Lyapunov exponent for the parabolic Anderson model in RdRd

We consider the asymptotic almost sure behavior of the solution of the equationu(t,x)=u0(x)+κ2∫0tΔu(s,x)ds+∫0tu(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→∞1tlogu(t,x). We also show that c1κ1/3⩽λ(κ)⩽c2κ1/5c1κ1/3⩽λ(κ)⩽c2κ1/5 as κ↘0κ↘0 under the assumption that the correlation function of the background field {Wx:x∈Rd} is CβCβ for 1<β⩽21<β⩽2.