Nonlinear SPDE driven by Levy noise: Well-posedness, optimal control and invariant measure

TL;DR

This paper investigates a nonlinear stochastic PDE driven by Levy noise, establishing well-posedness, existence of optimal controls, and invariant measures using advanced probabilistic and variational techniques.

Contribution

It introduces new methods to prove existence and uniqueness of solutions, optimal controls, and invariant measures for nonlinear SPDEs with Levy noise.

Findings

Proved existence of pathwise unique strong solutions.

Established existence of weak optimal controls.

Proved existence of invariant measures for the SPDE.

Abstract

In this article, we study a nonlinear stochastic control problem perturbed by multiplicative Levy noise, where the nonlinear operator in divergence form satisfies p type growth with coercivity assumptions. By using Aldous tightness criteria and Jakubowski version of the Skorokhod theorem on nonmetric spaces along with standard contraction method, we establish existence of pathwise unique strong solution. Formulating the associated control problem, and using variational approach together with the convexity property of cost functional in control variable, we establish existence of a weak optimal solution of the underlying problem. We use the technique of Maslowski and Seidler to prove existence of an invariant measure for uncontrolled SPDE driven with multiplicative Levy noise.