A note on stochastic semilinear equations and their associated Fokker-Planck equations

TL;DR

This paper extends the analysis of semilinear stochastic PDEs by broadening the functional framework and establishing existence and uniqueness of solutions for associated Fokker-Planck and stochastic equations with complex nonlinearities.

Contribution

It generalizes the framework from Hilbert spaces to Gelfand triples and proves existence of solutions for Fokker-Planck equations with complex nonlinearities, also establishing unique strong solutions for the SPDEs.

Findings

Existence of solutions for Fokker-Planck equations with polynomial and Burgers nonlinearities.

Existence and uniqueness of strong solutions for semilinear SPDEs driven by space-time white noise.

Extension of the framework to a Gelfand triple setting.

Abstract

In this paper we treat semilinear stochastic partial differential equations by two methods. First, we extend the framework of [BDR10] from a Hilbert space to a Gelfand triple and as an application we prove the existence of solutions for the Fokker-Planck equations associated to semilinear equations with space-time white noise and both with polynomially growing nonlinearities and Burgers type nonlinearities at the same time. Second we adopt the approximation technique from [BDR10] to obtain existence of unique strong solutions to semilinear stochastic partial differential equations driven by space-time white noise, generalizing corresponding known results from the literature.