Persistence exponent for random processes in Brownian scenery

Abstract

In this paper we consider the persistence properties of random processes in Brownian scenery, which are examples of non-Markovian and non-Gaussian processes. More precisely we study the asymptotic behaviour for large $T$, of the probability $P[ \sup\_{t\in[0,T]} \Delta\_t \leq 1] $ where $\Delta\_t = \int\_{\mathbb{R}} L\_t(x) \, dW(x).$ Here $W={W(x); x\in\mathbb{R}}$ is a two-sided standard real Brownian motion and ${L\_t(x); x\in\mathbb{R},t\geq 0}$ is the local time of some self-similar random process $Y$, independent from the process $W$. We thus generalize the results of \cite{BFFN} where the increments of $Y$ were assumed to be independent.