Space-time fractional SPDEs with locally Lipschitz coefficients: well-posedness

TL;DR

This paper establishes the well-posedness of a space-time fractional stochastic partial differential equation with locally Lipschitz coefficients, extending previous results to a more general fractional setting.

Contribution

It proves the existence and uniqueness of solutions for space-time fractional SPDEs with minimal assumptions on coefficients, generalizing prior work to fractional derivatives.

Findings

Proved well-posedness under minimal conditions

Extended results to space-time fractional SPDEs

Established solution properties with locally Lipschitz coefficients

Abstract

In this article, we study the space-time SPDE $$ \partial_t^\beta u=-(-\Delta)^{\alpha/2} u+I_t^{1-\beta}[b(u)+\sigma(u)\dot{W}],$$ where $u=u(t,x)$ is defined for $(t,x)\in\mathbb{R}_+\times \mathbb{R},$ $\beta\in(0,1), \alpha\in(0,2)$ and $\dot{W}$ denotes a space-time white noise. It has long been conjectured that this equation has a unique solution with finite moments under the minimal assumptions of locally Lipschitz coefficients $b$ and $\sigma$ with linear growth. We prove that this SPDE is well-posed under the assumptions that the initial condition $u_0$ is bounded and measurable, and the functions $b$ and $\sigma$ are locally Lipschitz and have at-most linear growth and some conditions on the Lipschitz constants on the truncated versions of $b$ and $\sigma$. Our results generalize the work of Foondun et al.(2025) to a space-time fractional setting.