随机微分方程及其应用会议日程安排

08月05号上午8:30—11:40

8:30--9:00 开幕式，主持人：吕广迎，河南大学 冯淑霞院长致词，照相

9:00--11:40    主持人： Jiang-Lun Wu(吴奖伦)、英国斯旺西大学

9:00-9:40 报告人： 董昭、中科院

报告题目：随机微分方程渐进稳定性研究感想浅谈

9:40-10:20 报告人：宋仁明、University of Illinois

报告题目：Dirichlet   heat kernel estimates of Subordinate Brownian Motions

10:20--10:50 茶歇

10:50--11:30报告人： 张希承、武汉大学

报告题目：Ergodicity of stochastic differential equations with jumps and   singular coefficients

08月05号下午 2:30—5:40

2:30--3:50 主持人：黄建华、国防科技大学

2:30--3:10 报告人：李用声、华南理工大学

报告题目：Wellposedness for a   Higher Order Shallow Water Type Equations in Low Regularity Spaces

3:10--3:50 报告人：翟建梁，中国科技大学

报告题目：2D Stochastic Chemotaxis-Navier-Stokes   System

3:50--4:20 茶歇

4:20--5:40 主持人：刘文军、南京信息工程大学

4:20--5:00 报告人：周国立、重庆大学

报告题目：Asymptotic behavior of 3-D stochastic   primitive equations of large-scale moist atmosphere with additive noise

5:00--5:40 报告人： 杨新光、河南师范大学

报告题目：Tempered backward behavior for   non-autonomous incompressible Navier-Stokes equation

08月06号8:30—11:30

8:30--9:50 主持人：黎育红、华中科技大学

8:30--9:10 报告人：张土生、英国曼彻斯特大学

报告题目：On global existence of solutions of   stochastic reaction equations

9:10--9:50 报告人：Jiang-Lun Wu(吴奖伦)、英国斯旺西大学

报告题目：Characterising   path-independence of Girsanov transform for stochastic differential equations

9:50--10:10 茶歇

10:10--11:30 主持人：王伟、南京大学

10:10--10:50 报告人：邹广安，河南大学

报告题目：Finite element   methods for time-fractional stochastic diffusion and diffusion-wave equations

10:50--11:30，报告人：吕广迎，河南大学

报告题目： $W^{1,r}_{loc}$-solutions for   stochastic transport equations

报告题目及摘要

随机微分方程渐进稳定性研究感想浅谈

董昭

中国科学院数学与系统科学院

通过两类具体方程，浅谈有关随机微分方程和随机偏微分方程渐进稳定性的研究感想。

Existence and uniqueness of $W^{1,r}_{loc}$-solutions for stochastic transport equations

吕广迎

河南大学

We investigate a stochastic transport equation driven by a multiplicative noise. For $L^q(0,T;W^{1,p}(\mR^d;\mR^d))$ drift and $W^{1,r}(\mR^d)$ initial data, we obtain the existence and uniqueness of stochastic strong solutions (in $W^{1,r}_{loc}(\mR^d))$. In particular, when $r=\infty$, we establish a Lipschitz estimate for solutions and this question is opened by Fedrizzi and Flandoli in case of  $L^q(0,T;L^p(\mR^d;\mR^d))$ drift. Moreover, opposite to the deterministic case where $L^q(0,T;W^{1,p}(\mR^d;\mR^d))$ drift and $W^{1,p}(\mR^d)$ initial data may induce non-existence for strong

solutions (in $W^{1,p}_{loc}(\mR^d)$), we prove that a multiplicative stochastic perturbation of Brownian type is enough to render the equation well-posed. It is an interesting example of a deterministic PDE that

becomes well-posed under the influence of a multiplicative Brownian

type noise. We extend the existing results \cite{FF2,FGP1} partially.

