TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 351, Number 10, Pages 3915–3934 S 0002-9947(99)02421-6 Article electronically published on May 21, 1999
RATES OF CONVERGENCE OF DIFFUSIONS WITH DRIFTED BROWNIAN POTENTIALS
YUEYUN HU, ZHAN SHI, AND MARC YOR
Abstract. We are interested in the asymptotic behaviour of a diffusion pro- cess with drifted Brownian potential. The model is a continuous time analogue to the random walk in random environment studied in the classical paper of Kesten, Kozlov, and Spitzer. We not only recover the convergence of the dif- fusion process which was previously established by Kawazu and Tanaka, but also obtain all the possible convergence rates. An interesting feature of our ap- proach is that it shows a clear relationship between drifted Brownian potentials and Bessel processes.
1. Introduction Modelling a class of random phenomena in physics and biology (cf. Bouchaud et al. [4]), the simple Random Walk in Random Environment (RWRE) is defined as def follows: Let Ξ = ξi i Z be a sequence of independent and identically distributed random variables{ taking} ∈ values in (0,1), and any realization of Ξ is labelled as “environment”. Define the RWRE Sn n 0 by S0 =0and { } ≥ ξi, if j = i +1, P Sn+1 = j Sn = i;Ξ = 1 ξ, if j = i 1, − i − 0, otherwise, for any n 1andi Z. Among the best known works on RWRE, Solomon [30] treats the≥ basic recurrence/transience∈ problem, Sinai [29] and Kesten et al. [21] es- tablish limit theorems respectively in recurrent and transient situations, Deheuvels and R´ev´esz [9] investigate the almost sure behaviour in Sinai’s setting, and Cs¨org˝o et al. [8] study the local time. We refer to the books of R´ev´esz [25] and Hughes [16] for a detailed account. There is also a renewed interest concerning various aspects of RWRE. Let us mention recent papers by Alili [1], Comets et al. [7], Dembo et al. [10], Gantert and Zeitouni [12], Greven and den Hollander [13], Hu and Shi [14]–[15], and Shi [28]. In the recurrent case treated in the celebrated paper of Sinai [29], it is shown that RWRE moves far slower than the usual random walk (in non-random environment). On the other hand, the convergence and rates of RWRE in the transient case are completely characterized by Kesten et al. [21]. In this paper, we are interested in the continuous time analogue of RWRE, namely, a diffusion process with random potential, introduced by Schumacher [27]
Received by the editors November 17, 1997 and, in revised form, July 3, 1998. 1991 Mathematics Subject Classification. Primary 60J60, 60F05. Key words and phrases. Diffusion with random potential, Bessel process, rate of convergence.
c 1999 American Mathematical Society
3915
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 3916 YUEYUN HU, ZHAN SHI, AND MARC YOR
and Brox [5]. In what follows, W (x); x R denotes a two-sided Brownian motion with W (0) = 0, and { ∈ } κ (1.1) V (x) def= W (x) x, x R, − 2 ∈ is a Brownian motion with drift ( κ/2). Consider the formal solution of the fol- lowing stochastic differential equation:− 1 dX(t)=dβ(t) V0(X(t)) d t, (1.2) −2 X(0) = 0, where β denotes a Brownian motion starting from 0, independent of W (hence of the random potential V ). We note that changing κ into ( κ) leads to the same formal equation for ( X), since β(t); t 0 and W ( −x); x R are independent Brownian motions.− Hence,{− we may,≥ without} loss{ of− generality,∈ } restrict ourselves to κ 0. In fact, we shall only discuss the case κ>0, which characterizes the transience≥ of X (see Brox [5] for the recurrence with κ = 0, and Kawazu and Tanaka [19] for the transience with κ>0). Rigorously speaking, instead of writing the formal derivative of V in (1.2), we should consider X as a diffusion process with generator 1 d d (1.3) eV (x) e V (x) . 2 d x − d x A representation of X(t); t 0 using time change is given in Section 3. We refer to Kawazu and Tanaka{ [19],≥ Mathieu} [23], and Carmona [6] for related studies and further extensions. This paper aims at completing previous work of Kawazu and Tanaka [19], [20], which respectively obtain the convergence of X, and its convergence rate in the particular case κ>2. Our main result is a complete characterization of all possible convergence rates, therefore providing the continuous time analogue of what Kesten et al. [21] established for transient RWRE. Our approach is new, mainly based on Bessel process techniques by showing how Bessel and Jacobi processes can be used in the study of random potentials. To state our result, we define
(1.4) H(r) def=inf t>0: X(t)>r ,r>0, { } the first hitting time process associated with X. Throughout the paper, stands ca N for a Gaussian (0, 1) variable, α (for 0 <α<2) a completely asymmetric N caS stable variable of index α,and λ (for λ>0) a (positive) completely asymmetric Cauchy variable of parameter Cλ (the superscript “ca” standing for “completely ca asymmetric”). When 0 <α<1, α is a positive variable. The distributions of ca and ca are characterized by theirS respective characteristic functions: for t R, Sα Cλ ∈ πα E exp(it ca)=exp t α 1 i sgn(t)tan , Sα −| | − 2 h 2 i E exp(it ca)=exp λ t +it log t . Cλ − || π | | h i The symbols “ law ”, “ P. ”and“a.s.” denote convergences, respectively, in distri- bution, in probability−→ −→ and with probability−→ one.
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use DIFFUSIONS WITH DRIFTED BROWNIAN POTENTIALS 3917
Throughout the paper, we write π (1.5) c def=2γ+log , 0 4 πx 1/x (1.6) ψ(x) def= , 0
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 3918 YUEYUN HU, ZHAN SHI, AND MARC YOR
Notation. The usual symbol a(x) b(x)(x x0) means limx x0 a(x)/b(x)= 1. We use BES(d) (resp. BESQ(d))∼ to denote→ a Bessel process→ (resp. squared
Bessel process) of dimension d. Furthermore, log2 stands for the iterated logarithm function. Unless stated otherwise, ε( ) denotes an asymptotically negligible random · term in the sense that ε(t) P. 0(ast ), whose value may change from one −→ →∞ line to another. Finally, we shall also use, whenever possible, the notation d0 for d/2, which will often be convenient in our formulae involving dimension parameters d1 and d2. Acknowledgment. We are grateful to the referee for valuable comments which helped us to improve the presentation of our paper and give more accurate refer- ences.
