The stochastic Klausmeier system and a stochastic Schauder-Tychonoff type theorem

TL;DR

This paper establishes the existence and uniqueness of solutions to a stochastic nonlinear advection-diffusion system inspired by Klausmeier, and introduces a stochastic Schauder-Tychonoff fixed point theorem to facilitate the analysis.

Contribution

It provides a novel stochastic fixed point theorem and applies it to prove solution existence for a stochastic Klausmeier system, combining variational and semigroup methods.

Findings

Existence of nonnegative martingale solutions

Pathwise uniqueness of solutions

Regularity properties of solutions

Abstract

On the one hand, we investigate the existence and pathwise uniqueness of a nonnegative martingale solution to the stochastic evolution system of nonlinear advection-diffusion equations proposed by Klausmeier with Gaussian multiplicative noise. On the other hand, we present and verify a general stochastic version of the Schauder-Tychonoff fixed point theorem, as its application is an essential step for showing existence of the solution to the stochastic Klausmeier system. The analysis of the system is based both on variational and semigroup techniques. We also discuss additional regularity properties of the solution.