Local existence and non-explosion of solutions for stochastic fractional partial differential equations driven by multiplicative noise

TL;DR

This paper proves local existence and uniqueness of solutions for stochastic fractional PDEs with multiplicative noise, showing that noise can prevent blow-up under certain conditions, with applications to various SPDEs.

Contribution

It establishes the local existence, uniqueness, and non-explosion results for a broad class of stochastic fractional PDEs driven by multiplicative noise, highlighting a regularizing effect.

Findings

Solutions exist locally and are unique.

Adding linear multiplicative noise prevents blow-up with high probability.

Results apply to multiple types of stochastic PDEs.

Abstract

In this paper we prove the local existence and uniqueness of solutions for a class of stochastic fractional partial differential equations driven by multiplicative noise. We also establish that for this class of equations adding linear multiplicative noise provides a regularizing effect: the solutions will not blow up with high probability if the initial data is sufficiently small, or if the noise coefficient is sufficiently large. As applications our main results are applied to various types of SPDE such as stochastic reaction-diffusion equations, stochastic fractional Burgers equation, stochastic fractional Navier-Stokes equation, stochastic quasi-geostrophic equations and stochastic surface growth PDE.