A large deviation principle for the stochastic heat equation with general rough noise

TL;DR

This paper establishes a large deviation principle for a one-dimensional stochastic heat equation driven by rough Gaussian noise, extending previous results to more general conditions and noise types.

Contribution

It introduces a new sufficient condition for the weak convergence criterion, broadening the applicability of large deviation principles to equations with fractional noise.

Findings

Proves a large deviation principle for the stochastic heat equation with fractional noise.

Extends well-posedness results to cases without the zero initial condition.

Utilizes a novel criterion for weak convergence in large deviations.

Abstract

We study Freidlin-Wentzell's large deviation principle for one dimensional nonlinear stochastic heat equation driven by a Gaussian noise: $$\frac{\partial u^\varepsilon(t,x)}{\partial t} = \frac{\partial^2 u^\varepsilon(t,x)}{\partial x^2}+\sqrt{\varepsilon} \sigma(t, x, u^\varepsilon(t,x))\dot{W}(t,x),\quad t> 0,\, x\in\mathbb{R},$$ where $\dot W$ is white in time and fractional in space with Hurst parameter $H\in(\frac 14,\frac 12)$. Recently, Hu and Wang ({\it Ann. Inst. Henri Poincar\'e Probab. Stat.} {\bf 58} (2022) 379-423) studied the well-posedness of this equation without the technical condition of $\sigma(0)=0$ which was previously assumed in Hu et al. ({\it Ann. Probab}. {\bf 45} (2017) 4561-4616). We adopt a new sufficient condition proposed by Matoussi et al. ({\it Appl. Math. Optim.} \textbf{83} (2021) 849-879) for the weak convergence criterion of the large deviation principle.