Large deviation principles for SPDEs with locally Lipschitz coefficients

Abstract

Consider the stochastic partial differential equation, \begin{align*} \partial_t u^{\varepsilon}(t\,,x) = \frac{1}{2} \partial^2_x u^{\varepsilon}(t\,,x) + b(t\,,u^{\varepsilon}(t\,,x)) + \sqrt{\varepsilon}\sigma(t\,,u^{\varepsilon}(t\,,x)) \dot{W}(t\,,x), \end{align*} where $(t\,,x)\in(0\,,\infty)\times\mathbb{R}$, and $\dot{W}$ denotes space-time white noise. Foondun, Khoshnevisan, and Nualart \cite{FKN24} showed that this stochastic partial differential equation is well-posed under the assumptions that the initial condition $u(0)$ is bounded and measurable, while $b$ and $\sigma$ are locally Lipschitz continuous functions with at most linear growth. A Freidlin-Wentzell large deviation principle for the stochastic partial differential equation is established by a weak convergence approach in this paper.