On averaging principle for diusion processes with null-recurrent fast component

Transcription

1 Stochastic Processes and their Applications 93 (21) On averaging principle for diusion processes with null-recurrent fast component R. Khasminskii a; ;1, N. Krylov b;2 a Department of Mathematics, Wayne State University, Detroit, MI 4822, USA b School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA Received 1 April 2; received in revised form 27 October 2; accepted 27 October 2 Abstract An averaging principle is proved for diusion processes of type (X (t);y (t)) with null-recurrent fast component X (t). In contrast with positive recurrent setting, the slow component Y (t) alone cannot be approximated by diusion processes. However, one can approximate the pair (X (t);y (t)) by a Markov diusion with coecients averaged in some sense. c 21 Elsevier Science B.V. All rights reserved. MSC: primary 6J6; secondary 6B1 Keywords: Averaging principle; Null-recurrent diusion; Arcsine law; Homogenization 1. Introduction The problem of limit behavior of a slow component of multidimensional Markov diusion process has many applications in physics, biology and other areas. One possible settingof this problem is as follows. Assume that we are given a Markov diusion process (X (t);y (t)) consistingof two components X (t) and Y (t) and dependingon the parameter which tends to zero. Then we are interested in what happens if, as ; X (t) changes faster and faster in time and Y (t) lives in the same time scale for all. This problem was originally considered in Khasminskii (1968). Roughly speaking, the main result of Khasminskii (1968) can be described as follows. Let the generator L (x; y) of the process (X (t);y (t)) have the form L (x; y)= 2 L 1 (x; y) 1 L 2 (x; y)l 3 (x; y); Correspondingauthor. addresses: rafail@math.wayne.edu (R. Khasminskii), krylov@math.umn.edu (N. Krylov). 1 Partially supported by NSF Grant DMS Partially supported by NSF Grant DMS /1/$ - see front matter c 21 Elsevier Science B.V. All rights reserved. PII: S ()97-1

2 23 R. Khasminskii, N. Krylov / Stochastic Processes and their Applications 93 (21) where L 1 (x; y)= l 1 i;j=1 a (1) ij (x; j l 1 b (1) i (x; i L 2 (x; y)= l 1 l 2 j=1 a (2) ij (x; j ; L 3 (x; y)= l 2 i;j=1 a (3) ij (x; j l 2 b (3) i (x; i Suppose that, for any xed y, the Markov process X x;y (t) with generator L 1 (x; y) and satisfying X x;y () = x is ergodic. Assume the density of its stationary distribution exists and denote it by (x; y). Also, denote L 3 (y)= L 3 (x; y)(x; y)dx and let Y y (t) be the Markov process startingat y with generator L 3. Under some additional conditions on the coecients, guaranteeing, in particular, compactness of the family of measures generated by Y x;y (t) for and weak uniqueness of the process Y y (t), the averaging principle for the slow component was proved in the followingform: Y x;y (t) distr Y y (t) for any x; y. More precisely, the averaging principle was proved in Khasminskii (1968) (see also Papanicolaou et al., 1977) under a slightly less restrictive assumption than ergodicity of X x;y (t). Namely, it is enough to assume existence for all x; y, non-randomness, and independence of x of the (a.s.) limit 1 T lim L 3 (X x;y (t;!);y)dt = L 3 (y): (1.1) T T Later the result of Khasminskii (1968) was extended in several papers and monographs, see e.g. Papanicolaou et al. (1977), Gikhman and Skorokhod (1982), Skorokhod (1989) and Freidlin and Wentzell (1998). The subject of the present paper is more delicate: we study the limit behavior of the slow component in the situation in which X x;y (t) is a null-recurrent one-dimensional diusion process for each y, and the limit in (1.1) does not exist. We prove that under appropriate conditions, the process (X (t);y (t)) converges weakly to a limit ( X (t); Y (t)), as. In comparison with the ergodic case, the essential dierence is the fact that Y (t) is not Markovian, and only the pair ( X (t); Y (t)) is a Markov process with discontinuous at x = drift and diusion coecients. These coecients are evaluated usingsome sort of averagingthe coecients in the original system. A caricature of the situation we are concerned with can be given by Y (t) = t (w (s)=)dw 1 (s), where w (t);w 1 (t) are independent one-dimensional Wiener processes. It is easy to understand that Y (t) converges in distribution as to the process ( I w(s) I w(s) )dw 1 (s)

