论文标题
与一般粗糙噪声的随机热方程的大偏差原理
A large deviation principle for the stochastic heat equation with general rough noise
论文作者
论文摘要
我们研究了由高斯噪声驱动的一维非线性随机热方程的Freidlin-Wentzell的大偏差原理：$$ \ frac {\ partial u^\ varepsilon（t，x，x，x）} {\ partial t} {\ partial t} = \ frac { x^2}+\sqrt{\varepsilon} σ(t, x, u^\varepsilon(t,x))\dot{W}(t,x),\quad t> 0,\, x\in\mathbb{R},$$ where $\dot W$ is white in time and fractional in space with Hurst parameter $ h \ in（\ frac 14，\ frac 12）$。最近，Hu and Wang（{\ itAnn。Inst。HenriPoincaréProbab。Stat。} {\ Bf 58}（2022）379-423）研究了该方程的良好性，而没有$σ（0）= 0 $的技术条件，该条件的技术条件= 0 $。 （{\ itann。probab}。{\ bf 45}（2017）4561-4616）。我们采用了Matoussi等人提出的新的充分条件。 （{\ itAppl。Math。optim。} \ TextBf {83}（2021）849-879）用于大偏差原理的弱收敛性标准。
We study Freidlin-Wentzell's large deviation principle for one dimensional nonlinear stochastic heat equation driven by a Gaussian noise: $$\frac{\partial u^\varepsilon(t,x)}{\partial t} = \frac{\partial^2 u^\varepsilon(t,x)}{\partial x^2}+\sqrt{\varepsilon} σ(t, x, u^\varepsilon(t,x))\dot{W}(t,x),\quad t> 0,\, x\in\mathbb{R},$$ 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 Poincaré Probab. Stat.} {\bf 58} (2022) 379-423) studied the well-posedness of this equation without the technical condition of $σ(0)=0$ 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.
