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

Abstract

We study the Freidlin–Wentzell large deviation principle for the nonlinear one-dimensional stochastic heat equation driven by a Gaussian noise:

$$\begin{aligned} \frac{\partial u^{{\varepsilon }}(t,x)}{\partial t}=\frac{\partial ^2 u^{{\varepsilon }}(t,x)}{\partial x^2}+\sqrt{{\varepsilon }}\sigma (t, x, u^{{\varepsilon }}(t,x))\dot{W}(t,x),\quad t> 0,\, x\in \mathbb {R}, \end{aligned}$$

where $\dot{W}$ is white in time and fractional in space with Hurst parameter $H\in \left( \frac{1}{4},\frac{1}{2}\right)$. 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 $\sigma (0)=0$ 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.

Appendix

In this section, we give some lemmas related to the heat kernel $p_t(x)$. Recall $\lambda (x)$ defined by (2.13), $D_{t}(x,h)$, $\Box _{t}(x,y,h)$ defined by (3.8) and (3.9), respectively.

Lemma 6.1

([22, Lemma 2.5]) For any $T>0$,

$$\begin{aligned} \sup _{t\in [0,T]}\sup _{x\in \mathbb {R}} \frac{1}{\lambda (x)}\int _{\mathbb {R}} p_{t}(x-y) \lambda (y) \textrm{d}y<\infty . \end{aligned}$$

(6.1)

Lemma 6.2

([22, Lemma 2.8]) For any $H\in (\frac{1}{4},\frac{1}{2})$, there exists some constant $C_H$ such that

$$\begin{aligned} \int _{\mathbb {R}^2}|D_{t}(x,h)|^2\cdot |h|^{2H-2}\textrm{d}h\textrm{d}x= C_Ht^{H-1}, \end{aligned}$$

(6.2)

and

$$\begin{aligned} \int _{\mathbb {R}^3}|\Box _{t}(x,y,h)|^2\cdot |h|^{2H-2}\cdot |y|^{2H-2}\textrm{d}y\textrm{d}h\textrm{d}x= C_H t^{2H-\frac{3}{2}}. \end{aligned}$$

(6.3)

Lemma 6.3

([22, Lemma 2.10]) For any $t>0$, there exists some constant $C_H$ such that

$$\begin{aligned} \int _{\mathbb {R}}|D_{t}(x,h)|^2\cdot |h|^{2H-2}\textrm{d}h\le C_H\left( t^{H-\frac{3}{2}}\wedge \frac{|x|^{2H-2}}{\sqrt{t}}\right) , \end{aligned}$$

(6.4)

where $0<H<\frac{1}{2}$.

Lemma 6.4

([22, Lemma 2.11]) For any $t>0$, there exists some constant $C_H$ such that

$$\begin{aligned} \int _{\mathbb {R}^2}|\Box _{t}(x,y,h)|^2\cdot |h|^{2H-2}\cdot |y|^{2H-2}\textrm{d}y\textrm{d}h\le C_H\left( t^{2H-2}\wedge \frac{|x|^{2H-2}}{t^{1-H}}\right) . \end{aligned}$$

(6.5)

Lemma 6.5

([22, Lemma 2.12]) For any $t>0$, there exists some constant $C_{T,H}$ such that

$$\begin{aligned} \int _{\mathbb {R}^2}|D_{t}(x,h)|^2\cdot |h|^{2H-2}\lambda (z-x)\textrm{d}x\textrm{d}h\le C_{T,H}t^{H-1}\lambda (z), \end{aligned}$$

(6.6)

and

$$\begin{aligned} \int _{\mathbb {R}^3}|\Box _{t}(x,y,h)|^2\cdot |h|^{2H-2}\cdot |y|^{2H-2}\lambda (z-x)\textrm{d}x\textrm{d}y\textrm{d}h\le C_{T,H}t^{2H-\frac{3}{2}}\lambda (z). \end{aligned}$$

(6.7)

Lemma 6.6

([22, (4.29)], [22, (4.32)]) For some fixed $\gamma \in (0,1)$ and $\alpha \in (0,1)$, the following two inequalities hold:

$$\begin{aligned} |(t+h)^{\alpha -1}-t^{\alpha -1}|\lesssim |t|^{\alpha -1-\gamma }h^\gamma , \end{aligned}$$

(6.8)

and

$$\begin{aligned} |p_{t+h}(x)-p_{t}(x)| \lesssim h^\gamma t^{-\gamma }\left[ p_{\frac{2}{\gamma }(t+h)}(x)+p_{\frac{2t}{\gamma }}(x)\right] . \end{aligned}$$

(6.9)

Lemma 6.7

([22, Theorem 4.4]) A sequence $\{\mathbb {P}_n\}^\infty _{n=1}$ of probability measures on $(\mathcal {C}([0,T]\times \mathbb {R}),\mathcal {B}(\mathcal {C}([0,T]\times \mathbb {R})))$ is tight if and only if the following conditions hold:

• (i). $\displaystyle \lim _{\lambda \uparrow \infty } \sup _{n\ge 1} \mathbb {P}_n\left( \{\omega \in \mathcal {C}([0,T]\times \mathbb {R}):|\omega (0,0)|>\lambda \}\right) =0$.

• (ii). For any $R>0$ and $\gamma >0$,

  $$\begin{aligned} \lim _{\delta \downarrow 0}\sup _{n\ge 1}\mathbb {P}_n\left( \left\{ \omega \in \mathcal {C}([0,T]\times \mathbb {R}):m^{T,R}(\omega ,\theta )>\gamma \right\} \right) =0{,} \end{aligned}$$

  where

  $$\begin{aligned} m^{T,R}(\omega ,\theta ):=\max \limits _{\begin{array}{c} |t-s|+|x-y|\le \theta , \\ 0\le t,s\le T;\,-R\le x, y\le R \end{array}}\left| \omega (t,x)-\omega (s,y)\right| \end{aligned}$$

  is the modulus of continuity on $[0,T]\times [-R,R]$.