Lyapunov exponent for the parabolic anderson model with lévy noise

Abstract

We consider the asymptotic almost sure behavior of the solution of the equationwhere {Y x : x ∈ Z d } is a field of independent Lévy processes and is the discrete Laplacian.

On large deviations for the parabolic Anderson model

“…This, by translation invariance and independence of the field {W x : x ∈ Z}, yields that with probability at least 1 − 10 , for all of the (no more than) 4 points…”

“…[3]. It is known that the (random in {W x : x ∈ Z d }) solution satisfies lim t→∞ log u(x, t) t = λ(κ) > 0, P a.s., for nonrandom λ(κ), see [2,4,9,15] and [17] for more details. This raises the question of the large deviations regimes for these solutions: for > 0, how rapidly do the probabilities of events P( log u(x,t) t > λ(κ) + ) and P( log u(x,t) t < λ(κ) − ) tend to zero as t tends to infinity?…”

“…In [4], it was shown that for some c( ) > 0, P log u(x, t) t > λ(κ) + ≤ e −c( )t , as t → ∞, but the probabilities of small values of u(x, t) were not considered. We remark here that the random variables {u(x, t), x ∈ Z d } are identically distributed due to the fact that the field {W x : x ∈ Z d } is iid, but these random variables are correlated.…”

“…A closely related question (see [2] or [4]) concerns an additive functional associated to the Brownian field {W x : x ∈ Z d } on ( , F, P) which we now more cleary define. Throughout most of this paper (the exception being the remark following the theorems in this section), we shall consider the space of right continuous, left limit paths which have only jumps of size one.…”

“…We now sketch the proof of Proposition 1.2 drawing on results in [4] using the following steps. First we note that, since we are dealing with a fixed time t, we may consider…”