Fuzzy-Stochastic Partial Differential Equations

Calderon-Type Uniqueness Theorem for Stochastic Partial Differential Equations

In this Note, we present a Calderón-type uniqueness theorem on the Cauchy problem of stochastic partial differential equations. To this aim, we introduce the concept of stochastic pseudo-differential operators, and establish their boundedness and other fundamental properties. The proof of our uniqueness theorem is based on a new Carleman-type estimate.

非线性随机偏微分方程的光滑解

Smooth Solutions of Non-linear Stochastic Partial Differential Equations

本文在Hilbert标度的框架下研究了非线性随机偏微分方程解的存在性。然后将所得结果应用于几类典型的非线性随机偏微分方程，如实数上的随机Burgers方程和Ginzburg-Landau方程，全空间上的随机二维Navier-斯托克斯方程以及全空间上的随机驯服三维Navier-斯托克斯方程，得到了它们各自光滑解的存在性.

On the Dirichlet Problem for Backward Parabolic Stochastic Partial Differential Equations in General Smooth Domains

Backward stochastic partial differential equations of parabolic type with variable coefficients are considered in smooth domains. Existence and uniqueness results are given in weighted Sobolev spaces allowing the derivatives of the solutions to blow up near the boundary.

一类带非局部扩散的随机偏微分方程

On a stochastic partial differential equation with non-local diffusion

本文证明了一维时空白噪声驱动下一类含分数拉普拉斯算子的随机偏微分方程解的存在性、唯一性和正则性。我们考虑的方程也可能包括一个反应项。

巴拿赫空间中的随机Volterra方程与随机偏微分方程

Stochastic Volterra Equations in Banach Spaces and Stochastic Partial Differential Equations

本文首先研究了2-光滑巴拿赫空间中具有奇异核的随机Volterra积分方程解的存在唯一性和大偏差估计。然后，我们将它们应用于一类由布朗运动和分数布朗运动驱动的半线性随机偏微分方程，在局部利普希茨条件下，得到了该方程的唯一极大强解（在偏微分方程和偏微分方程意义下）的存在性.最后，研究了欧氏空间有界区域上的高阶随机偏微分方程、完备黎曼流形上的二阶随机偏微分方程以及随机Navier-斯托克斯方程。

路径随机控制与一类随机偏微分方程

Pathwise stochastic control and a class of stochastic partial differential equations

本文研究路径随机最优控制问题，证明值过程是相关Hamilton-Jacobi-Bellman随机偏微分方程在粘性意义下的唯一解，并讨论随机路径Noether定理的相应版本。

Random attractors for locally monotone stochastic partial differential equations

We prove the existence of random dynamical systems and random attractors for a large class of locally monotone stochastic partial differential equations perturbed by additive Lévy noise. The main result is applicable to various types of SPDE such as stochastic Burgers type equations, stochastic 2D Navier-Stokes equations, the stochastic 3D Leray-$α$ model, stochastic power law fluids, the stochastic Ladyzhenskaya model, stochastic Cahn-Hilliard type equations, stochastic Kuramoto-Sivashinsky type equations, stochastic porous media equations and stochastic $p$-Laplace equations.

随机和随机偏微分方程的不变流形

Invariant manifolds for random and stochastic partial differential equations

随机不变流形是理解随机影响下的复杂动力学的有用几何对象。在非一致双曲性或非一致指数二分性条件下，证明了一类\emph{random}偏微分方程和\emph{stochastic}偏微分方程的随机伪稳定和伪非稳定流形的存在性.与随机\emph{ordinary}微分方程的不变流形理论不同，不使用随机范数。然后将所得结果应用于带线性乘性噪声的非线性随机偏微分方程。

随机解与奇异偏微分方程

Stochastic solutions and singular partial differential equations

本文提出将随机解技术推广至分布值噪声驱动的偏微分方程，作为其求解方法。