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

Abstract

We study Freidlin-Wentzell's large deviation principle for one dimensional nonlinear stochastic heat equation driven by a Gaussian noise: $$\frac{\partial u^{\epsilon }(t,x)}{\partial t}=\frac{{\partial }^2{u}^{\epsilon }(t,x)}{\partial x^2}+\sqrt{\epsilon }\sigma (t,x,u^{\epsilon }(t,x))\dot{W}(t,x),t>0,x\in \mathbb{R},$$ where $\dot{W}$ is white in time and fractional in space with Hurst parameter $H\in (\frac{1}{4},\frac{1}{2})$. Recently, Hu and Wang ({\it Ann. Inst. Henri Poincaré 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.

43 pages