On the well-posedness of SPDEs with locally Lipschitz coefficients

TL;DR

This paper establishes the well-posedness of a class of stochastic partial differential equations with locally Lipschitz coefficients, using truncation and moment bounds, and extends results to time-dependent coefficients and stochastic wave equations.

Contribution

It proves well-posedness of SPDEs with locally Lipschitz coefficients under minimal assumptions, extending to time-dependent coefficients and stochastic wave equations.

Findings

SPDEs with locally Lipschitz coefficients are well-posed under bounded initial conditions.

Method based on truncation, moment bounds, and tail estimates.

Results extend to time-dependent coefficients and stochastic wave equations.

Abstract

We consider the stochastic partial differential equation, $\partial_t u = \tfrac12 \partial^2_x u + b(u) + \sigma(u) \dot{W},$ where $u=u(t\,,x)$ is defined for $(t\,,x)\in(0\,,\infty)\times\mathbb{R}$, and $\dot{W}$ denotes space-time white noise. We prove that this SPDE is well posed solely under the assumptions that the initial condition $u(0)$ is bounded and measurable, and $b$ and $\sigma$ are locally Lipschitz continuous functions and have at most linear growth. Our method is based on a truncation argument together with moment bounds and tail estimates of the truncated solution. The results naturally generalize to the case where $b$ and $\sigma$ are time dependent with uniform-in-time growth and oscillation properties. Additionally, our method can be extended to the stochastic wave equation.