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

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.