arXiv 2212.102822022-12math.PR

由梯度依赖噪声驱动的完全局部单调随机偏微分方程的大偏差

Large deviations of fully local monotone stochastic partial differential equations driven by gradient-dependent noise

Tianyi Pan, Shijie Shang, Jianliang Zhai, Tusheng Zhang

备注：: 36 pages

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AI总结：

本文研究了完全局部单调系数且扩散系数依赖解梯度的随机偏微分方程，结合伪单调技巧与紧性论证建立了其小噪声大偏差原理，框架涵盖多种现有工作无法包含的模型。

摘要：

考虑在 Gelfand 三元组 $V\subseteq H\subseteq V^*$ 中具有完全局部单调系数的随机偏微分方程（SPDEs） $$ \left\{ \begin{align} &dX_t=A(t,X_t)dt+B(t,X_t)dW_t,\\ t\in (0,T]\\\\& X_0=x\in H, \end{align} \right. $$ 其中 $$A: [0,T] \times V\rightarrow V^*,\\ \\ B:[0,T]\times V\rightarrow\\ L_2(U,H)$$ 是可测映射，$L_2(U,H)$ 是从 $U$ 到 $H$ 的 Hilbert-Schmidt 算子空间，$W$ 是一个 $U$-柱状 Wiener 过程。本文为上述 SPDEs 的解 {$u^\varepsilon$}$_{\varepsilon>0}$ 建立了小噪声大偏差原理（LDP）。本文的主要贡献在于，我们的框架比现有结果具有更广泛的通用性。特别地，扩散系数 $B(t,\cdot)$ 可以依赖于解的梯度，这在 SPDEs 领域引起了极大关注，但关于 LDP 主题的现有结果很少。完全局部单调设定下更广的范围促使我们采用了不同的策略和技术。伪单调技巧与紧性论证的结合在整篇论文中发挥了关键作用。我们的框架非常通用，涵盖了现有工作无法涵盖的许多有趣模型，包括随机拟线性 SPDEs、随机对流扩散方程、随机 2D 液晶方程、带有梯度依赖噪声的随机 $p$-Laplace 方程、带有梯度依赖噪声的随机 2D Navier-Stokes 方程等。

英文摘要：

Consider stochastic partial differential equations (SPDEs) with fully local monotone coefficients in a Gelfand triple $V\subseteq H\subseteq V^*$ $$ \left\{ \begin{align} &dX_t=A(t,X_t)dt+B(t,X_t)dW_t,\ t\in (0,T]\\\\& X_0=x\in H, \end{align} \right. $$ where $$A: [0,T] \times V\rightarrow V^*,\ \ B:[0,T]\times V\rightarrow\ L_2(U,H)$$ are measurable maps, $L_2(U,H)$ is the space of Hilbert-Schmidt operators from $U$ to $H$ and $W$ is a $U$-cylindrical Wiener process.\par In this paper, we establish a small noise large deviation principle(LDP) for the solutions {$u^\varepsilon$}$_{\varepsilon>0}$ of the above SPDEs. The main contribution of this paper is the much more generality of our framework than that of the existing results. In particular, the diffusion coefficient $B(t,\cdot)$ may depend on the gradient of the solutions, which is of great interest in the field of SPDEs, but there are few existing results on the topic of LDP. The broader scope of the fully local monotone setting leads us to use different strategies and techniques. A combination of the pseudomonotone technique and compactness arguement plays a crucial role in the whole paper. Our framework is very general to include many interesting models that could not be covered by existing work, including stochastic quasilinear SPDEs, stochastic convection diffusion equation, stochastic 2D Liquid crystal equation, stochastic $p$-Laplace equation with gradient-dependent noise, stochastic 2D Navier-Stokes equation with gradient-dependent noise etc.

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