =_E,d,j ~o=, . Then p(t, y, z) satisfies Op(t, y, z) ---[L*p(t, y, . )](z), (2.4) a (t, y, z) =[Lp(t, • z)](y). (2.5) at Ot (2.4) and (2.5) are called the forward Fokker-Planck equation and the backward FokkerPlanck equation, respectiveIy. For any given y E R a, let 1 d L,,. a (9 E d - $~3 (9 , S I Eao(.)~p(1-t,y,.) + p(1 - t, y , . ) . . S~3 (gxj 0 " o~zl Operator (2.6) is called the time reversed operator of L starting form y. (2.6) 150 ACTA MATHEMATICAE APPLICATAE SINICA Vol.10 Let ft = C([0,1]), endowed with the topology of uniformly convergences over the time interval [0,1], z(t) be the coordinate process on f/ and 9rt, .T be the natural a-filtration over ~. By Aronson's estimate (2.2), we know that there is a unique probability Py over (f/, ~') for each y E R d, such that (Iota, 9r , ~'t, z(t), Py) is a strongly Feller continuous, Maxkov process with the transition density function p(t, y, z). Denote by P~,z the law of the process (z(t), Py) under the condition that x(1) = Z. Then for any t : 0 < t < 1, dPy,z I p(1 - t, x(t), z) dPy Ilz, = p(1,y,z) (2.7) Denote by Qz,~ the unique solution of the Lt,y-martingale problem starting from z (see D.W. Stroock and S.R.S. Varadhan [8]), that is, Qz,y is the unique probability on (~2,5r), such that Q~,~(x(O) = z) = 1, f(x(t) ) - ~ot Ls,yf(z(s) ) ds (2.8) is a martingale under the law Qz,u on the time interval [0,1) for any f e C~ ° (Ra). Then A. Millet, D. Nualart and M. Sanz [3] have proved the following T h e o r e m 2.1. (See Theorem 2.3 in [3]) Under Q~,y the probability law of (x(1 -t))t<l is the same as the law of (z(t))t<l under P~,~. That is, if ~'1 denotes the time reversed operator: (x o Vl)(t) = x(1 - t) for any 0 < t < 1, then (2.9) P.,.(F) = Qz,.(F o for any bounded function F E ~'. But the time reversed diffusion Qz,y is not good enough for our purpose. In fact, Aroson's estimate for Lt,y-diffusion is unavailable. We consider the A-diffusion process, where A= - ,. zt3 (2.101 $ Denote by q(t, y, z) and (Py)yeR~ the transition function and the diffusion measure family corresponding to the operator A, respectively. Then q(t, y, z) also satisfies Aronson's estimate (2.2) for the same constant M. For any given points y and z, denote by Pz,y the probability on (f~,br), such that for any t : 0 < t < 1, d~,y ] = p(1-t,y,z(t)) p(1,y,z) (\- jot f vb(x(s))ds ) (2.11) Pz,~ is well defined provided that div b is bounded. In fact, P(1--t'Y'Z(t))exp(--~oP(1, y,z) t divb(x(s))ds) is a martingale under Pz on the time interval [0, 1) (using Ito's formula and the forward Fokker-Planck equation (2.4)). Using Aronson's estimate (2.2), we know that 15~,y is well defined if div b is bounded. T h e o r e m 2.2. For any y, z E R d, Px,v = Q=,u, (2.12) No.2 ARONSON'S ESTIMATES AND CONDITIONAL DIFFUSION PROCESSES 151 that is, Q=,y is the unique probability on (ft, ~'), such that d-Px p(1, y, z) exp (/o' - ) div b(z(8)) d8 (2.13) for any t : 0 < t < l . Proof. By the assmnptions made on a and b, and Aronson's estin2ate , we know that the L,,u-maxtingale problem is well-posed. So it suffices to show that P=a is a solution of the Lt,v-maxtingale problem starting from z. For any f E C~°(Rd), f(x(t)) - fo A.f(z(s))d8 is a P--z-martingaleon [0, i). Using Girsanov'stheorem,one knows that f(z(t)) - j ( Af(z(s))ds - f ' -~a