The nonlinear Schrödinger equation driven by jump processes

Abstract

The main result of the paper is the existence of a martingale solution of the nonlinear Schrödinger equation with a Lévy noise with infinite activity. To be more precise, let A=Δ be the Laplace operator with D(A)={u∈L2(Rd):Δu∈L2(Rd)}. Let Z↪L2(Rd) be a function space and η be a Poisson random measure on Z with intensity measure ν. Let g:R→C and h:R→C be some given functions, satisfying certain conditions specified later. Let α≥1 and λ≥0. We are interested in the solution of the following equation(0.1){idu(t,x)−Δu(t,x)dt+λ|u(t,x)|α−1u(t,x)dtu(0)=∫Zu(t,x)g(z(x))η˜(dz,dt)+∫Zu(t,x)h(z(x))ν(dz)dtu(0)=u0, where ν(dz)dt denotes the compensator of the random measure η. First, we consider the case where the Lévy process is a compound Poisson process, then we use this result to show that (0.1) has a solution in the general case.