3 R. Khasminskii, N. Krylov / Stochastic Processes and their Applications 93 (21) if the limits ± := lim x ± (x) exist. But what happens if at least one of them does not exist or the process Y (t) is not so simple? This is the question we are tryingto answer. 2. Assumptions and the main result We consider a Markov process Z (t)=(x (t);y (t)) with a fast component X (t) R 1 and a slow component Y (t) R d. Assume that this process is a solution of Itô s dierential equations dx (t)= 1 (X (t);y (t)) dw(t); dy (t)=b(x (t);y (t)) dt (X (t);y (t)) dw(t); X () = x; Y () = y: (2.1) Here (:;:)=( ij (:;:)) is d k matrix, (:;:)=( 1 (:;:);:::; k (:;:)); w(t)=(w 1 (t);:::;w k (t)) ; where w i (t) are independent standard Wiener processes. It is well known that the generator of Z (t) is given by with L (x; y) = 2 a a ij (x; j 2 a i (x; i b i i := 2 L 1 (x; y) 1 L 2 (x; y)l 3 (x; y) (2.2) A =(a ij ) i;j=1;:::;d = 1 2 ; a = 1 2 k 2 i ; a i = 1 2 k i ij : We suppose that the followingconditions concerningthe coecients of (2.1) are satised. (A1) The coecients ; b; are Lipschitz continuous in (x; y) and; for each x; their derivatives in y up to and including second-order derivatives are bounded continuous functions of y. (A2) There are positive constants c 1 ;c 2 ;c 3 such that c 1 6a (x; y)6c 2 ; (a ii (x; y)b 2 i (x; y))6c 3 (1 y 2 ): Write for convenience p(x; y)=(a (x; y)) 1.

4 232 R. Khasminskii, N. Krylov / Stochastic Processes and their Applications 93 (21) (B1) The function p(x; y) has a limit p ± (y) in Cesaro sense, as x ± : lim x ± x 1 p(t; y)dt = p ± (y) uniformly in y R d and, moreover, also uniformly in y R d, lim x ± x 1 D y p(t; y)dt = D y p ± (y); lim x ± x 1 Dyp(t; 2 y)dt = Dyp 2 ± (y): Here and below D y u and Dyu 2 are the gradient vector and the matrix of second derivatives in y of u. The same notation is applied to the x variable. We suppose further that the coecients a ij (x; y);b i (x; y); ;:::;d; j =;:::;d and all their derivatives in y up to the second order also have averages in x, as x ± with the weight p(x; y). To write these conditions in a compact form we use the followingnotation: for any function K(x; y) havingthe limit in Cesaro sense as x ± ; K (y) := lim x x 1 K(t; y)dt; K (y) := lim K ± (x; y):=k (y)1 {x } K (y)1 {x6} : x x 1 For a function K(x; y) we write K K if K ± exists and x 1 K(t; y)dt K ± (x; y)=(1 y 2 )(x; y); where the function is bounded and satises K(t; y)dt; lim sup (x; y) =: (2.3) x y R d (B2) For i =1;:::;d; j=;:::;d, we have pb i ;D y (pb i );D 2 y(pb i );pa ij ;D y (pa ij );D 2 y(pa ij ) K: Remark 2.1. It is easy to see that the conditions A ensure the null-recurrence of the process X x;y (t), dened for each y as the Markov process with generator L 3 (x; y) startingat x. Hence, the limit in (1.1), as a rule, does not exist. As is mentioned in Introduction, the limit behavior of the process Z (t)=(x (t);y (t)) =( X (t); Ỹ (t)) will be studied here. It follows from (2.1) that dx (t)= ( X (t)=; Ỹ (t)) dw(t); dỹ (t)=b( X (t)=; Ỹ (t)) dt ( X (t); Ỹ (t)) dw(t); (2.4) X () = x; Ỹ () = y (2.5) and the generator of Z (t) is L (x; y)=l 1 (x=; y)l 2 (x=; y)l 3 (x=; y): (2.6)

