### 狄氏型及随机分析学术研讨会

2018-05-11 ~ 2018-05-13 798

### 会议日程安排

 5月11日（星期五） 10:00—23:00 报到—三亚清华数学论坛 5月12日（星期六） 08:00-08:30 开幕式 主持人：向开南 08:30-08:55 张希承 Strong Uniqueness of Degenerate SDEs with Sobolev Diffusion   coefficients 08:55-09:20 刘伟 Quasi-Linear SPDE with Time-Fractional Derivatives 09:20-09:45 闫理坦 Cauchy principal values of some integral functionals driven by a   fractional Brownian motion 09:45-10:10 茶歇 主持人：张希承 10:10-10:30 翟建梁 Well-posed and large deviations for 2-D Stochastic   Navier-Stokes equations with jumps 10:30-10:50 周国立 Random   attractor for the 2D stochastic nematic liquid crystals flows with multiplicative   noise 10:50-11:10 巫静 On   Approximations of the Euler-Peano scheme for Multivalued stochastic   differential equations 11:10-11:30 武伟娜 Stochastic porous media equation on general measure spaces   with increasing Lipschitz nonlinearities 11:30-11:50 黎怀谦 Large-time   behaviors of the heat equation under the generalized curvature-dimension   inequality 12:00-14:30 午餐休息 主持人：刘伟 14:30-14:55 李应求 Application of Poisson approach in occupation times 14:55-15:20 胡泽春 随机变量的收敛性和大数定律 15：20-15:45 向开南 The first order correction to harmonic measure for random walks   of rotationally invariant step distribution 15:45-16:10 宋永生 Properties of martingales under sublinear expectations 16:10-16:30 茶歇 主持人：胡泽春 16:30-16:50 毕秀春 Continuous-Time   Portfolio and Consumption Decisions under Loss Aversion 16:50-17:10 刘党政 Finite   rank perturbations in products of coupled random matrices: From one   correlated to two Wishart ensembles 17:10-17:30 李利平 On   stiff problems via Dirichlet forms 17:30-17:50 马春华 Long-term   behaviors of stochastic flows of continuous-state branching processes and   extremal processes 18:00-19:00 晚餐 5月13日（星期日） 主持人：闫理坦 08:00-08:25 金鹏 Uniqueness in law for stable-like processes of variable   order 08:25-08:50 罗德军 White noise solution to 2D stochastic Euler equations 08:50-09:15 邵井海 Heavy tail and light tail of Cox-Ingersoll-Ross processes   with regime-switching 09:15-09:40 席福宝 Stabilization of regime-switching processes by feedback   control based on discrete time observations II: state-dependent case 09:40-10:00 茶歇 主持人：金鹏 10:00-10:20 张振中 Ergodicity for SDEs Driven by α-Stable  processes with  Markov switching 10:20-10:40 陈昕 Functional inequalities on loop space over a general   non-compact manifold 10:40-11:00 胡二彦 Dirichlet heat kernel estimates for a vector of multiple   independent one-dimensional stable processes 11:00-11:20 邓昌松 Exact Asymptotic Formulas for the Heat Kernels of Space   and Time-Fractional Equations 11:20-11:40 马丽 一类随机泛函微分方程的Em逼近 11:30-15:00 午餐休息

### 报告摘要

Strong Uniqueness of Degenerate SDEs with Sobolev Diffusion coefficients

AbstractWe prove a new strong uniqueness result and a weak existence result for possibly degenerate multidimensional stochastic differential equations with Sobolev diffusion coefficients and rough drifts. Examples with Holder diffusion coefficients are also provided to show our results.

Quasi-Linear SPDE with Time-Fractional Derivatives

Abstract: In this talk we present a method to solve (stochastic) evolution equations on Gelfand triples with time-fractional derivative based on monotonicity techniques. Applications include deterministic and stochastic quasi-linear partial differential equations with time-fractional derivatives, including time-fractional (stochastic) porous media equations (including the case where the Laplace operator is also fractional) and p-Laplace equations as special cases. This talk is mainly based on the joint work with Michael Roeckner and Jose Luis da Silva.