2. Bessel and Jacobi processes
Throughout this section, R1(t); t 0 and R2(t); t 0 denote independent Bessel processes, of dimensions{ d and≥d }respectively.{ We≥ assume} R (0) = r 0 1 2 1 1 ≥ and R2(0) = r2 > 0. Our aim is to study the asymptotic behaviour of t R2(s) (2.1) Ξ(t) def= 1 d s, R4(s) Z0 2 as t tends to infinity. Observe that, in order to ensure the finiteness of Ξ(t), we have to take d2 2, a hypothesis which, unless stated otherwise, we assume throughout this section.≥ Here is the main estimate of this section, which plays an important role in the def def proof of Theorem 1.1 in Section 3. Recall that d10 = d1/2andd20 = d2/2.
Proposition 2.1. Let d1 > 0 and d2 > 2, and let Ξ(t) be as in (2.1).Whent tends to infinity (writing h(t) def= d (log t)/(d 2)(d 4) for brevity), 1 2 − 2 − Ξ(t) law ca (2.2) c 2 , 2
1/(d0 1) def 1 8 2− c = ψ(d0 1) , 2 2 2 − (d 2)2 B(d ,d 1) 2 − 10 20 − 2 1 h i def d1 d0 1 d1(c0 log(d1 +2)) c = y 1− log(1 y)dy+ − , 3 8 − 4 Z0 1/2 def d1 1 1 2 d2 2 c4 = + + + − , (d20 1)(d2 4) d2 2 d2 4 d2 6 d1(d2 6) − − h − − − − i and B( , ) denotes the beta function. · ·
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use DIFFUSIONS WITH DRIFTED BROWNIAN POTENTIALS 3919
Remark 2.2. We shall not use (2.6) in the paper. It is presented here only for the sake of completeness. It is also possible to treat the case d1 > 0, d2 = 2, for which one can prove that log Ξ(t) 1 law ,t , log t −→ →∞ E where denotes an exponential variable with mean 1. E The proof of Proposition 2.1 relies on some preliminaries, stated below as Lem- mas 2.3 and 2.8. Lemma 2.3. Let β be a standard Brownian motion, with local time Lx; t 0,x { t ≥ ∈ def 0 R .Letτ(r)=inf s>0: Ls >r (for r>0) denote its inverse local time at 0. Fix}a>0,b>0,µ{1and λ>0.Wedefinethefunction} ≥ y d x S(y)def= ,y(0, 1), xa(1 x)b+1 ∈ Z1/2 − and the process 1 def a+µ 1 b µ 1 S(y)/t J(t) = y − (1 y) − − L d y, t > 0. − τ(λ) Z0 Then we have λB(a+µ, b µ)+δ(t),b>µ, − 1 Lx λ Lx 1τ(λ)− ∞ τ(λ) J(t)=c5λ+bλlog t + 0 x d x + 1 x d x + ε(t),b=µ, µ/b 2 ∞ µ/b 2 x µ/b 1 b − 0 x − RLτ (λ) d x + ε(t) tR − ,b<µ, a.s. where δ(t) 0(as t R ),B( , )is the usual beta function, and −→ →∞ · · 1 def log b a+b 2 c = +(a+b 1) y − log(1 y)dy. 5 b − − Z0 Proof of Lemma 2.3. Looking at the integral expression in J(t), it is seen that the only possible explosion comes from the neighbourhood of 1. However, if b>µ,the 1 a+µ 1 b µ 1 integral 0 y − (1 y) − − d y being finite, Lemma 2.3 follows trivially from the dominated convergence− theorem. It remainsR to treat the case b µ.Write ≤ 1/2 1 a+µ 1 b µ 1 S(y)/t J(t)= + y − (1 y) − − Lτ(λ) d y (2.7) 0 1/2 − Z Z def = J1(t)+J2(t).
Again, we can apply the dominated convergence theorem to J1(t) to arrive at 1/2 a+µ 1 b µ 1 (2.8) J (t)=λ y − (1 y) − − d y + ε(t). 1 − Z0 1 We have to deal with J2(t). Writing S− for the inverse function of S,andbya change of variables x = S(y)/t,
∞ 1 2a+µ 1 1 2b µ x (2.9) J (t)=t (S− (tx)) − (1 S− (tx)) − L d x. 2 − τ (λ) Z0
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 3920 YUEYUN HU, ZHAN SHI, AND MARC YOR
Observe that 1 1/b (2.10) 1 S− (v) (bv)− ,v . − ∼ →∞ Therefore, in case b<µ, we have, by the dominated convergence theorem, 1 2b µ µ/b 1 µ/b 2 ∞ 1 2a+µ 1 (1 S− (tx)) − x J (t)=t − b − (S− (tx)) − − L d x 2 (bt)µ/b 2 τ (λ) Z0 − Lx = tµ/b 1bµ/b 2 ∞ τ (λ) d x + ε(t) , − − 2 µ/b 0 x − Z which, in view of (2.7)–(2.8), yields Lemma 2.3 (the constant term in J1(t)being absorbed by ε(t)). Finally, let us assume b = µ. By (2.9),
∞ 1 2a+b 1 1 b x J2(t)=t (S− (tx)) − (1 S− (tx)) Lτ (λ) d x 1 − Z 1 1 2a+b 1 1 b x + t (S− (tx)) − (1 S− (tx)) (Lτ (λ) λ)dx 0 − − Z 1 1 2a+b 1 1 b +λt (S− (tx)) − (1 S− (tx)) d x. − Z0 Now, on the right hand side, we apply the dominated convergence theorem (via observation (2.10), of course) to the first two terms, and use a change of variables x = S(y)/t for the third term. Accordingly, x 1 x 1 ∞ Lτ(λ) 1 Lτ(λ) λ J2(t)= dx+ − dx+ε(t) b 1 x b 0 x (2.11) Z 1 Z S− (t) ya+b 1 +λ − dy. 1 y Z1/2 − To treat the last integral term on the right hand side, let us recall (2.10) and use integration by parts, which leads to: 1 S− (t) ya+b 1 λ log t λ − d y = + o(1) 1 y b (2.12) Z1/2 − log b log 2 1 + λ +(a+b 1) ya+b 2 log(1 y)dy . a+b 1 − b − 2 − − 1/2 − Z Combining (2.7)–(2.8) and (2.11)–(2.12) completes the proof of Lemma 2.3. To get our second preliminary estimate, we have to recall the definition of the Jacobi processes (cf., for example, Karlin and Taylor [18]): d Y (t)=2 Y(t)(1 Y (t)) d B(t)+(d (d +d )Y(t)) d t, − 1 − 1 2 Y (0) = a (0, 1), p∈ where B is a standard Brownian motion. In the literature, Y (t); t 0 is some- { ≥ } times referred to as a Jacobi process of dimensions (d1,d2), starting from a. We need the following known results for Jacobi processes and Brownian local times. Fact 2.4 is due to Warren and Yor [33]. Fact 2.5 is a consequence of the ergodic theorem on path space, cf. for example Revuz and Yor [26, p. 411]. Fact 2.7 is due to McKean [22] and Ray [24]; see, e.g. Itˆo and McKean [17, p. 65 et seq.]; this estimate has been extended by Barlow and Marcus–Rosen to a large class of
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use DIFFUSIONS WITH DRIFTED BROWNIAN POTENTIALS 3921
L´evy processes. Fact 2.6, which clearly shows where the stable laws in Theorem 1.1 come from, can be found in Biane and Yor [3], except for (2.18), whose proof is postponed until Section 4. We point out that the value of c0 in (2.16) is given in (1.5); this corrects a misprint made in Biane and Yor [3, p. 66], in passing from the formula on their (correct) 4th line to their (wrong) 8th line.
