persistence exponent for random processes in brownian ?· persistence exponent for random...

Download Persistence exponent for random processes in Brownian ?· Persistence exponent for random processes…

Post on 14-Aug-2018

212 views

Category:

Documents

0 download

Embed Size (px)

TRANSCRIPT

  • ALEA, Lat. Am. J. Probab. Math. Stat. 13, 7994 (2016)

    Persistence exponent for random processesin Brownian scenery

    Fabienne Castell, Nadine Guillotin-Plantard andFrederique Watbled

    Aix-Marseille Universite, CNRS, Centrale Marseille, Institut de Mathematiques de Mar-seille, UMR 7373, 13 453 Marseille. France.E-mail address: fabienne.castell@univ-amu.fr

    Institut Camille Jordan, CNRS UMR 5208, Universite de Lyon, Universite Lyon 1, 43,Boulevard du 11 novembre 1918, 69622 Villeurbanne, France.E-mail address: nadine.guillotin@univ-lyon1.fr

    LMBA, CNRS UMR 6205, Universite de Bretagne Sud, Campus de Tohannic, BP 573,56017 Vannes, France.E-mail address: watbled@univ-ubs.fr

    Abstract. In this paper we consider the persistence properties of random processesin Brownian scenery, which are examples of non-Markovian and non-Gaussian pro-cesses. More precisely we study the asymptotic behaviour for large T , of the prob-ability P[supt[0,T ] t 1] where t =

    R Lt(x) dW (x). Here W = {W (x);x R}

    is a two-sided standard real Brownian motion and {Lt(x);x R, t 0} is the localtime of some self-similar random process Y , independent from the process W . Wethus generalize the results of Castell et al. (2013) where the increments of Y wereassumed to be independent.

    1. Introduction

    Let W = {W (x);x R} be a standard two-sided real Brownian motion andY = {Y (t); t 0} be a real-valued self-similar process of index (0, 2) (i.e. forany c > 0, {Y (ct); t 0} (d)= {cY (t); t 0}), with stationary increments. Whenit exists, we will denote by {Lt(x);x R, t 0} a version of the local time of theprocess {Y (t); t 0}. The process L satisfies the occupation density formula: for

    Received by the editors January 15, 2015; accepted November 6, 2015.

    2010 Mathematics Subject Classification. 60F05; 60F17; 60G15; 60G18; 60K37.Key words and phrases. Random scenery, First passage time, One-sided barrier problem, One-

    sided exit problem, Survival exponent, Persistence.This research was supported by the french ANR project MEMEMO2 2010 BLAN 0125. F. Watbled

    thanks the IRMAR, CNRS UMR 6625, University of Rennes 1, for its hospitality.

    79

    http://alea.impa.br/english/index_v13.htm

  • 80 Castell et al.

    each bounded measurable function f : R R and for each t 0, t0

    f(Y (s))ds =

    Rf(x)Lt(x)dx. (1.1)

    The processes W and Y are defined on the same probability space and are assumedto be independent. We consider the random process in Brownian scenery {t; t 0}defined as

    t =

    RLt(x) dW (x).

    The process is itself a self-similar process of index h with stationary increments,with

    h := 1 2.

    This process can be seen as a mixture of Gaussian processes, but it is neitherGaussian nor Markovian. It appeared independently in the mathematical litera-ture (see Kesten and Spitzer (1979); Borodin (1979)), and in the physics literature(see Matheron and de Marsily (1980)) where it was originally introduced to modeldiffusion in layered white-noise velocity field. Note indeed that t can formally

    be written as t

    0W (Y (s))ds. Thus, it describes the horizontal motion of a tracer

    particle in a (1+1)-dimensional medium, where the motion of the particle along thevertical direction y is described by the process Y , while along the horizontal direc-tion, the particle is driven by the white noise velocity field W (y), that depends onlyon the vertical coordinate y. The process {t; t 0} provides a simple example ofanomalous super-diffusion, being for instance of order t3/4 when Y is a Brownianmotion. In this paper, we are interested in the persistence probability of the process, i.e. the asymptotic behaviour of

    F(T ) := P[

    supt[0,T ]

    t 1]

    as T +. The study of the one-sided exit problem of random processes is aclassical issue in probability. One usually gets polynomial decay of the persistenceprobability: F(T ) T for a non-negative , often called persistence exponent,or survival exponent. Classical examples, where this exponent can be computed,include random walks or Levy processes, and we refer the reader to the recent surveypapers Aurzada and Simon (2012); Bray et al. (2013) for an account of models wherethe persistence exponent is known. However, there are relevant physical situationswhere this exponent remains unknown (see for instance Majumdar (1999); Brayet al. (2013)). The persistence exponent of the Brownian motion in Brownianscenery was studied by Redner (1990, 1997), and Majumdar (2003). Based onphysical arguments, numerical simulations and analogy with fractional Brownianmotion, Redner and Majumdar conjectured the value of the persistence exponent.In Castell et al. (2013), their conjecture was proved up to logarithmic factors, whenthe process Y is a stable Levy process with index (1, 2]. The proof of Castellet al. (2013) depends heavily on the increments independence of the process Y andthe question raises if it is possible to compute the persistence exponent without it.The aim of this paper is to answer this question, and to provide assumptions on Yallowing to compute the persistence exponent of the random process in Brownianscenery.

