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

TL;DR

The paper addresses well-posedness of a space-time fractional SPDE with Caputo derivative $∂_t^β$ and fractional Laplacian $−(Δ)^{α/2}$, driven by space-time white noise. A truncation scheme for coefficients $b_N$ and $σ_N$ is used to obtain uniform moment and tail estimates, enabling passage to a limit that yields a unique random-field solution with finite moments. Two coefficient regimes, $L_σ>0$ and $σ∈L^∞$, are treated with BDG-based estimates and heat-kernel bounds to prove existence and uniqueness. This extends prior work to space-time fractional settings and provides quantitative controls, contributing to the theory of well-posed SPDEs with locally Lipschitz coefficients.

Abstract

In this article, we study the space-time SPDE $$ \partial_t^βu=-(-Δ)^{α/2} u+I_t^{1-β}[b(u)+σ(u)\dot{W}],$$ where $u=u(t,x)$ is defined for $(t,x)\in\mathbb{R}_+\times \mathbb{R},$ $β\in(0,1), α\in(0,2)$ 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 $σ$ 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 $σ$ are locally Lipschitz and have at-most linear growth and some conditions on the Lipschitz constants on the truncated versions of $b$ and $σ$. Our results generalize the work of Foondun et al.(2025) to a space-time fractional setting.

Theorems & Definitions (7)

• Remark 2.1
• Theorem 2.5
• Proposition 2.6
• proof
• Proposition 2.7
• proof
• Proposition 2.8: FooSanLart, Lemma 2.5