Fact 2.4 (Warren and Yor). Let R1(t); t 0 and R2(t); t 0 be independent Bessel processes of dimensions d and{ d respectively,≥ } with{ d +d≥ }2, R (0) = r 1 2 1 2 ≥ 1 1 ≥ 0andR2(0) = r2 > 0. There exists a Jacobi process Y (t); t 0 of dimensions 2 2 { 2 ≥2 } (d1,d2), starting from r1/ r1 + r2, independent of R1(t)+R2(t); t 0 ,such that for all t 0, { ≥ } ≥ p 2 t R1(t) ds (2.13) 2 2 =Y 2 2 . R1(t)+R2(t) 0 R1(s)+R2(s) Z Fact 2.5 (Revuz and Yor). If R(t); t 0 is a Bessel process of dimension d>2, starting from r>0, then { ≥ } 1 t d u 1 (2.14) a.s. ,t . log t R2(u) −→ d 2 →∞ Z0 − Moreover, 1 t d u log t 2 (2.15) law ,t . 1/2 2 3/2 (log t) 0 R (u) − d 2 −→ ( d 2) N →∞ Z − − Fact 2.6 (Biane and Yor). In the notation of Lemma 2.3, for any λ>0, 0
x y 1 sup Ls Ls Θ(T ) x y log , 0 s T | − |≤ s | − | x y ≤ ≤ | − | for all (x, y) R2 with x y 1/2. ∈ | − |≤ Our second preliminary estimate concerns a class of processes which includes Ξ(t) (cf. (2.1)) as a special case.
Lemma 2.8. Let R1 and R2 be independent Bessel processes of dimensions d1 > 0 and d > 2 respectively, with R (0) = r 0 and R (0) = r > 0.Fixµ 1and 2 1 1 ≥ 2 2 ≥ t 2µ def R1 (s) Ξµ(t) = d s, t > 0. R2µ+2(s) Z0 2
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 3922 YUEYUN HU, ZHAN SHI, AND MARC YOR
We have, as t goes to infinity,
Ξµ(t) a.s. B ( d 10 + µ, d20 µ 1) (2.19) − − ,d0>µ+1, log t −→ (d + d 2)B(d ,d ) 2 1 2 − 10 20 Ξµ(t) c6 (log t)(log2 t) law ca (2.20) − c 7 + c 8 ,d20=µ+1, log t −→ C 4/B(d10 ,d20 )
Ξµ(t) law ca (2.21) c 9 , 1
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use DIFFUSIONS WITH DRIFTED BROWNIAN POTENTIALS 3923
where t ds (2.29) U(t) def=4 , Yd1 1(s)(1 Y(s))d2 1 Z0 − − − with U( ) = inf t>0: β(t)=S(0) (hence U( )= if S(0) = ). Consider the inverse∞ process{ of U: } ∞ ∞ −∞
1 def U − (t) =inf s>0: U(s)>t
nt d o1 d 1 1 1 1− 1 2− = S− (β(s)) 1 S− (β(s)) d s, 4 − Z0 1 x where S− is the inverse of S.Write Lt(β); t 0,x R for the bicontinuous local time process of β. Applying the occupation{ ≥ density∈ formula} gives 1 1 1 d0 1 d0 1 S(x) (2.30) U − (t)= x 1− (1 x) 2− L (β)dx. 4 − t Z0 Recall Υµ(v) from (2.23). By (2.29), v Yµ(s) 1 Υ (v)= Y d1 1(s)(1 Y (s))d2 1 d U(s) µ µ+1 − − 0 (1 Y (s)) 4 − Z v − 1 d +µ 1 d µ 2 = Y 1 − (s)(1 Y (s)) 2− − d U(s) 4 − Z0 U(v) d +µ 1 d µ 2 1 1 1 − 1 2− − = S− (β(u)) 1 S− (β(u)) d u, 4 0 − Z where in the last equality we used the representation (2.28). By the occupation density formula,
1 ∞ 1 d +µ 1 1 d µ 2 y Υ (v)= (S− (y)) 1 − (1 S− (y)) 2− − L (β)dy µ 4 − U(v) (2.31) ZS(0) 1 1 d0 +µ 1 d0 µ 2 S(x) = x 1 − (1 x) 2− − L (β)dx. 4 − U(v) Z0 Now, for each fixed v>0, we introduce the Brownian motion: 1 (2.32) β (s) def= β(v2s),s0. v v ≥
Denote its local time process by L(βv), and its inverse local time at 0 by τβv ( ). According to (2.30), · 1 1 1 2 1 d0 1 d0 1 S(x)/v U − (v s)= x 1− (1 x) 2− L (β )dx v 4 − s v Z0 1 = B(d ,d )L0(β )+ε , 4 10 20 s v 1 where B( , ) is as before the beta function, and · · 1 def 1 d10 1 d20 1 S(x)/v 0 ε1 = ε1(v, s) = x − (1 x) − Ls (βv) Ls(βv) d x. 4 0 − − Z Writing, for brevity, 4 (2.33) λ def= , B(d10 ,d20 )
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 3924 YUEYUN HU, ZHAN SHI, AND MARC YOR
P. it is easily seen using Fact 2.7 that sup0 s τ (2λ) ε1 0, as v tends to infinity. ≤ ≤ βv | | −→ Let δ>0 be a small number. On the event sup0 s τ (2λ) ε1 <δ , { ≤ ≤ βv | | } U(v) (2.34) τ ((1 δ)λ) τ ((1 + δ)λ). βv − ≤ v2 ≤ βv
Going back to (2.31), we have, on sup0 s τ (2λ) ε1 <δ , { ≤ ≤ βv | | } 1 v S(x)/v d10+µ 1 d20 µ 2 x − (1 x) − − Lτ ((1 δ)λ)(βv)dx Υµ(v) 4 − βv ≤ (2.35) 0 − Z 1 v S(x)/v d10 +µ 1 d20 µ 2 x − (1 x) − − Lτ ((1+δ)λ)(βv)dx. ≤ 4 − βv Z0 We now complete the proof of Lemma 2.8 by discussing three possible situations.
