The nonlinear Schr\"odinger Equation driven by jump processes

Abstract

The main result of the paper is the existence of a solution of the nonlinear Schr\"odinger 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 % $$ i \, d u(t,x) - \Delta u(t,x)\,dt +\lambda |u(t,x)|^{\alpha-1} u(t,x) \, dt $$ $$= \int_Z u(t,x)\, g(z(x))\,\tilde \eta (dz,dt)+\int_Z u(t,x)\, h (z(x))\, \gamma (dz, dt), $$ $$ u(0)= u_0. $$ 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.