Almost sure asymptotic for Ornstein-Uhlenbeck processes of Poisson potential

The objective of this paper is to study the large time asymptotic of the following exponential moment: E-x exp{+/- integral(1)(0) V(X(s)) ds}, where {X(s)) is a d-dimensional Ornstein-Uhlenbeck process and {V(x)}(x is an element of R)(d) is a homogeneous ergodic random Poisson potential. It turns out that the positive/negative exponential moment has e(ct) growth/decay rate, which is different from the Brownian motion model studied by Carmona and Molchanov (1995) for positive exponential moment and Sznitman (1993) for negative exponential moment. (C) 2012 Elsevier B.V. All rights reserved.