5 R. Khasminskii, N. Krylov / Stochastic Processes and their Applications 93 (21) Below we also consider the process (2.4) for the initial conditions X () = x; Ỹ () = y: (2.7) Introduce the notation a ij (x; y)= (a ijp) ± (x; y) p ± ; b(x; y)= (bp)± (x; y) (x; y) p ± (x; y) : (2.8) A(x; y)=(a ij (x; y)) i;j=;1;:::;d ; (x; y)=(a(x; y)) 1=2 : Observe, in passing, that a (x; y)=(p ± (x; y)) 1, and consider the Markov diusion process Z(t)=( X (t); Ỹ (t)), described by the stochastic dierential equation d Z(t) = b( Z(t)) dt ( Z(t)) dw(t); Z() =(x; y); (2.9) with b(z)= b(x; y)=(; b(x; y)). The coecients of (2.9) are smooth functions only in each halfspace {x } and {x } and can have jumps at the hyperplane x =. This circumstance makes not self-evident the uniqueness even of a weak solution of (2.9), which is essential for the approach below. We list some known sucient conditions guaranteeing the weak uniqueness. In 1 3 below A is assumed to be uniformly nondegenerate. 1. In the case p (y)=p (y); (a ij p) (y)=(a ij p) (y); i;j=; 1;:::;d; weak uniqueness follows from Stroock and Varadhan (1979), Chapter For d = 1 (the component Y is also one-dimensional) the uniqueness of the weak solution of (2.9) follows from Krylov (1969), see also exercise in Stroock and Varadhan (1979). 3. It follows from Bass and Pardoux (1987) that weak uniqueness holds if the coecients a ij ; b i are constant in each halfspace {x } and {x }. We believe that weak uniqueness can be proved for essentially more general situations. Therefore, instead of consideration these special cases only, we introduce the followingcondition. C. If problem (2.9) has a solution, it is weakly unique. The main result of this paper is the followingtheorem. Theorem 2.2. Let conditions A C be satised and let Z (t) =( X (t); Ỹ (t)) be a solution of (2:4) and (2:7). Then the process Z (t) converges in distribution to the Markov process Z (t)=( X (t); Ỹ (t)); as. Moreover; the solution of (2:4) and (2:5) converges in distribution to the solution of (2:9) with Z() =(;y). 3. Proof of Theorem 2.2 To prove Theorem 2.2 we shall use several simple lemmas. In these lemmas c; c i ;k stand for generic positive constants and (x; y) stands for generic scalar, vector or matrix functions which are bounded and satisfy (2.3).

6 234 R. Khasminskii, N. Krylov / Stochastic Processes and their Applications 93 (21) Lemma 3.1. Let a function (x; y) f(x; y):r 1 R d R 1 be a Borel function satisfying Assume f(x; y) 6c(1 y k ): (3.1) F(x; y):=x 1 f(t; y)dt =(1 y k )(x; y); (3.2) and let u be a unique solution of the problem Then D 2 xu = f(x=; y); u (;y)=d x u (;y)=: (3.3) D x u (x; y)=x(1 y k )(x=; y); u (x; y)=x 2 (1 y k )(x=; y): (3.4) If; in addition; for each x; the function F has two continuous derivatives in y satisfying D y F(x; y)=(1 y k )(x; y); D 2 yf(x; y)=(1 y k )(x; y); (3.5) then similar bounds for the derivatives in y hold: D y u (x; y)=x 2 (1 y k )(x=; y); D 2 yu (x; y)=x 2 (1 y k )(x=; y); D x D y u (x; y)=x(1 y k )(x=; y): (3.6) Proof. Integrating the equation in (3.3) leads to D x u (x; y)=xf(x=; y). So (3.2) yields the rst equation in (3.4). By integrating it, we get ( x= ) x= u (x; y)= 2 tf(t; y)dt =(1 y k )x 2 (=x) 2 t(t; y)dt (3.7) and the second equation in (3.4) follows. Similarly, by dierentiatingin y Eq. (3.7) and using(3.5), we obtain (3.6). Lemma 3.2. Let conditions A and B be satised. Let V (x; y) be a solution to any one of the following problems (3:8) for i =1;:::;d or (3:9) for i; j =; 1;:::;d: Then a (x=; y)d 2 xu = b i (x=; y) b i (x; y); u(;y)=d x u(;y)=; (3.8) a (x=; y)d 2 xu = a ij (x=; y) a ij (x; y); u(;y)=d x u(;y)=: (3.9) V (x; y)=x 2 (1 y 2 )(x=; y); D y V (x; y)=x 2 (1 y 2 )(x=; y); D 2 yv (x; y)=x 2 (1 y 2 )(x=; y); D x V (x; y)=x(1 y 2 )(x=; y); (3.1) D x D y V (x; y)=x(1 y 2 )(x=; y): Here; as before; (x; y) are various bounded functions with property (2:3).