Wellposedness for a Higher Order Shallow Water Type Equations in Low Regularity Spaces Wellposedness for a Higher Order Shallow Water Type Equations in Low Regularity Spaces Wellposedness for a Higher Order Shallow Water Type Equations in Low Regularity Spaces

李用声

华南理工大学

Abstract

In this talk we discuss the wellpossedness of  the following shallow water type equation
\begin{align*}
 u_{t}+\partial_{x}^{3}u
      + \frac{1}{2}\partial_{x}(u^{2})
      + \partial_{x}(1-\partial_{x}^{2})^{-1}\left[u^{2}
      + \frac{1}{2} u_{x}^{2}\right]=0,
        \quad x\in {\mathbf T}=\R/2\pi \lambda
\end{align*}
By applying the bilinear estimate in the space $W^{s}$,
Himonas and Misiolek (CPDE, 1998) proved that the problem
is locally well-posed in $H^{s}([0,2\pi))$ with $s\geq {1}/{2}$
for small initial data. We introduce a new function
space $Z^{s}$ to show that, when $s<{1}/{2}$, the bilinear estimate
in $W^{s}$ is invalid. We also demonstrate that the bilinear estimate
in $Z^{s}$ is indeed valid for ${1}/{6}<s<{1}/{2}$. This enables us to
prove that the problem is locally well-posed in $H^{s}(\mathbf{T})$
with ${1}/{6}<s<{1}/{2}$ for small initial data. Abstract

In this talk we discuss the wellpossedness of  the following shallow water type equation
\begin{align*}
 u_{t}+\partial_{x}^{3}u
      + \frac{1}{2}\partial_{x}(u^{2})
      + \partial_{x}(1-\partial_{x}^{2})^{-1}\left[u^{2}
      + \frac{1}{2} u_{x}^{2}\right]=0,
        \quad x\in {\mathbf T}=\R/2\pi \lambda
\end{align*}
By applying the bilinear estimate in the space $W^{s}$,
Himonas and Misiolek (CPDE, 1998) proved that the problem
is locally well-posed in $H^{s}([0,2\pi))$ with $s\geq {1}/{2}$
for small initial data. We introduce a new function
space $Z^{s}$ to show that, when $s<{1}/{2}$, the bilinear estimate
in $W^{s}$ is invalid. We also demonstrate that the bilinear estimate
in $Z^{s}$ is indeed valid for ${1}/{6}<s<{1}/{2}$. This enables us to
prove that the problem is locally well-posed in $H^{s}(\mathbf{T})$
with ${1}/{6}<s<{1}/{2}$ for small initial data. AbstractIn this talk we discuss the wellpossedness of  the following shallow water type equation  \begin{align*}

u_{t}+\partial_{x}^{3}u+\frac{1}{2}\partial_{x}(u^{2})+ \partial_{x}(1-\partial_{x}^{2})^{-1}\left[u^{2}+ \frac{1}{2} u_{x}^{2}\right]=0,        \quad x\in {\mathbf T}=\R/2\pi \lambda\end{align*} By applying the bilinear estimate in the space $W^{s}$, Himonas and Misiolek (CPDE, 1998) proved that the problemis locally well-posed in $H^{s}([0,2\pi))$ with $s\geq {1}/{2}$ for small initial data. We introduce a new function space $Z^{s}$ to show that, when $s<{1}/{2}$, the bilinear estimate in $W^{s}$ is invalid. We also demonstrate that the bilinear estimate in $Z^{s}$ is indeed valid for ${1}/{6}<s<{1}/{2}$. This enables us to prove that the problem is locally well-posed in $H^{s}(\mathbf{T})$ with ${1}/{6}<s<{1}/{2}$ for small initial data.

Dirichlet heat kernel estimates of Subordinate Brownian Motions
Renming Song

University of Illinois

Abstract:  A subordinate Brownian motion can be obtained by replacing the time parameter of a Brownian motion by an independent increasing Levy process(i. e., a subordinator). Subordinate Brownian motions form a large subclass of Levy processes and  they are very important in various applications. The generator of of a subordinate Brownian motion is a function of the Laplacian. In this talk, I will give a survey of some of the recent results in the study of the potential theory of subordinate Brownian motions. In particular, I will present recent results on sharp two-sided estimates on the transition densities of killed subordinate Brownian motions in smooth open sets, or equivalently, sharp two-sided estimates on the Dirichlet heat kernels of the generators of subordinate Brownian motions.