(i) In case d20 >µ+ 1, this is an immediate consequence of the ergodic theorem on path space.
(ii) Assume d20 = µ + 1. In view of (2.35), applying Lemma 2.3 to a = d10 and b = d0 1 gives (recalling λ from (2.33)) 2 − (2.36) 1 λv log(d0 1) 2 d10 +d20 3 Υµ(v)= − +(d10 +d20 2) y − log(1 y)dy 4 d20 1 − 0 − − Z v 1 Lx λ Lx + λ log v + τ(λ) − d x + ∞ τ(λ) d x + vε(v). 2(d2 2) 0 x 1 x − Z Z In light of (2.16), combining (2.36) with (2.26) (we do need (2.26) here, instead of (2.25) only, to ensure that the error term in (2.25) is negligible) and (2.22) yields (2.20).
(iii) Finally, let us treat the case d20 <µ+ 1. Again, we can use (2.35) and Lemma2.3toseethat
Υµ(v) law 1 ∞ µ/(d20 1) 2 µ/(d20 1) 2 x ( d 20 1) − − x − − Lτ(λ) d x. vµ/(d20 1) −→ 4 − − Z0 Applying (2.17) to p =(d20 1)/µ and using (2.25) and (2.22), we readily get (2.21). −
Proof of Proposition 2.1. The estimates (2.2)–(2.3) being already proved in Lemma 2.8 (taking µ = 1 there); we only have to check (2.4)–(2.6). So let us assume d2 > 4. We write the stochastic differential equations satisfied by R1 and R2: t 2 2 (2.37) R1(t)=r1 +2 R1(s)dβ1(s)+d1t, 0 Z t 2 2 (2.38) R2(t)=r2 +2 R2(s)dβ2(s)+d2t, Z0 where β1 and β2 are independent real-valued Brownian motions, both starting from 2 2 0. Applying Itˆo’s formula to R1(t)/R2(t)gives t ds r2 R2(t) (d 4)Ξ(t)=d + 1 1 2− 1 R2(s) r2 −R2(t) (2.39) Z0 2 2 2 t 2 R1(s)dβ1(s) R1(s)dβ2(s) +2 2 3 . 0 R2(s) − R2(s) Z
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use DIFFUSIONS WITH DRIFTED BROWNIAN POTENTIALS 3925
On the other hand, observe that
t t d β2(s) d s (2.40) log R (t) log r = +(d0 1) , 2 − 2 R (s) 2 − R2(s) Z0 2 Z0 2 which, in view of (2.39), implies
2 2 def (d2 4)(d20 1) d20 1 r1 R1(t) (2.41) Z(t) = − − Ξ(t) log R2(t) + log r2 − 2 2 d1 − − d1 r2 − R2(t) d 2 t R (s)dβ (s) t 1 (d 2)R2(s) (2.42) = 2 − 1 1 + 2 − 1 d β (s) d R2(s) − R (s) d R3(s) 2 1 Z0 2 Z0 2 1 2 (2.43) = γ(∆(t)).
Let us explain the identity (2.43). The expression (2.42) being a continuous local martingale, it follows from the well known Dubins–Schwarz representation theorem that (2.43) holds for some Brownian motion γ and for ∆(t) equal to the increasing process associated with this local martingale, i.e.
2 t 2 t 2 2 d2 2 R1(s) 1 (d2 2)R1(s) ∆(t)= − 4 ds+ + − 3 d s d1 0 R2(s) 0 R2(s) d1R2(s) Z Z t d s (d 2)(d 2+2d ) t R2(s) = + 2 − 2 − 1 1 ds R2(s) d2 R4(s) Z0 2 1 Z0 2 d 2 2 t R4(s) + 2 − 1 ds d R6(s) 1 Z0 2 def (2.44) =∆1(t)+∆2(t)+∆3(t).
Before estimating ∆j (t)(for1 j 3), let us look at Z(t) in (2.41). By the usual law of the iterated logarithm≤ and≤ the Dvoretzky–Erd˝os [11] integral test for the 2 2 rate of escape of transient Bessel processes, one can easily check that R1(t)/R2(t)= 1/2 o((log t) ) and that log R2(t)=(logt)/2+O(log2 t) almost surely. Therefore, (d 4)(d 2) 1 (2.45) Z(t)= 2− 2 − Ξ(t) log t + o (log t)1/2 , a. s. 2d1 − 2 Applying (2.43) and (2.44) yields
(d 4)(d 2) 1 3 (2.46) 2 − 2 − Ξ(t)= log t + γ ∆ (t) + o (log t)1/2 , a. s. 2d 2 j 1 j=1 X Let us estimate the ∆j (t)’s. By Fact 2.5, ∆ (t) 1 (2.47) 1 a.s. . log t −→ d 2 2 − t 2 4 Applying (2.19) to 0 (R1(s)/R2(s)) d s gives that, as t tends to infinity,
R ∆ (t) a.s. 2 d + d 2 (2.48) 2 1 2 − , a. s. log t −→ d ( d 4) 1 2 − We now complete the proof of Proposition 2.1 by verifying (2.6), (2.5) and (2.4) separately.