7 R. Khasminskii, N. Krylov / Stochastic Processes and their Applications 93 (21) Proof. Upon dividingeqs. (3.8) and (3.9) by a (x=; y) we rewrite them in the form (3.3). Note now that by denition a ij (x; y)= a ij (x=; y); p ± (x; y)=p ± (x=; y); etc: To prove (3.1) for the solution of (3.9) it is enough to check the conditions (3.1), (3.2), and (3.5) for the function (x; y)=p(x; y)(a ij (x; y) a ij (x; y)): Condition (3.1) follows from A2. Further, takinginto account conditions B1 and B2, we have that, for x, (x=; y)=x 1 (t=; y)dt = x 1 p(t=; y)a ij (t=; y)dt (a ij p) (y)(a ij p) (y) [1 (p (y)) 1 x 1 ] p(t=; y)dt =(1 y 2 ) 1 (x=; y)(1 y 2 ) 2 (x=; y) and hence (3.2) holds as well. In the case x, the proof is exactly the same. Let us check now conditions (3.5). The condition y(x; y) c follows from A1. Then, conningourselves for brevity only to the case x leads to D y (x; y)=x 1 D y (t; y)dt = x 1 D y [p(t; y)(a ij (t; y) a ij (t; y))] dt = x 1 D y [p(t; y)(a ij )(t; y)] dt D y ((a ij p) (y)) [ ] D y (a ij p) (y)(p (y)) 1 (p (y) x 1 p(t; y)dt) =(1 y 2 )(x; y) due to conditions B1, B2. Verication of (3.5) for Dy 2 and Dy 2 is completely similar. In exactly the same way one can prove (3.1) for solutions of (3.8). Lemma 3.3. Take a and let ( X (t); Ỹ (t)) be a solution of the problem (2:4) and (2:7). Then; for any k 1; there is a constant c k independent of and such that; for t ; { E } sup X (s) 2k Ỹ (s) 2k 6( x 2k y 2k 1) exp{c k t}: (3.11) s6t For any T ; the inequality E X (t h) X (t) 4 E Ỹ (t h) Ỹ (t) 4 6ch 2 (3.12) holds if t; t h [;T] with constant c independent of t; h; and.

8 236 R. Khasminskii, N. Krylov / Stochastic Processes and their Applications 93 (21) Proof. It is easy to see from conditions A2 that L [( x 2k y 2k 1) exp{ c k t}]6 for c k suciently large. It guarantees that the process ( X (t) 2k Ỹ (t) 2k 1) exp{ c k t} is a local supermartingale, and (3.11) follows. The inequality (3.12) follows from (3.11) and properties of stochastic integrals. Remark 3.4. Lemma 3.3 guarantees compactness of the distributions of the processes ( X (t); Ỹ (t)) in C[;T] for any T, see Prokhorov, Let F t be the monotone family of -algebras generated by Z (s) for 6s6t. Lemma 3.5. For any T ; ;:::;d; and j; k =; 1;:::;d; we have (b i ( X (s)=; Ỹ (s)) b i ( X (s); Ỹ (s))) ds ; (3.13) sup t6t sup t6t as in probability. (a jk ( X (s)=; Ỹ (s)) a jk ( X (s); Ỹ (s))) ds (3.14) Proof. The proofs of assertions (3.13) and (3.14) are similar, so we prove only the rst one. Let V be the solution of (3.8). It has two derivatives in (x; y), which are perhaps discontinuous only at x=. Due to results of Krylov (1977) and nondegeneracy condition in A2 one can still apply Itô s formula to V ( X (t); Ỹ (t)) and get V ( X (t); Ỹ (t)) = V (x; y) where, for y := x, A t = B t = j k j=1 (b i ( X (s)=; Ỹ (s)) b i ( X (s); Ỹ (s))) ds A t B t M t ; a ij ( X (s)=; Ỹ j ( X (s); Ỹ (s)) ds; b j ( X (s)=; j ( X (s); Ỹ (s)) M t ( X (s); Ỹ (s)) ( X (s)=; Ỹ (s)) dw s D y V ( X (s); Ỹ (s))( X (s); Ỹ (s)) dw s : (3.15)

