ON THE LOCAL TIME OF RANDOM PROCESSES IN RANDOM SCENERY

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ON THE LOCAL TIME OF RANDOM PROCESSES IN RANDOM SCENERY FABIENNE CASTELL, NADINE GUILLOTIN-PLANTARD, FRANÇOISE PÈNE, AND BRUNO SCHAPIRA Abstract. Random walks in random scenery are processes defined by Zn := ∑n k=1 ?X1+...+Xk , where basically (Xk, k ≥ 1) and (?y, y ? Z) are two independent sequences of i.i.d. random variables. We assume here that X1 is Z-valued, centered and with finite moments of all orders. We also assume that ?0 is Z-valued, centered and square integrable. In this case H. Kesten and F. Spitzer proved that (n?3/4Z[nt], t ≥ 0) converges in distribution as n?∞ toward some self-similar process (∆t, t ≥ 0) called Brownian motion in random scenery. In a previous paper, we established that P(Zn = 0) behaves asymptotically like a constant times n?3/4, as n ? ∞. We extend here this local limit theorem: we give a precise asymptotic result for the probability for Z to return to zero simultaneously at several times. As a byproduct of our computations, we show that ∆ admits a bi-continuous version of its local time process which is locally Hölder continuous of order 1/4 ? ? and 1/6 ? ?, respectively in the time and space variables, for any ? > 0.

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• Publié par: profil-urra-2012
• Nombre de lectures: 19
• Langue: English

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ON THE LOCAL TIME OF RANDOM PROCESSES IN RANDOM SCENERY

FABIENNE CASTELL, NADINE GUILLOTIN-PLANTARD, FRANÇOISE PÈNE, AND BRUNO SCHAPIRA

Abstract.Random walks in random scenery are processes defined byZ n := P n k=1 ξ X 1 +...+X k , where basically(X k , k≥1)and(ξ y , y∈Z) randomare two independent sequences of i.i.d. variables. We assume here thatX 1 isZ-valued, centered and with finite moments of all orders. We also assume thatξ 0 isZ In-valued, centered and square integrable. this case H. Kesten and F. Spitzer proved that(n −3/4 Z [nt] , t≥0)converges in distribution asn→ ∞toward some self-similar process(Δ t , t≥0) a previous paper, Incalled Brownian motion in random scenery. we established thatP(Z n = 0)behaves asymptotically like a constant timesn −3/4 , asn→ ∞. We extend here this local limit theorem: we give a precise asymptotic result for the probability forZ a byproduct of our computations, Asto return to zero simultaneously at several times. we show thatΔadmits a bi-continuous version of its local time process which is locally Hölder continuous of order1/4−δand1/6−δ, respectively in the time and space variables, for any δ >0 particular, this gives a new proof of the fact, previously obtained by Khoshnevisan,. In that the level sets ofΔ to equalhave Hausdorff dimension a.s.1/4 also get the convergence. We of every moment of the normalized local time ofZtoward its continuous counterpart.

1.Introduction

1.1.the model and of some earlier results.Description of We consider two independent sequences(X k , k≥1)and(ξ y , y∈Z)of independent identically distributedZ-valued random variables. We assume in this paper thatX 1 is centered, with finite moments of all orders, and that its support generatesZ consider the. Werandom walk(S n , n≥0)defined by n S 0 := 0andS n := X X i for alln≥1. i=1 We suppose thatξ 0 is centered, with finite second momentσ 2 :=E[ξ 02 ]. The sequenceξis called therandom scenery.

Therandom walk in random sceneryZis then defined for alln≥1by n−1 Z n := X ξ S k . k=0

For motivation in studying this process and in particular for a description of its connections with many other models, we refer to [5, 10, 14] and references therein. Denoting byN n (y)the local time of the random walkS:

N n (y) = #{k= 0, . . . , n−1 :S k =y}, it is straightforward, and important, to see thatZ n can be rewritten asZ n = P y ξ y N n (y).

2000Mathematics Subject Classification.60F05; 60F17; 60G15; 60G18; 60K37. Key words and phrases.Random walk in random scenery; local limit theorem; local time; level sets This research was supported by the french ANR project MEMEMO2. 1

ON THE LOCAL TIME OF RANDOM PROCESSES IN RANDOM SCENERY

Kesten and Spitzer [10] and Borodin [2] proved the following functional limit theorem :  n −3/4 Z nt , t≥0  n→∞ = (L ⇒ ) (σΔ t , t≥0), where

2

(1)

•Z s :=Z n + (s−n)(Z n+1 −Z n ), for alln≤s≤n+ 1, •Δis defined by Δ t := Z +∞ L t (x) dβ x , −∞ with(β x ) x∈R a standard Brownian motion and(L t (x), t≥0, x∈R)a jointly continuous intandxversion of the local time process of some other standard Brownian motion (B t ) t≥0 independent ofβ. The processΔis known to be a continuous(3/4)-self-similar process with stationary increments, and is calledBrownian motion in random scenery can be seen as a mixture of stable processes,. It but it is not a stable process. Let nowϕ ξ denote the characteristic function ofξ 0 and letdbe such that{u:|ϕ ξ (u)|= 1}= (2π/d)Z [5] we established the following local limit theorem :. In P  Z n = j n 3 4 x k = ( 0 dσ −1 p 1,1 (x/σ)n − 43 +o(n − 3 4 )if hto P e  rw n i ξ 0 − j n 3 4 x k ∈dZ  = 1 (2) se, with p 1,1 (x) := √ 2 1 π E h ||L 1 || 2 −1 e −||L 1 || 22 x 2 /2 i , and||L 1 || 2 := R R L 1 2 (y)dy  1/2 theL 2 -norm ofL 1  the particular case when. Inx= 0, we get P(Z n = 0) =  d 0 σ −1 p 1,1 (0)n − 43 +o(n − 43 ) hot fin er ∈ w d i 0 se Z , (3) withd 0 := min{m≥1 :ϕ ξ (2π/d) m = 1}. 1 Actually the results mentioned above were proved in the more general case when the distributions of theξ y ’s andX k ’s are only supposed to be in the basin of attraction of stable laws (see [4], [5] and [10] for details).

1.2.Statement of the results.

1.2.1.Local time of Brownian motion in random scenery.LetT 1 , . . . , T k , bek Setpositive reals. D T 1 ,...,T k := det(M T 1 ,...,T k )withM T 1 ,...,T k =  hL T i , L T j i  1≤i,j≤k , whereh∙,∙idenotes the usual scalar product onL 2 (R), and C T 1 ,...,T k :=E h D T − 1 1 .,.. /2 ,T k i . Our first result is the following

1 Recall that, for everyn≥0, we have

P(nξ 0 ∈dZ)>0⇐⇒P(nξ 0 ∈dZ) = 1⇐⇒n∈d 0 Z.