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: Exexp{±∫0tV(X(s))ds}, where {X(s)}{X(s)} is a dd-dimensional Ornstein–Uhlenbeck process and {V(x)}x∈Rd{V(x)}x∈Rd is a homogeneous ergodic random Poisson potential. It turns out that the positive/negative exponential moment has ectect 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.