d(M,, f(z(s)), is a P=,~-martingMe on [0, 1), where for simplicity we denote by Mt the martingale Mt = P(1-p(q,y,z) t' Y'Z(t)) exp ( - fo Set N~ = p ( 1 - t , v,~(O) exp (f - divb(x(s))ds). divb(~(s))& ) (2.14) (2.15) . We note that f divb(z(s))ds), f(x(s))) = lP(1-- t, Y,Z(t)) exp ( - jo '(/o"divb(z(s))ds ) =Is exp -- ~Z d(p(1- " exp s, y,z(s)), f(z(s))) , - s, u , x ( 8 ) ) a,jtzCs))@P(10z, i j dO of(~(8)) Ozj ' where we use Ito's formula for semimaxtingMes p(1 - t, y, =(t)) and z(t), and the fact that p(1 - t, '(/o" = ~ exp (/0' y, zCt)) exp - div b(z(s)) divb(zCs))ds ) ) d8 dp(1 - s, y, z(8)) + bounded variation process. Hence we get = f t l~ d(N., f(z(,))) = ~ijfo* p(1 1 Op(I-s,y,z(a))~(z(S))ai.CzCs))d 8 - s, y,z(s)) b'-x: ~ ' " 152 ACTA MATHEMATICAE Thus A P P L I C A T A E SINICA Vol.10 t f(z(t)) - fo Ls,yf(z(s)) ds is a martingale under the law Pffi,y. On the other hand, it is easy to see P~,v(z(0) = z) = 1. I The proof is completed. Remark. If aij and bi are Lipschitz, divb is bounded, and p(t,y,z) is C a, then Theorems 2.1 and 2.2 also hold. §3. C o n d i t i o n a l Diffusion in a Small Time Interval In this section, we give several results on the compact properties of a conditional dit~sion process in a small time interval as applications of Theorem 2.2 and Aronson's estimate. We use the notations and assumptions established in Section 2. For each ~ E (0,1], denote p(~,y,z) by p~(t,y,z). Then pe(t,y,z) is the transition density function corresponding to the infinitesimal generator eL. Denote by P~, P~,z and Q~.y the probability measures on (f~,yr) corresponding to the operator eL as in Section 2 for the generator L, and by ~ the ~A-diffusion measure family on (f~,~'). In particnlar, under Q~,~, the law of (z(t))t<_l is the same as the law of (z~l - t))t<_l under P~,z, and dQez'~lytPc(1-t'Y'z(t))d~'~z = p~(1, y, z) exp ( - e t d i v b ( z ( s ) ) d S ) f o . (3.1) We note that the ~A-~ffusion process has the transition density function q~(t,y,z) q( et, y, z), satisfying M(~'t)~ exp - ~t ) -< qs(t' y, z) < --(et)~ exp - = (3.2) for(t,y,z) EC0,11XR dXR d. Of course, (3.2) also holds for pc(l, y, z). As a consequence of (3.2) and the strongly Markovian property, we have (see Z. Qian and G. Wei [4]) L e m m a 3.1. There is a constant ct, depending only on A, d and [bloo and Idivb[oo, such t h a t for any ~ E (0,1], ( sup Ix(8) \~eC0,tl _ - yl -> " ) -< exp (3.3) for any (t, y, r) E (0,1] x R d x (0, co). (3.3) also holds for ~ with the same constant t~. T h e o r e m 3.2. There is a positive constant C, depending only on A, d, [b[~o and Idivbloo, such that for any ~ E (0, 1], (\0<_t_<l sup I ' ( 0 - yl -> " < - for any r :> 0, y, z E R d such that r :> 2 [ y - z[. p~(1,y,z) exp (3.4) No.2 ARONSON'S ESTIMATES A N D CONDITIONAL DIFFUSION PROCESSES 153 Proof. Since P~,z (sup ]z(t)- y[ _~r) \0<_t_<l for the firstterm on the rightside,using (3.3)and Aronson's estimate (3.2),we get ,z(t)-y[>.r) />~.z\o<,<½(sup (ii =~ p.(1,y,z) <(~)~ M : sup ) I~(O-~l>_r 0<t<½ (3.5) p.,. (sup I~(O-vl>~) p,(1,v,z) ~ \0<L<½ -- p~(1,y,z) exp \ ae / " For the second term, using Theorem 2.2 and Aronson's estimate, we get P~y,. ( s u p .z(t)--y.>r) \½<t<l =Q:,.(\½_<t<: sup ,z(l-t)-V,>_r) _Q.,y< " (sup \o<t< ½Ix(t)-zl_>2) 1 I :P-':z~ ..