A Large Deviation Principle for the Stochastic Heat Equation with General Rough Noise

We study the Freidlin–Wentzell large deviation principle for the nonlinear one-dimensional stochastic heat equation driven by a Gaussian noise: $$\begin{aligned} \frac{\partial u^{{\varepsilon }}(t,x)}{\partial t}=\frac{\partial ^2 u^{{\varepsilon }}(t,x)}{\partial x^2}+\sqrt{{\varepsilon }}\sigma (t, x, u^{{\varepsilon }}(t,x))\dot{W}(t,x),\quad t> 0,\, x\in \mathbb {R}, \end{aligned}$$ ∂ u ε ( t , x ) ∂ t = ∂ 2 u ε ( t , x ) ∂ x 2 + ε σ ( t , x , u ε ( t , x ) ) W ˙ ( t , x ) , t > 0 , x ∈ R , where $$\dot{W}$$ W ˙ is white in time and fractional in space with Hurst parameter $$H\in \left( \frac{1}{4},\frac{1}{2}\right) $$ H ∈ 1 4 , 1 2 . Recently, Hu and Wang (Ann Inst Henri Poincaré Probab Stat 58(1):379–423, 2022) have studied the well-posedness of this equation without the technical condition of $$\sigma (0)=0$$ σ ( 0 ) = 0 which was previously assumed in Hu et al. (Ann Probab 45(6):4561–4616, 2017). We adopt a new sufficient condition proposed by Matoussi et al. (Appl Math Optim 83(2):849–879, 2021) for the weak convergence criterion of the large deviation principle.