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∈[0,T]Δ_t≤1] where Δ_t=∫_RL_t(x)dW(x). Here W=W(x);x∈R is a
two-sided standard real Brownian motion and L_t(x);x∈R,t≥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