The nonlinear Schrödinger Equation driven by jump processes

Abstract

The main result of the paper is the existence of a solution of the nonlinear Schrödinger equation with a \levy noise with infinite activity. To be more precise, let $A=\Delta$ be the Laplace operator with $D(A)=\{u\in L^2({\mathbb{R}}^d):\Delta u\in L^2({\mathbb{R}}^d)\}$. Let $Z\hookrightarrow L^2({\mathbb{R}}^d)$ be a function space and $\eta$ be a Poisson random measure on $Z$, let $g:\mathbb{R}\to \mathbb{C}$ and $h:\mathbb{R}\to \mathbb{C}$ be some given functions, satisfying certain conditions specified later. Let $\alpha \ge 1$ and $\lambda \ge 0$. We are interested in the solution of the following equation % $$idu(t,x)-\Delta u(t,x)dt+\lambda |u(t,x){|}^{\alpha -1}u(t,x)dt$$ $$={\int }_{Z}u(t,x)g(z(x))\tilde{\eta }(dz,dt)+{\int }_{Z}u(t,x)h(z(x))\gamma (dz,dt),$$ $$u(0)=u_0.$$ First we consider the case, where the \levy process is a compound Poisson process. With the help of this result we can tackle the general case, and show that the equation above has a solution.