(1,y,z) ~ exp(-~o < (2),Mexp(le]divb[oo,._~z e z p , (1, y, z) ____(2) _ ' aM exp(1]divbloc) : o<t<½sup ( sup .z(t) _ \o<t< ½ _ (3.6) ) z]> 2) _ r2 Combining (3.5) and (3.6), we get (3.4) immediately. | For any z , y E R d, denote by d(z,y) the geodesic metric from x to y. By definition, we have (see D.W. Stroock [6]) d(~,v) = sup {1¢(-)- ¢(v)l : ¢ e O1(Rd; R) where and r(¢) -I(V¢, ..v¢)(~)l~ < ~o. r(¢) < I}, (3.7) 154 ACTA MATHEMATICAE APPLICATAE SINICA Vol.10 It is well known that t-'-*OO 2, in p(t,z,V) = -d2Cz, v) (3.8) uniformly over x, V such that Ix - Yl is bounded. Moreover, by using (1.3), we know that there is a constant ~ E (0,1], such that ~1~ - ~1 -< d(x, v) _< ~-~I~- vl for any x, y E R d. For the proof of (3.8), refer to S.I~.S. Varadhan [9] and Z. Qian and G. Wei [4]. As a consequence of Theorem 3.2 and (3.8), we have C o r o l l a r y 3.3. For any y, z E R d, we have li.,~up limsup ~ In ~.. r-*oo ¢-*0 ( sup ~0<t<l IxCt)- yl-> ,"] = -~. / (3.9) Corollary 3.3 shows that (P$,z)¢e(0,1] is sequence compact for each pair y , z G R d. The following Theorem 3.4 is a key to establish the large deviation properties for the conditional diffusion processes in a small time interval. T h e o r e m 3.4. There is a constant C, depending only on A, d, [bloo and Idivbloo such that ) P;" ~,o_<,<, ( sup IxCt)-xCto)l >_* __ (ao)-Jp.Cl, v,. ) e ~ ('2) 4~.(T--to) (3.10) for any & > O, O < to < l and y, z E R d. Proof. Using Theorem 2.2, we get P;,z ( sup I x ( t ) - x(to)l > 6'~ / \to___*<l =Q:.~ ( sup Ix(1- t)- ~(1-~)1-> ~]/ \*o_~*$1 =Q*z,y ( sup \0_<t<l--to <. _2Q,., Ix(t)-z(1- (... Ix(t)- h0St<l-to t0)l > ~ / z, ___ .2~_..: (,¢ (~;O. ~/..(1Z to))exp ( ~l-todivb(x(s))da ) : \ p,(1,y,z) --, < p¢(1, y , z ) P--~z ( sup \0<t_<l--t0 sup I x C t ) - , l 0_<,_<1-~o _> ~) IxCt)-zl_> §4. Large Deviation Property In this section, we will establish a large deviation principle for a conditional diffusion process in a small time interval. Throughout this section, we assume a(- ) is a continuous, symmetric and matrix-valued function satisfying (1.3), b(. ) is a bounded measurable R dvalued function. We use the notations developed in the above three sections. It was proved in [4] that (P~) has large deviation properties with the rate 5ruction 1 I(~) = ~ [ jo 1 <~(t), a - ' (~(t)).~(t)> at. No.2 ARONSON'S ESTIMATES AND CONDITIONAL DIFFUSION PROCESSES 155 That is, for any closed set C C fl and open set G C fl, nm s u p , I- P;((7) < - ~ , *--*O I(~,), (4.1) n m i ~ ~ In P$(G) > -- ¢--*0 inf I ( ~ ) , ~EGy where, denote by Cy the set {w 6 C : w(0) = y}. The main result in this section is the following T h e o r e m 4.1. Under the above assumptions made on a and b, if we further have lira sup limsup 6 In P~,z ( sup Iz(t)- x(to)l > 6) = -co ~o?1 for any 6 > 0, then / \~o <t:<l e--*0 (P~,x) has the large deviation property Jy,=Cw) = I(w) - (4.2) if e --* 0, with the rate function d2(y,z), that is, for any closed set C C fl and open set G C n, we have limsup e In P$,,(C) < - inf Ju,z(w), ~-.0 ~ec,,,, liminf e In P~,,(G) > - in[ Jv,zCw). • --*0 -caEGy,. (4.3) Here we denote by Or,, the set {w 6 (7 : w(0) = y and w(1) = z}. Using Theorem 3.4 and Theorem 4.1, we get T h e o r e m 4.2. If a(. ) and b(. ) are C~, then (py. e, = ) satisfies (4.2), and hence (P~,=) has the large deviation property, i.e., (4.3) holds. Proof of Theorem 4.1. Because much of the proof is similar to the proof of Theorem 2.1 in [5], we give details only when new dit~culties are likely to rise. For each positive integer n > 3, denote by lr, the partition: 0 = *0 < tl < --- < t,~ = 1 with ti = i/n, by Tn the mapping: fl -* R dx'* defined by T.w = {w(to),"" , w ( t . - 1)}, (4.4) and by 8 . the mapping: n -+ R dX("+l) 8nw = {wCto),'" ,w(*.)