Large deviation principle for stochastic reaction-diffusion equations with super-linear drift on $\mathbb{R}$ driven by space-time white noise

TL;DR

This paper establishes a large deviation principle for stochastic reaction-diffusion equations with super-linear drift on the real line driven by space-time white noise, addressing challenges of unbounded domain and non-dissipative drift.

Contribution

It develops a modified weak convergence method and specialized norms to prove the large deviation principle for complex stochastic PDEs with super-linear drift on unbounded domains.

Findings

Large deviation principle successfully established.

Novel analytical techniques for unbounded domain and super-linear drift.

Key estimates and inequalities developed for stochastic convolution.

Abstract

In this paper, we consider stochastic reaction-diffusion equations with super-linear drift on the real line $\mathbb{R}$ driven by space-time white noise. A Freidlin-Wentzell large deviation principle is established by a modified weak convergence method on the space $C([0,T], C_{tem}(\mathbb{R}))$. Obtaining the main result in this paper is challenging due to the setting of unbounded domain, the space-time white noise, and the superlinear drift term without dissipation. To overcome these difficulties, the special designed norm on $C([0,T], C_{tem}(\mathbb{R}))$, one order moment estimates of the stochastic convolution and two nonlinear Gronwall-type inequalities play an important role.