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 3926 YUEYUN HU, ZHAN SHI, AND MARC YOR
Proof of (2.6). Assume d2 > 6. In this case, Lemma 2.8 (with µ = 2) confirms that
∆ (t) a.s. ( d +2)(d 2) 3 1 2 − , a. s., log t −→ d (d 4)(d 6) 1 2 − 2 − which, in view of (2.46)–(2.48), yields (2.6).
def Proof of (2.5). Let d2 = 6. By Lemma 2.8 (recalling that λ =4/B(d10 ,d20 )), ∆ (t) d +1 3 P. 10 . (log t)(log2 t) −→ d1 This, jointly considered with (2.46)–(2.48), implies (2.5).
Proof of (2.4). Assume now 4
M(t) law ca (2.49) ( d 0 2) c2 ,t , 1/(d 1) 2 d20 1 (log t) 20 − −→ − S − →∞ where t 2 def R1(s) R1(s) M(t) = 2 d β1(s) 3 d β2(s) . 0 R2(s) − R2(s) Z To this end, we use an argument similar to that in the proof of Lemma 2.8. Indeed, Fact 2.4 guarantees the existence of a Jacobi process Y of dimensions (d1,d2) with 2 2 Y (0) = r1/ r1 + r2 satisfying (2.13). Recall Λ(t) from (2.24). By the Dubins– Schwarz theorem, there exists a Brownian motion B such that p t R (s)dβ (s) R (s)dβ (s) (2.50) 2 1 − 1 2 =B(Λ(t)). R2(s)+R2(s) Z0 1 2 We point out that, according to Knight’s theorem on continuous orthogonal mar- tingales and in view of (2.37)–(2.38), the Brownian motion B is independent of 2 2 t R1(t)+R2(t), therefore of t Λ(t). 7→It is an immediate consequence7→ of (2.50) that t R (s)(R2(s)+R2(s)) R (s)dβ (s) R (s)dβ (s) M(t)= 1 1 2 2 1 − 1 2 R3(s) R2(s)+R2(s) Z0 2 1 2 Λ(t) Y 1/2(u) = d B(u), (1 Y (u))3/2 Z0 −
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use DIFFUSIONS WITH DRIFTED BROWNIAN POTENTIALS 3927
where the last equality follows from (2.13). For notational convenience, write
v 1/2 def Y (u) N+(v) = 1 Y (u)>1/2 d B(u), (1 Y (u))3/2 { } Z0 − v 1/2 def Y (u) N (v) = 1 Y (u) 1/2 d B(u). − (1 Y (u))3/2 { ≤ } Z0 − (Therefore M(t)=N+(Λ(t))+N (Λ(t)) for all t.) The increasing process associated − v 3 with the continuous martingale N is 0 Y (u)(1 Y (u))− 1 Y (u) 1/2 d u,which is smaller than 4v. It follows from− the Dubins–Schwarz− theorem{ ≤ and} the usual iterated logarithm law for Brownian motionR that for all ε>0, N (v)/v1/2+ε a.s. 0 a.s. − −→ ( v ). Since M(t)=N(Λ(t)), with Λ(t)/ log t 1 / ( d 1 + d 2 2) (cf. (2.25)), the→∞ proof of (2.49) is reduced to showing the following−→ estimate: −
N+(v) law 1/(d0 1) ca (2.51) ( d 0 2)(d1 + d2 2) 2− c2 ,v . 1/(d 1) 2 d20 1 v 20 − −→ − − S − →∞ A key observation is that the Brownian motion B introduced in (2.50) is also the Brownian motion in the stochastic differential equation satisfied by Y , i.e. d Y (t)=2 Y(t)(1 Y (t)) d B(t)+(d (d +d )Y(t)) d t. − 1 − 1 2 (Indeed, this observationp yields the skew-product result presented in Fact 2.4.) Recall S from (2.27), which is a scale function of Y .Wehave
(1 d )/2 (1 d )/2 dS(Y(t)) = 2Y − 1 (t)(1 Y (t)) − 2 d B(t). − Consequently,
N+(v) v Y 1/2(u) 1 (d1 1)/2 (d2 1)/2 = Y − (u)(1 Y (u)) − 1 Y (u)>1/2 d S(Y (u)). (1 Y (u))3/2 2 − { } Z0 − According to (2.28)–(2.29) (for some Brownian motion β), this yields v 1 d0 d0 2 N+(v)= Y 1(u)(1 Y (u)) 2− 1 Y (u)>1/2 d β(U(u)) 2 − { } Z0 U(v) d0 d0 2 1 1 1 1 2− = S− (β(s)) 1 S− (β(s)) 1 β(s)>0 d β(s), 2 0 − { } Z 1 where S− is as before the inverse function of S. Write, for brevity,
d0 d0 2 def 1 1 1 2− g(x) = S− (x) 1 S− (x) 1 x>0 ,xR, − { } ∈ which is a deterministic function. Define as in (2.32) the Brownian motion βv( )= β(v2 )/v,andτ ( ), its inverse local time at 0. We have · · βv · 2 1 U(v) v U(v)/v (2.52) N (v)= g(β(s)) d β(s)= g(vβ (s)) d β (s). + 2 2 v v Z0 Z0 def Let λ =4/B(d10 ,d20 ) as in (2.33). It is easily checked that (writing q = (d 4)/(d 2)) 2 − 2 − q g(x) (d0 1)x − ,x . ∼ 2 − →∞
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 3928 YUEYUN HU, ZHAN SHI, AND MARC YOR
Let τ be the inverse local time at 0 of β. It follows from a martingale convergence theorem (cf. Revuz and Yor [26, Exercise IV.4.14]) that, as v tends to infinity, τ(λ) τ(λ) q P. q q v g(vβ(s)) d β(s) ( d 20 1)− β− (s) 1 β(s)>0 d β(s). −→ − { } Z0 Z0 Note from (2.18) that the law of the last variable (recalling that λ =4/B(d10 ,d20 )) ca is c10 d 1,where S 20 − 1/(d0 1) def 4 2− c10 =(d20 2) ψ(d20 1). − (d20 1) B(d10 ,d20 ) − − Since
τβ (λ) τ(λ) v law g(vβv(s)) d βv(s) = g(vβ(s)) d β(s), Z0 Z0 we obtain
τβv (λ) q law ca v g(vβv(s)) d βv(s) c 10 d 1,v . −→ S 20 − →∞ Z0 In view of (2.34), we have actually proved the following convergence: U(v)/v2 q law ca v g(vβv(s)) d βv(s) c 10 d 1,v . −→ S 20 − →∞ Z0 Going back to (2.52) gives N (v) c + law 10 ca ,v . 1/(d 1) d20 1 v 20 − −→ 2 S − →∞ This yields (2.51), and therefore completes the proof of (2.4).