}. Using the fact t h a t for any Borel set A C R dx", T~IA 6 ~ . - x , Markov property, we get P~'2(T~I A) = P~ ( p'( l - " - l ' y,z(t"-l ~) )' (4.5) and hence using (2.7) and : T~I A ) i • -1-= p . (1.V, zl,P.(Sg A), where i t . = {@o,--- , v . - 1 ) e A : No = v} a n d A = {(No,... ,v,-~,~) : (No,..-,u,-1) e A}. 156 ACTA MATHEMATICAE APPLICATAE SINICA Vol.10 Noting that each fixed y, (f~),e(o,z] has the large deviation property with rate function I, and using (3.8), we have L e m m A 4.3. If A C R dxn is a closed set, C = TjXA, then 1~., sup ~ m J~,=(~). P ~ , = ( C ) _< - ¢-'*0 /'root:. (4.7) Obviously A is a dosed set; hence ~ s u p ~ I. ~ ( s = ' ] ) _< e-.,.O ~ _ I(~), wECI, (4.8) where U = S~'IA. It is easy to check that Uu = Cu,z. Hence by Using (3.8) and (4.6), we get the lemma. II For each positive integer n _> 3, let zn(t) denote a process on (f~,£:) defined by: zn(t~) = =(t~) if 0 < i < n - 1; z(t~) and z(t~+l) are joined by one of geodesics on [t~,t4+,] if 0 < i < n - 2; and z.(t) = z(tn-x) if t E [t~_x, 1]. By the same method, for each w E l l , define an w. E f~ such that zn(t;w) = x(t;w.). Then by definition, we have sup O<t<l L e m m A 4.4. d(z(t),z.(t)) < 2 sup 0~<~--I sup t$ <:t~tj+l d(z(t),z(tj)). (4.9) For any 5 > 0, / sup e--*O • in ~-.-~0 [ sup \ O < t _1 >_ \ / = -oo (4.10) Proof. By/(4.9), we get sup o<_t<1 d(z(t),zn(t)) ~ 26) sup sup 0<k<n-1 t~<t<t~+, d(z(t),z(tj))> 6! n--1 -<E j=0 n--2 j=0 \ t i <t<t~+, p~ (p~(1 - t j + l , x(tj+l) , z) : t~<t<tj+, sup pe(1, y, z) +P~v,-(ka~<,<lsup d(=(t),=0~)) > 6) d(z(t),z(tn -1)) > 6) . Using Amnson's estimate and Lemr-a 3.3, we know that the first term on the right side is less than a ( n - 1)M From the assumption (4.2), (3.8) and the above inequality, we deduce (4.10). [[ Using Lemmas 4.3 and 4.4, and by the same argument as given in the proof of Theorem 2.6 in [5], we can prove the upper bound of (4.3). On the other hand, from the proofs of No.2 A R O N S O N ' S ESTIMATES AND CONDITIONAL DIFFUSION PROCESSES 157 L e m m a 2.7 and Theorem 2.8 in [5],we know that the lower bo-nd of (4.3) is a consequence of L e m m a 4.4 and the upper bound of (4.3). Thus we complete the proof of Theorem 4.1. Acknowledgements. T h e a u t h o r w o u l d like t o tban]c P r o f e s s o r K . A . Yen for useful c o n v e r s a t i o n s o n t h i s p a p e r , a s well as P r o f e s s o r S h e n g - w u H e for helpful suggestions. References [1] Bismut, J.M. Large Deviations and the Malllavin Calculus. Birkh~user, 1984. [2] Hsu, P. Brownian Bridges on RiemAnn~.u manifolds. Prob. Tb. Re]. Fields, 1990, 84: 103-118. [3] Millet, A., Nualart, D. and Sanz, M. Integration by Parts and Time Reversal for Diffusion Processes. The Ann. o f Prob., 1989, 17 (I): 208--238. [4] Qian, Z. and Wel, G. Large Deviations for Symmetric Diffusion Process. Chin. Ann. of Math., 1992, 13B (4): 430-439. 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