Lyapunov exponent for the parabolic Anderson model in Rd ✩

We consider the asymptotic almost sure behavior of the solution of the equation u(t, x) = u0(x) + κ 2 t 0 u(s, x) ds + t 0 u(s, x)∂Wx (s), where {Wx : x ∈ Rd } is a field of Brownian motions. In fact, we establish existence of the Lyapunov exponent, λ(κ) = limt→∞ 1 t log u(t, x). We also show that c1κ1/3 λ(κ) c2κ1/5 as κ 0 under the assumption that the correlation function of the background field {Wx : x ∈ Rd } is Cβ for 1<β 2. © 2006 Elsevier Inc. All rights reserved