Cauchy principal values of some integral functionals driven by a fractional Brownian motion

AbstractIn this talk, we consider the Cauchy principal values of some integral functionals driven by a fractional Brownian motion with arbitrary Hurst index. Some calculus and limit theorems associated with these integral functionals are introduced.

Well-posed and large deviations for 2-D Stochastic Navier-Stokes equations with jumps

Abstract: Under the classical Lipschitz and linear growth assumptions, we established the existence and uniqueness of strong (in probability sense and PDE sense) solutions for 2-D Stochastic Navier-Stokes equations driven by multiplicative Lévy noise. And we established Wentzell-Freidlin type large deviation principles for the solutions.

Random attractor for the 2D stochastic nematic liquid crystals flows with multiplicative noise

Abstract: Under non-periodic boundary conditions, we consider the long-time behavior for stochastic 2D nematic liquid crystals flows with velocity and orientations perturbed by additive noise and multiplicative noise respectively.

The presence of the noises destroys the basic balance law of the  nematic liquid crystals flows, so we can not follow the standard argument to obtain uniform a priori estimates for the stochastic flow  under Dirichlet boundary condition and Numann boundary condition for velocity field and orientation field respectively. To overcome the difficulty our idea is to use some kind of logarithmic energy estimates and Ito formula in some Banach space to obtain the uniform estimates which improve the previous result for the orientation field that grows exponentially w.r.t.time t. In order to study the existence of random attractor, we need to show the solution is a stochastic flow.  But this is not obvious because of the emergence of this kind of multiplicative noise in the orientation field. We give a short proof which is highly non-trivial to show the flow property of the orientation field.  Our idea is to construct several linear stochastic partial differential equations whose scalar valued solutions are stochastic flow, then by discussing the relationship between these scalar  equations and orientation field equation we prove that the each component of orientation field is indeed a stochastic flow  .  Since the global well-posedness is only established for the weak solution, to consider the existence of random attractor, the common method is to derive  uniform a priori estimates in functional space which is more regular than the weak solution space. However, the common method fails because of the ill-posedness of the strong solution. Here, our idea is that by proving the compactness property of the stochastic flow and regularity of the solutions we construct a compact absorbing ball in the weak solution space which implies the existence of the random attractor.  It is\ $\mathbf{the}$\ $\mathbf{first}$ $\mathbf{result}$ for the long-time behavior of stochastic nematic liquid crystals under Dirichlet boundary condition for velocity field and Neumann boundary condition for orientation field.

On Approximations of the Euler-Peano scheme for Multivalued stochastic differential equations

Abstract: In this work we apply the Euler-Peano scheme to show that existence of weak solution and pathwise uniqueness still hold for multivalued stochastic differential equations when the coefficients are random and satisfy one-sided locally Lipschitz continuous and an integral condition. When the coefficients are nonrandom and possibly discontinuous, the sequence of solutions of the Euler-Peano scheme converges weakly, and the limit is a weak solution of the corresponding MSDE. We also obtain  a global semi-flow for stochastic differential equations reflected in closed, convex domains.

Stochastic porous media equation on general measure spaces with increasing Lipschitz nonlinearities

Abstract: We prove the existence and uniqueness of probabilistically strong solutions to stochastic porous media equations driven by time-dependent multiplicative noise on general measure spaces, and the Laplacian replaced by a negative definite self-adjoint operator L. In the case of Lipschitz nonlinearities $\Psi$, we in particular generalize previous results for open domain in $\mathbb{R}^d$ and L=Laplacian to fractional Laplacians. We also generalize known results on general measure spaces, where we succeeded in dropping the transience assumption on L, in extending the set of allowed initial data and in avoiding the restriction to superlinear behavior of $\Psi$ at infinity for $L^2$-initial data.

