Almost sure asymptotics for the continuous parabolic Anderson model
The authors study the almost sure growth of \(u(t,0)\), where \(u(t,x)\) solves the parabolic Anderson equation \[ \partial_t u(t,x)=\kappa \Delta u(t,x)+\xi(x) u(t,x),\quad u(0,x)=1,\quad s \in {\mathbb R}^d, t>0, \] where \(\kappa>0\) and \(\xi\) is either a stationary Gaussian or a shot--noise Poisson field. The main result states that under slight regularity assumptions \((1/t)\log u(t,0)=h_t-(\chi + o(1))\sqrt{h_t}\) almost surely as \(t \to \infty\), where \(\chi\) is an explicit constant and \(h_t\) solves \(L(h_t)=d \log t\), where \(L\) is the Legendre transform of the cumulant generating function of \(\xi(0)\).
scientific article; zbMATH DE number 1563075