    (H1): There exists a continuous version {Lt(x);x R, t 0} of the local timeof Y ;

  • Persistence exponent for random processes in Brownian scenery 81

    (H2): Y is a self-similar process of index (0, 2);(H3): For every T > 0, {Y (Tt)Y (T ); t [0, T ]} has the same distribution

    as either {Y (t); t [0, T ]} or {Y (t); t [0, T ]};(H4): Let V1 :=

    L21(x) dx be the self-intersection local time of Y . There

    exist a real number > 1, and positive constants C, c such that for anyx 0,

    P [V1 x] C exp(cx) ;(H5): There exist a real number > 0, and positive constants C, c such that

    for any x > 0,P [V1 x] C exp(cx) .

    Our main result is the following one.

    Theorem 1.1. Assume (H1) to (H5) hold. There exists a constant c > 0, suchthat for large enough T ,

    T/2(lnT )c P[

    supt[0,T ]

    t 1] T/2(lnT )+c. (1.2)

    Our main achievement in this paper is to remove the increments independenceassumption on {Y (t); t 0}, which was crucial in Castell et al. (2013). Anotherinteresting issue would be to obtain the persistence exponent for {W (x);x R}being a stable Levy process. This seems delicate, our proof relying on the fact thatthe process {(t); t 0} is a Gaussian process, conditionally to {Y (t); t 0}.

    The paper is organized as follows. Section 2 provides three examples of processessatisfying (H1) to (H5), including the stable Levy process with index (1, 2], andthus generalizing the result of Castell et al. (2013). Section 3 states some usefulproperties of the process . Section 4 is devoted to the proof of Theorem 1.1.

    2. Some examples

    2.1. A sufficient condition for (H5). In this section we assume that (H1) holds andwe provide a sufficient condition on {Y (t); t 0} allowing to check (H5). Let usfirst compare V1 with max

    s[0,1]|Y (s)|. By noting that

    1 =

    {|x|maxs[0,1] |Y (s)|}

    L1(x)dx V 1/21

    2 maxs[0,1]

    |Y (s)|,

    we deduce that1

    2 maxs[0,1]

    |Y (s)| V1. (2.1)

    Lemma 2.1. Assume that (H1) holds and that there exist positive constants C andc, and a real number > 0 such that for all x > 0,

    P[

    maxs[0,1]

    |Y (s)| x] C exp

    (cx

    ).

    Then (H5) holds true.

    Proof : It is a direct consequence of (2.1) that for all x > 0,

    P [V1 x] P[

    maxs[0,1]

    |Y (s)| 12x

    ] C exp(c2x) .

  • 82 Castell et al.

    2.2. Stable Levy processes. A process Y = {Y (t); t 0} is a strictly stable Levyprocess with index (1, 2] if it is a process with stationary and independentincrements, such that Y (0) = 0 and for any t 0 and u R,

    E[eiuY (t)] = exp{ c|u|t

    (1 + i sgn(u) tan(/2)

    )}, (2.2)

    where [1, 1], and c is a positive scale parameter.

    Lemma 2.2. The stable Levy process with index (1, 2] satisfies (H1) to (H5)with = 1 , = , and =

    21 .

    Proof : It is immediate that the Levy process is self-similar of index 1/ and thatfor every T > 0,

    {Y (T t) Y (T ); t [0, T ]} (d)= {Y (t); t [0, T ]}

    then (H2) and (H3) are satisfied.A continuous version of the local time exists, since > 1 (this was proved by Boylan(1964)), so it satisfies (H1). Moreover, Corollary 5.6 in Khoshnevisan and Lewis(1998) states that there exist positive constants C and s.t. for every x > 0,

    P[V1 x] Cex

    ,

    which gives (H4). To prove (H5) we let Vt :=L2t (x) dx be the self-intersection

    local time of Y up to time t for t 0, and we show that there exists l > 0 suchthat

    P [Vt 1] elt for t large enough. (2.3)As {Vt; t 0} is self-similar of index 21 , (2.3) implies that for every positive small enough,

    P[V1 ] = P[V

    21 1] el

    21

    ,

    which gives (H5). To prove (2.3) we show that the function defined on R+ by

    f(t) = logP[Vt 1]

    is subadditive. Let us fix s, t in [0,+). We consider the process Y (s) := {Y (s)u ;u 0} defined by

    Y (s)u = Y (u+ s) Y (s),

    its local time {L(s)u (x);x R, u 0}, and its self-intersection local time {V (s)u ;u 0}. Note that

    Lt+s(x) = Ls(x) + L(s)t (x Y (s)).

    Hence

    Vt+s Vs + V (s)t .

    The process Y being a Levy process, V(s)t and Vs are independent and V

    (s)t has the

    same law as Vt. Therefore

    P[Vt+s 1] P[Vs 1;V (s)t 1] = P[Vs 1]P[Vt 1].

    Thus f is subadditive, which implies that f(t)t converges, as t +, towards alimit l = inft>0 f(t)t (see Hammersley (1962)). It remains to show that l is strictly

  • Persistence exponent for random processes in Brownian scenery 83

    positive. As l is less than or equal