The nonlinear Schr\"odinger Equation driven by jump processes

TL;DR

This paper proves the existence of solutions for a nonlinear Schrödinger equation driven by jump processes, including infinite activity Lévy noise, extending previous results to more complex stochastic influences.

Contribution

The paper introduces a method to establish solutions for nonlinear Schrödinger equations with Lévy noise of infinite activity, broadening the scope of stochastic PDE analysis.

Findings

Existence of solutions for equations driven by compound Poisson Lévy processes.

Extension to general Lévy processes with infinite activity.

Framework applicable to nonlinear Schrödinger equations with jump noise.

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.