3. Proof of Theorem 1.1 Since (1.13) is proved by Kawazu and Tanaka [20] (cf. Remark 1.2), we shall only check (1.10)–(1.12), and also obtain (1.7)–(1.8) as a by-product. Let X(t); t 0 be a diffusion process with drifted Brownian potential as in (1.2). Let{ ≥ } x (3.1) A(x) def= eV (y) d y, x R, ∈ Z0 which is a scale function of X.NotethatA( )< almost surely, since κ> 0. Using diffusion (or martingale) theory, one∞ can construct∞ a Brownian motion B(t); t 0 starting from 0, independent of the drifted Brownian potential V , such{ that≥ } (3.2) A(X(t)) = B(E(t)),
where E(t) def= A X (t) denotes the increasing process associated with the local martingale A Xh .◦ Notei that the representation (3.2) involves only B(u); u< E( ). Define◦ the first hitting times of B: { ∞} σ(r) def=inf t>0: B(t)>r ,r>0. { } Then E( )=σ(A( )). Now, let ∞ ∞ r def 1 1 T (r) = E− (r)= exp( 2V (A− (B(s)))) d s, 0 r License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use DIFFUSIONS WITH DRIFTED BROWNIAN POTENTIALS 3929 and we get the following representation for X (cf. Brox [5]): 1 1 (3.3) X(t)=A− (B(T− (t))),t0. ≥ Let H be as in (1.4), denoting the first hitting time process associated with X. Let Lx(B); t 0,x R be the local time of B. By (3.3), for any r>0, { t ≥ ∈ } H(r)=T(σ(A(r))) σ(A(r)) 1 = exp 2V (A− (B(s))) d s − Z0 A(r) 1 2V (A− (x)) x = e− Lσ(A(r))(B)dx Z−∞ (3.4) =I1(r)+I2(r), where 0 1 def 2V (A− (x)) x (3.5) I1(r) = e− Lσ(A(r))(B)dx, Z−∞ A(r) 1 def 2V (A− (x)) x (3.6) I2(r) = e− Lσ(A(r))(B)dx. Z0 We treat the terms I1(r)andI2(r) separately. For the former, let us use a change of variables x = A(y) to arrive at 0 V (y) A(y) I1(r)= e− Lσ(A(r))(B)dy. Z−∞ 0 V (y) Recall that A( ) < and that e− d y< almost surely. Therefore, ∞ ∞ −∞ ∞ (3.7) sup I1(r)=R I1( )< , a.s. r 0 ∞ ∞ ≥ It remains to handle the second term I2(r). Recall that V is a Brownian motion with negative drift, cf. (1.1). According to the well known (Lamperti) relation between drifted Brownian motion and Bessel processes (cf. Yor [34]), there exists aBES(2 2κ), denoted by R(t); t 0 , with R(0) = 2, such that − { ≥ } 1 (3.8) eV (t)/2 = R(A(t)),t0, 2 ≥ where A is as in (3.1). Moreover, A( )(< , almost surely) is the first hitting time of R at 0 (i.e. the life time of R).∞ Define∞ (t) def= R(A( ) t), 0 t A( ). < ∞ − ≤ ≤ ∞ By Williams’ time reversal theorem for Bessel processes (cf. for example Revuz and Yor [26, Exercise XI.1.23]), is a (transient) BES(2 + 2κ), starting from 0, such that A( ) is the last exit< time of from 2. Accordingly, (3.8) becomes 1 ∞ < exp(V (A− (t))/2) = (A( ) t)/2 for all 0 A( ) s A( ) L ∞ − (B) I (r)=16 ∞ σ(A(r)) d s. 2 4 A( ) A(r) (s) Z ∞ − < License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 3930 YUEYUN HU, ZHAN SHI, AND MARC YOR 1 (1 t)A(r) According to the first Ray–Knight theorem, L − (B); 0 t 1 is a { A(r) σ(A(r)) ≤ ≤ } BESQ(2) starting from 0, independent of V . Therefore, letting R1 denote a BES(2) with R1(0) = 0, A( ) 2 ∞ R(A(r)+s A( )) I (r) law=16 1 d s 2 4 − ∞ A( ) A(r) (s) Z ∞− < A(r) R2(v) (3.9) =16 1 d v, 4(v+A( ) A(r)) Z0 < ∞ − by a change of variables v = s + A(r) A( ). Write D(r) def= A( ) A(r)= − ∞ ∞ − ∞ V(y) r e dy. With the aide of the usual law of the iterated logarithm, it is easily checked that R κ (3.10) log D(r)= r+O (rlog r)1/2 , a. s. −2 2 Going back to (3.9), we have A( ) D(r) R2(v) I (r)=16 ∞ − 1 d v 2 4(v+D(r)) Z0 < A( ) D(r) 1 R2(v) =16 ∞ − + 1 d v 4 1 0 (v+D(r)) Z Z < (3.11) = I3(r)+I4(D(r)), where A( ) D(r) 2 ∞− R(v) (3.12) I (r) def=16 1 d v is a bounded term 3 4(v+D(r)) Z1 < when r tends to infinity (recalling that A( ) is the last exit time of from 2), and for small δ>0, ∞ < 1 R2(v) I (δ) def=16 1 dv 4 4(v+δ) Z0 < 1/δ R2(uδ) =16δ 1 d u 4((1 + u)δ) Z0 < 1/δ R2(u) law=16 1 d u. 4(1 + u) Z0 < 1/2 ν Observe that for any fixed ν>0, (1 + u)/R(u)=1+o(u − ) almost surely < (as u ). Applying Proposition 2.1 to I4(δ)(withd1 =2andd2 =2+2κ), assembling→∞ (3.4), (3.7) and (3.10)–(3.12), and using the monotonicity of r I (r)+ 7→ 1 I2(r), we readily get (1.7)–(1.8) and (1.10)–(1.12), with (3.13) c1 =8ψ(κ). This completes the proof of Theorem 1.1. Remark 3.1. The value of c1 found by Kawazu and Tanaka [19] is implicit. The form (3.13) is new. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use DIFFUSIONS WITH DRIFTED BROWNIAN POTENTIALS 3931 4. Principal values and stable Levy´ processes Let β be a standard Brownian motion. Its local time is denoted by L, and inverse local time at 0 by τ. It has been remarked in Biane and Yor [3] that all possible stable L´evy processes may be obtained as suitable combinations of principal values of Brownian local times, taken at τ(λ); λ 0 . However, the actual computations are made precisely only in the case{ of the≥ stable} index α =1. ca Since the realization of α ,forα=1/(1 q)(0 def ∞ d x x X±(λ) = (L± λ), x1+q τ(λ) − Z0 + and if we can show that X (λ); λ 0 and X−(λ); λ 0 are two independent, completely asymmetric stable{ processes≥ } of index{ α =1/≥(1 } q) (1, 2), then we will have proved (2.18). − ∈ The independence follows from excursion theory, while the stability follows from + the scaling property of Brownian motion. That X and X− are completely asym- metric is due to the fact that they have no negative jumps. Thus (cf. Bertoin [2, p. 217]), there exists a constant c = c (α) such that for all t R, 11 11 ∈ πα E exp itX+(λ) =exp λc t α 1 i sgn(t)tan . − 11 | | − 2 + h i From the independence of X and X−, we deduce that + α E exp(it(X (λ) X−(λ))) = exp( 2λc t ), − − 11 | | which we now compare with the computations of Biane and Yor [3]; this yields 2 α 2 c11 = 2α ψ(α) /α . We now complete the discussion with the following estimate which is well known in the literature (cf. Bertoin [2, p. 221]). Lemma 4.1. For each λ>0, 1 E X+(λ)a < if and only if a< . | | ∞ 1 q h i − The proof of Lemma 4.1 relies on some well known results on Brownian motion and Bessel processes. Throughout this section, (4.1) U(t); t 0 def= Bessel process of dimension 0, with U(0) = 1, { ≥ } (4.2) ζ def=inf t>0: U(t)=0 , { } (4.3) R(t); t 0 def= Bessel process of dimension 4, with R(0) = 0, { ≥ } (4.4) L def=supt>0: R(t)=1 . { } In words, ζ denotes the life time of U,andLthe Last exit time of R from 1. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 3932 YUEYUN HU, ZHAN SHI, AND MARC YOR Fact 4.2 (Yor [35, p. 52]). Let R(t); t 0 and L be as in (4.3) and (4.4) respec- tively. Define the normalized Bessel{ process≥ } R(s L) (4.5) ρ(s) def= , 0 s 1. √L ≤ ≤ For any bounded functional F ,wehave 2 (4.6) E F ρ(s); 0 s 1 = E F R(s); 0 s 1 . ≤ ≤ R2(1) ≤ ≤ h i h i Proof of Lemma 4.1. By the second Ray–Knight theorem, Lx ; x 0 is a { τ(λ) ≥ } BESQ(0), starting from λ; thus, in the notation of (4.1)–(4.2), it is distributed as λU 2(x/λ); x 0 . Moreover, U(x/λ) = 0 for all x λζ. Accordingly, { ≥ } ≥ 2 ζ 2 + law ∞ U (x/λ) 1 1 q U (t) 1 (4.7) X (λ) = λ − d x = λ − − d t. x1+q t1+q Z0 Z0 2 t Recall that U (t)=1+2 0U(s)dB(s) for some Brownian motion B,andthat 2 q with probability one, limt 0(U (t) 1)/t = 0 (noting that q<1/2). Hence, by → + R − 1 q ζ q integration by parts, X (λ) is distributed as (2λ − /q) 0 (U(s)/s )dB(s). Apply- ing the Burkholder–Davis–Gundy inequality yields that, for some finite constant K(a), depending only on the value of a, R λ1 q a ζ U2(s) a/2 (4.8) E X+(λ) a K(a) − E ds . 2q | | ≤ q 0 s h i hZ i In the notation of (4.1)–(4.4), and using Williams’ time reversal theorem for Bessel processes (cf. for example Revuz and Yor [26, Exercise XI.1.23]), we get (4.9) U(ζ t); 0 t ζ law= R(t); 0 t L . − ≤ ≤ ≤ ≤ Accordingly, n o n o ζ U 2(s) ζ1 2q L1 2q d s − sup U 2(t) law= − sup R2(t). 2q 0 s ≤ 1 2q 0 t ζ 1 2q 0 t L Z − ≤ ≤ − ≤ ≤ Let ρ be as in (4.5). The expression on the right hand side is L2(1 q) − sup ρ2(s). 1 2q 0 s 1 − ≤ ≤ By means of (4.6), ζ U2(s) a/2 2 E ds E R 2 2a(1 q)(1) sup Ra(s) . 2q a/2 − − − 0 s ≤ (1 2q) 0 s 1 hZ i − h ≤ ≤ i b b Since E(R (1)) < for all b> 4andEsup0 s 1 R (s) < for all b 0, it follows from H¨older’s∞ inequality that− the expectation≤ ≤ term on the∞ right hand≥ side is finite as long as 2 + 2a(1 q) < 4, which means a<1/(1 q). In view of (4.8), we have proved the “if ” part− of Lemma 4.1. − To check the “only if ” part, let us go back to (4.7) and use (4.9) to arrive at L 2 1 2 + law 1 q R (t) 1 1 q ρ (s) L X (λ) = λ − − d t =(λL) − − d s, (L t)1+q (1 s)1+q Z0 − Z0 − License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use DIFFUSIONS WITH DRIFTED BROWNIAN POTENTIALS 3933 where ρ is as in (4.5). Again, we can apply Fact 4.2 to see that (writing G def= 1((R2(s) R2(1))/(1 s)1+q)ds for brevity) 0 − − + a a(1 q) 2 2a(1 q) a R E X (λ) =2λ − E R− − − (1) G | | | | h i a(1 q) h 2 2a(1 q) i a 2λ − E R− − − (1) 1 R(1) 1/2 G . ≥ { ≤ } | | Now take a =1/(1 q) and considerh the function i − def 1/(1 q) ϕ(y) = E G − R(1) = y , 0 y 1/2. | | ≤ ≤ It is easily seen using the Bessel bridge that ϕ is a continuous and (strictly) positive function on [0, 1/2], which yields inf0 y 1/2 ϕ(y) c>0forsomeconstantc depending on q. As a consequence, ≤ ≤ ≥ + 1/(1 q) 4 E X (λ) − 2cλ E R− (1) 1 R(1) 1/2 = , | | ≥ { ≤ } ∞ completing theh proof of the “onlyi if ” parth of Lemma 4.1. i References [1] Alili, S.: Asymptotic behaviour for random walks in random environments. (preprint) [2] Bertoin, J.: L´evy Processes. Cambridge University Press, Cambridge, 1996. MR 98e:60117 [3] Biane, P. and Yor, M.: Valeurs principales associ´ees aux temps locaux browniens. Bull. Sci. Math. 111 (1987) 23–101. MR 88g:60188 [4] Bouchaud, J.-P., Comtet, A., Georges, A. and Le Doussel, P.: Classical diffusion of a particle in a one-dimensional random force field. Ann. Phys. 201 (1990) 285–341. MR 91k:82047 [5] Brox, T.: A one-dimensional diffusion process in a Wiener medium. Ann. Probab. 