Characterising path-independence of Girsanov transform

for stochastic differential equations

Jiang-lun Wu(吴奖伦)

Swansea University

This talk will address a new link from stochastic differential equations (SDEs) to nonlinear parabolic PDEs. Starting from the necessary and sufficient condition of the path-independence of the density of Girsanov transform for SDEs, we derive characterisation by nonlinear parabolic equations of Burgers-KPZ type. Extensions to the case of SDEs on differential manifolds and the case od SDEs with jumps as well as to that of (infinite dimensional) SDEs on separable Hilbert spaces will be discussed. A perspective to stochastically deformed dynamical systems will be briefly considered.

Tempered backward behavior for non-autonomous incompressible Navier-Stokes equation

杨新光

河南师范大学

ABSTRACT: This paper concerns the long-time backward dynamics of two-dimensional incompressible Navier-Stokes equations. Under the presence of a time-dependent external force f(t), we establish the existence of a minimal/fixed/unique compact backward attractors of finite fractal dimension for the corresponding non-autonomous dynamical system, with respect to different universe of tempered sets.

The title：2D Stochastic Chemotaxis-Navier-Stokes System
>
>Abstract：We establish the existence and uniqueness of both mild(/variational) solutions and weak (in the sense of PDE) solutions of coupled system of 2D stochastic Chemotaxis-Navier-Stokes equations. The mild/variational solution is obtained through a fixed point argument in a purposely constructed Banach space. To get the weak solution we  first prove the existence of a martingale weak solution and then we show that the pathwise uniqueness holds for the martingale solution.The title：2D Stochastic Chemotaxis-Navier-Stokes System
>
>Abstract：We establish the existence and uniqueness of both mild(/variational) solutions and weak (in the sense of PDE) solutions of coupled system of 2D stochastic Chemotaxis-Navier-Stokes equations. The mild/variational solution is obtained through a fixed point argument in a purposely constructed Banach space. To get the weak solution we  first prove the existence of a martingale weak solution and then we show that the pathwise uniqueness holds for the martingale solution.2D Stochastic Chemotaxis-Navier-Stokes System

翟建梁

中国科技大学

Abstract：We establish the existence and uniqueness of both mild(/variational) solutions and weak (in the sense of PDE) solutions of coupled system of 2D stochastic Chemotaxis-Navier-Stokes equations. The mild/variational solution is obtained through a fixed point argument in a purposely constructed Banach space. To get the weak solution we  first prove the existence of a martingale weak solution and then we show that the pathwise uniqueness holds for the martingale solution

On global existence of solutions of stochastic reaction equations

张土生

中国科技大学 and University of Manchester

In this talk I will present some recent  results on the global existence of stochastic reaction equations with superlinear drifts, driven by multiplicative space-time noise

Ergodicity of stochastic differential equations with jumps and
singular coefficients
张希承

武汉大学

摘要：We show the strong well-posedness of SDEs driven by general
multiplicative L\'evy noises with Sobolev diffusion and jump coefficients and integrable drift. Moreover, we also study
the strong Feller property, irreducibility as well as the exponential ergodicity of the corresponding semigroup when the
coefficients are time-independent and singular dissipative.
In particular, the large jump is allowed in the equation. To achieve
our main results, we present a general approach
for treating the SDEs with jumps and singular coefficients so that one
just needs to focus on Krylov's {\it apriori} estimates for SDEs

Asymptotic behavior of 3-D stochastic primitive equations of large-scale moist atmosphere with additive noise

周国立

重庆大学、zhouguoli736@126.com

Abstract: Using a new and general method, we prove the existence of random attractor for the three dimensional stochastic primitive equations defined on a  manifold $\D\subset\R^3$ improving the existence of weak attractor for the deterministic model. Furthermore, we show the existence of the invariant measure.

Finite element methods for time-fractional stochastic diffusion and diffusion-wave equations

Guang-an Zou

School of Mathematics and Statistics, Henan University

Abstract: This talk is concerned with the stochastic fractional diffusion and diffusion-wave equations driven by multiplicative noise. We prove the existence, uniqueness and the pathwise spatial-temporal regularity properties of mild solutions to these types of fractional SPDEs in a semigroup framework. A standard linear finite element approximation is used in space as well as a spatial-temporal discretization which is achieved by finite difference method in time direction. We prove the strong convergence error estimates for both semidiscrete and fully discrete schemes. Finally, numerical examples are presented to verify the theoretical results.