9 R. Khasminskii, N. Krylov / Stochastic Processes and their Applications 93 (21) It follows from (3.1) that V (x; y). By usingagain (3.1) and also Lemma 3.3 and bearingin mind the equality 1=I { X (t) } I { X (t) } ; we obtain E sup t6t V ( In the same way, E sup t6t X (t); Ỹ (t)) = E sup t6t 6 N sup x A t 6NE sup(1 Ỹ (t) 2 ) t6t X 2 (t)(1 Ỹ (t)) k ( X (t)=; Ỹ (t)) sup (x=; y) M : y ( D x D y V ( X (t); Ỹ (t)) D 2 yv ( X (t); Ỹ (t)) ) ; E sup B t : t6t It follows from (3.15) that to prove (3.13) it only remains to prove that sup t6t M t in probability. This is known to be equivalent to provingthe convergence of the quadratic variations of the martingales M t to zero. The latter is done followingthe same pattern as above. Now we can prove the theorem. First we outline the main ideas. Notice that the process ( X )(t); Ỹ (t) b( X (s)=; Ỹ (s)) ds is a martingale. Due to Lemma 3.3 by utilizingskorokhod s theorem from [13], for any sequence n going to zero one can choose a subsequence n such that the followingholds: 1. There are processes Z n (t), dened on some probability space havingthe same nite-dimensional distributions as Z n (t). 2. Z (t) n Z(t) =( X (t); Y (t)) in probability, where Z(t) is an a.s. continuous stochastic process. Further, similarly to the proof of Lemma 4:1 in Khasminskii (1968) one can establish that ( X (t); Y (t) b( X (s); Y (s)) ds) is a martingale, and Z(t) satises Eq. (2.9). An implementation of this argument in a much more general setting can be found in Liptser and Shiryayev, 1989 (see Theorem 8:3:3 there) under the additional assumption that a and b are continuous. In our situation, generally speaking these coecients are discontinuous at x =, which is a set of Lebesgue measure zero. However, the x component of our processes is uniformly nondegenerate and this allows one to carry out the program in our case as well. The details of passing to the limit in the case of coecients with Lebesgue measure zero of the set of their discontinuity can be found in Chao, Yi-Ju (1999). Now, the assertion of Theorem follows from Condition C. 4. Corollaryand example Theorem 2:1 provides some information on the limit behavior of solutions to elliptic and parabolic dierential equations with dierent scales of variables. We formulate here one result of this type. Below by W 2;1 d1;loc we mean the space of functions u(x; y; t) on

10 238 R. Khasminskii, N. Krylov / Stochastic Processes and their Applications 93 (21) [; ) R d1 which are bounded and continuous and have generalized derivatives of rst and second orders in (x; y) variables, as well as the rst generalized derivative in t, locally summable to the power d 1 in (; ) R d1. Corollary4.1. Let the coecients of the operator L (x; y) satisfy conditions A and B and let the matrix (a ij ) d i;j= be uniformly nondegenerate. Denote L(x; y)= L(x; y)= i;j= i;j= a i;j (x; y) a i;j (x; j b i i ; b i i (here; as before; we denote x = y ; the coecients a i;j ; b i are dened in (2:8)). Also assume that; for any bounded and innitely dierentiable function (x; y); the u = L(x; y)u; u(x; y; ) = (x; y) has a unique bounded solution u(x; y; t) W 2;1 d1;loc. Then; for any bounded and in- nitely dierentiable function (x; y); bounded solutions u (x; y; t) W 2;1 d1;loc of the = L(x=; y)u ; u (x; y; ) = (x; satisfy lim u (x; y; t)=u(x; y; t): The proof follows immediately from the well-known probabilistic representations of u (x; y; t) and u(x; y; t) (see, e.g. Section 2:1 of Krylov, 1977), Theorem 2.2, and the fact that condition C is equivalent to uniqueness of solution to (4.1). Remark 4.1. One can get similar results for inhomogeneous parabolic equations. For instance, let f(x; y) be a function such that the function f(x; y)p(x; y) and its derivatives in y up to and includingthe second order derivatives have limits in Cesaro sense for x ± uniformly in y. Denote f(x; y)= (fp) (y) p (y) 1 {x } (fp) (y) p (y) 1 {x6}: Also assume that the conditions of Corollary hold. Then the solutions v (x; y; t) of the problem W 2;1 = L(x=; y)v f(x=; y); v (x; y; ) = (x; converge for to the solution v(x; y; t) W 2;1 d1;loc of the = L(x; y)v f(x; y); v(x; y; ) = (x; y): The proof follows from the probabilistic representation of v (x; y; t) and the arguments used in Section 3.

