A stochastic Schauder-Tychonoff type theorem and its applications

TL;DR

This paper introduces a stochastic version of the Schauder-Tychonoff fixed-point theorem and demonstrates its application in establishing existence results for nonlinear stochastic diffusion equations with non-Lipschitz perturbations.

Contribution

It presents a novel stochastic fixed-point theorem extending classical results and applies it to complex stochastic PDEs with non-Lipschitz nonlinearities.

Findings

Established a stochastic Schauder-Tychonoff theorem

Proved existence of solutions for certain nonlinear stochastic PDEs

Extended fixed-point methods to stochastic settings

Abstract

One standard way to prove existence for deterministic, highly nonlinear PDEs is to use the Schauder-Tychonoff fixed-point theorem. In what follows, we introduce and verify a stochastic variant of the Schauder-Tychonoff theorem. We apply our existence result to nonlinear stochastic diffusion equations with non-Lipschitz perturbations