14 (1986) 1206–1218. MR 88f:60132 [6] Carmona, P.: The mean velocity of a Brownian motion in a random L´evy potential. Ann. Probab. 25 (1997) 1774–1788. MR 99b:60126 [7] Comets, F., Menshikov, M.V. and Popov, S.Yu.: Lyapunov functions for random walks and strings in random environment. Ann. Probab. 26 (1998) 1433–1445. CMP 99:09 [8] Cs¨org˝o, M., Horv´ath, L., and R´ev´esz, P.: Stability and instability of local time of random walk in random environment. Stoch. Proc. Appl. 25 (1987) 185–202. MR 89c:60078 [9] Deheuvels, P. and R´ev´esz, P.: Simple random walk on the line in random environment. Probab. Th. Rel. Fields 72 (1986) 215–230. MR 87m:60193 [10] Dembo, A., Peres, Y. and Zeitouni, O.: Tail estimates for one-dimensional random walk in random environment. Comm. Math. Phys. 181 (1996) 667–683. MR 97h:60074 [11] Dvoretzky, A. and Erd˝os, P.: Some problems on random walk in space. Proc. 2nd Berkeley Symp. Math. Statist. Probab. 353–367, University of California Press, Berkeley, 1951. MR 13:852b [12] Gantert, N. and Zeitouni, O.: Quenched sub-exponential tail estimates for one- dimensional random walk in random environment. Comm. Math. Phys. 194 (1998) 177–190. CMP 98:14 [13] Greven, A. and den Hollander, F.: Large deviations for a random walk in random environ- ment. Ann. Probab. 22 (1994) 1381–1428. MR 95m:60101 [14] Hu, Y. and Shi, Z.: The local time of simple random walk in random environment. J. Theo- retical Probab. 11 (1998) 765–793. CMP 98:15 [15] Hu, Y. and Shi, Z.: The limits of Sinai’s simple random walk in random environment. Ann. Probab. 26 (1998) 1477–1521. CMP 99:09 [16] Hughes, B.D.: Random Walks and Random Environments. Vol. II: Random Environments. Oxford Science Publications, Oxford, 1996. MR 98d:60139 [17] Itˆo, K. and McKean, H.P.: Diffusion Processes and Their Sample Paths. Springer, Berlin, 1965. MR 33:8031 [18] Karlin, S. and Taylor, H.M.: A Second Course in Stochastic Processes. Academic Press, New York, 1981. MR 82j:60003 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 3934 YUEYUN HU, ZHAN SHI, AND MARC YOR [19] Kawazu, K. and Tanaka, H.: A diffusion process in a Brownian environment with drift. J. Math. Soc. Japan 49 (1997) 189–211. MR 99c:60170 [20] Kawazu, K. and Tanaka, H.: Invariance principle for a Brownian motion with large drift in a white noise environment. Hiroshima Math. J. 28 (1998) 129–137. [21] Kesten, H., Kozlov, M.V. and Spitzer, F.: A limit law for random walk in a random environ- ment. Compositio Math. 30 (1975) 145–168. MR 52:1895 [22] McKean, H.P.: A H¨older condition for Brownian local time. J. Math. Kyoto Univ. 1 (1962) 195–201. MR 26:4421 [23] Mathieu, P.: On random perturbations of dynamical systems and diffusions with a Brownian potential in dimension one. Stoch. Proc. Appl. 77 (1998) 53–67. CMP 98:01 [24] Ray, D.: Sojourn times of a diffusion process. Illinois J. Math. 7 (1963) 615–630. MR 27:6306 [25] R´ev´esz, P.: Random Walk in Random and Non-Random Environments. World Scienfitic, Singapore, 1990. MR 92c:60096 [26] Revuz, D. and Yor, M.: Continuous Martingales and Brownian Motion. (Second Edition). Springer, Berlin, 1994. MR 95h:60072 [27] Schumacher, S.: Diffusions with random coefficients. Contemp. Math. 41 (1985) 351–356. MR 88k:60045 [28] Shi, Z.: A local time curiosity in random environment. Stoch. Proc. Appl. 76 (1998) 231–250. CMP 98:17 [29] Sinai, Ya.G.: The limiting behavior of a one-dimensional random walk in a random medium. Th. Probab. Appl. 27 (1982) 256–268. MR 83k:60078 [30] Solomon, F.: Random walks in a random environment. Ann. Probab. 3 (1975) 1–31. MR 50:14943 [31] Tanaka, H.: Diffusion processes in random environments. Proc. ICM (S.D. Chatterji, ed.) 1047–1054. Birkh¨auser, Basel, 1995. MR 97g:60112 [32] Tanaka, H.: Limit theorems for a Brownian motion with drift in a white noise environment. Chaos Solitons Fractals 11 (1997) 1807–1816. MR 99a:60085 [33] Warren, J. and Yor, M.: Skew-products involving Bessel and Jacobi processes. Technical report, Department of Mathematics, University of Bath, 1997. To appear in: S´eminaire de Probabilit´es XXXIII. Lecture Notes in Mathematics, Springer, 1999. [34] Yor, M.: Sur certaines fonctionnelles exponentielles du mouvement brownien r´eel. J. Appl. Probab. 29 (1992) 202–208. MR 93g:60179 [35] Yor, M.: Local Times and Excursions for Brownian Motion: A Concise Introduction. Lec- ciones en Matem´aticas, Universidad Central de Venezuela, 1995. [36] Yor, M.: Some Aspects of Brownian Motion. Part II: Some Recent Martingale Problems. Birkh¨auser, Basel, 1997. MR 98e:60140 Laboratoire de Probabilites,´ Universite´ Paris VI, 4 Place Jussieu, 75252 Paris Cedex 05, France E-mail address: [email protected] E-mail address: [email protected] E-mail address: [email protected] License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use