11 R. Khasminskii, N. Krylov / Stochastic Processes and their Applications 93 (21) Example 4.1. Consider the simplest situation where the conditions of Theorem 2:1 are satised, p ± ; (bp) ± ; (a 11 p) ± ; (a 1 p) ± are constants, and (a 1 p) ± =; (a 11 p) =(a 11 p) = a 11 p: (4.2) In this case, Eq. (2.9) can be solved explicitly. So we have from Theorem 2:1: the solution (X (t);y (t)) of (2.1) has the followinglimit behavior: X (t) distr X (p ;p ;t):=x (t); ( ) Y (t) distr (bp) (bp) ( ) 1=2 p t p (bp) 2a11 p p 1 {X(s) } ds w 1 (t): p Here X (t) is a unique continuous solution of the SDE dx (t) = [(2=p ) 1=2 1 {X(s) } (2=p ) 1=2 1 {X(s)6}]dw (t); X ()=; w (t);w 1 (t) are independent standard Brownian motions. It is easy to see from Portenko (199, p. 143) that the transition probability density for X (t) can be written as follows: Here G a ;a (t; x; y)= (2a ) 1=2 G c (t; x= 2a ;y= 2a ) if x ; y ; (2a ) 1=2 G c (t; x= 2a ;y= 2a ) if x ; y ; (2a ) 1=2 G c (t; x= 2a ;y= 2a ) if x ; y ; (2a ) 1=2 G c (t; x= 2a ;y= 2a ) if x ; y : a a G c (t; x; y)=g(t; x y)1 {y } cg(t; x y ); c= a a and g(t; x y) is the transition probability density of standard Brownian motion. It is also known (see Watanabe, 1999; Khasminskii, 1999) that the random variable (t)= 1 {X (s) } ds has the generalized Arcsine distribution: for x P{(t)=t x} = 2 p x sin 1 p (p p )x : So we see, in particular, that, up to some constants, the distribution of the slow component for xed t in this example converges to the convolution of the generalized Arcsine and Gaussian distributions if (bp) =p (bp) =p. References Bass, R., Pardoux, E., Uniqueness for diusions with piecewise constant coecients. Probab. Theory Rel. Fields 76, Chao, Yi-Ju, Diusion approximation of a sequence of semimartingales and its application in exploring the asymptotic behavior of some queueingnetworks. Ph.D. Thesis, University of Minnesota. Freidlin, M., Wentzell, A., Random Perturbations of Dynamical Systems, 2nd Edition. Springer, New York. Gikhman, I., Skorokhod, A., Stochastic Dierential Equations and their applications. Naukova Dumka, Kiev (in Russian).

12 24 R. Khasminskii, N. Krylov / Stochastic Processes and their Applications 93 (21) Khasminskii, R., On averaging principle for Itô stochastic dierential equations. Kybernetika, Chekhoslovakia 4 (3), (in Russian). Khasminskii, R., Arcsine law and one generalization. Acta Appl. Math. 58, Krylov, N.V., Diusion in the plane with reection. Construction of the process, Sibirsk. Mat. J. 1 (2), (in Russian); English translation in Siberian Math. J., 1 (2), Krylov, N.V., Controlled diusion processes. Nauka, Moscow, 1977 (in Russian; English translation), Springer, Berlin, 198. Liptser, R.S., Shiryayev, A.N., Theory of Martingales. Kluwer, Dordrecht, Boston. Papanicolaou, G.C., Stroock, D., Varadhan, S.R., Martingale approach to some limit theorems. Duke Univ. Math. Ser. III Portenko, N., 199. Generalized diusion processes. Translations of Mathematical Monographs, Vol. 83, AMS, Providence, RI. Prokhorov, Yu., Convergence of stochastic processes and the limit theorems. Theory Probab. Appl. I (2), Skorokhod, A., Asymptotic Methods in the Theory of Stochastic Dierential Equations. Translations of Mathematical Monographs, Vol. 78, AMS, Providence, RI. Stroock, D., Varadhan, S., Multidimensional Diusion Processes. Springer, New York. Watanabe, S., Generalized arc-sine laws for one-dimensional diusion processes and random walks. Proceedings of the Symposium in Pure Mathematics, Vol. 57, pp