论文标题
完全局部单调随机偏微分方程的大偏差由梯度依赖性噪声驱动
Large deviations of fully local monotone stochastic partial differential equations driven by gradient-dependent noise
论文作者
论文摘要
考虑使用Gelfand Triple $ v \ subseteq H \ subseteq v^*$ $$中的完全本地单调系数的随机部分微分方程（SPDE） \左边\{ \ begin {align} ＆dx_t = a（t，x_t）dt+b（t，x_t）dw_t，\ t \ in（0，t] \\\\\＆ x_0 = x \ in h， \ end {align} \正确的。 $$ 在哪里 $ a：[0，t] \ times v \ rightarrow v^*，\ \ b：[0，t] \ times v \ rightArrow \ l_2（u，u，h）$$是可衡量的地图，$ l_2（u，h）$是Hilbert-schmidt Operational of Hilbert-Schmidt Operators of Hilbert-Schmidt ocartors of Hilbert-schmidt ocartors of Hilbert-schmidt ocartors from $ u $ u $ u $ u $ h $ w- $ w- 在本文中，我们为解决方案建立了一个小的噪声大偏差原理（LDP）{$ u^\ varepsilon $} $ _ {\ varepsilon> 0} $。 本文的主要贡献是我们框架的一般性比现有结果的贡献要多得多。特别是，扩散系数$ b（t，\ cdot）$可能取决于解决方案的梯度，这在SPDE领域引起了极大的兴趣，但是关于LDP主题的现有结果很少。完全本地单调设置的更广泛的范围使我们使用不同的策略和技术。假单胞酮技术和紧凑性论证的结合在整个论文中起着至关重要的作用。 我们的框架非常普遍，包括许多有趣的模型，这些模型无法被现有工作所涵盖，包括随机的准式SPDES，随机对流扩散方程，随机2D液晶方程，随机$ P $ - laplace方程，具有梯度依赖性噪声，随机依赖性噪声，随机2D Navier-StokeS navier-Stokes Equient naviententent noise等，
Consider stochastic partial differential equations (SPDEs) with fully local monotone coefficients in a Gelfand triple $V\subseteq H\subseteq V^*$ $$ \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.