Large-time behaviors of the heat equation under the generalized curvature-dimension inequality

Abstract: We mainly talk about the large-time behaviors of the heat kernel and the solution to the heat equation under the generalized curvature-dimension inequality introduced by Baudoin–Garofalo. For the later, the moment bound type of estimates for the nonnegative solution to the heat equation in the spirit of J. Nash are established. Also, the mean value property of bounded harmonic functions at infinity is studied. All the results parallel or partially generalize the main ones in the particular context of Riemannian manifold with nonnegative Ricci curvature.

Application of Poisson approach in occupation times

Abstract: We adopt the Poisson approach of Li and Zhou(2014) to consider the joint Laplace transform of occupation times for SNLP and diffusion processes. We discuss the occupation times related first exiting time and last exiting time, and obtain some results: 1)Occupation times over intervals (a,r)$and (r,b) before it first exits from either a or b. 2) Potential measures that are discounted by joint occupation times over semi-infinite intervals$(-\infty, a)$and$(a, +\infty)$. 3)Joint Laplace transforms involving the last exit time(from a semi-infinite interval), the value of the process at the last exit time and associate occupation time. 随机变量的收敛性和大数定律 胡泽春 四川大学 Abstract本报告将基于以下2篇论文介绍我们在随机变量的收敛性和大数定律方面的几点工作： 1.Jiyanglin Li, Ze-Chun Hu: Toeplitz lemma, complete convergence and complete moment convergence, Communication in Statistics - Theory and Methods, 46(4), 1731-1743 (2017). 2.Ze-Chun Hu, Xue Peng, Wei Sun: Convergence of random variables and the law of large numbers, In preparation. The first order correction to harmonic measure for random walks of rotationally invariant step distribution 向开南 南开大学 AbstractIn this talk, I will describe a progress on a universality conjecture on the first order correction to Brownian harmonic measure for random walks with bounded jumps. The talk is based on a joint work with Wang Longmin（王龙敏）and Zou Lang（邹浪）. Properties of martingales under sublinear expectations 宋永生 中国科学院 AbstractIn this talk, I shall give an introduction to the structure and properties of martingales under sublinear expectations. Continuous-Time Portfolio and Consumption Decisions under Loss Aversion 毕秀春 中国科学技术大学 Abstract: We investigate continuous-time optimal portfolio and consumption problems under loss aversion in an infinite horizon. The investor's goal is to choose optimal portfolio and consumption policies to maximize total discounted S-shaped utility from consumption. The optimal consumption and portfolio policies are obtained through the martingale method and replication technique. Numerical results indicate the differences between the loss averse investor and the constant relative risk averse (CRRA) investor on the optimal consumption and portfolio policies: the loss averse investor likes consuming more money but exposing less to risk than that of the CRRA investor, and the optimal wealth, as a function of state price density, drops faster for the CRRA investor than that for the loss averse investor. Finite rank perturbations in products of coupled random matrices: From one correlated to two Wishart ensembles 刘党政 中国科学技术大学 Abstract: We compare finite rank perturbations of the following three ensembles of complex rectangular random matrices: First, a generalised Wishart ensemble with one random and two fixed correlation matrices introduced by Borodin and Peche, second, the product of two independent random matrices where one has correlated entries, and third, the case when the two random matrices become also coupled through a fixed matrix. The singular value statistics of all three ensembles is shown to be determinantal and we derive double contour integral representations for their respective kernels. Three different kernels are found in the limit of infinite matrix dimension at the origin of the spectrum. They depend on finite rank perturbations of the correlation and coupling matrices and are shown to be integrable. The first kernel (I) is found for two independent matrices from the second, and two weakly coupled matrices from the third ensemble. It generalises the Meijer G-kernel for two independent and uncorrelated matrices. The third kernel (III) is obtained for the generalised Wishart ensemble and for two strongly coupled matrices. It further generalises the perturbed Bessel kernel of Desrosiers and Forrester. Finally, kernel (II), found for the ensemble of two coupled matrices, provides an interpolation between the kernels (I) and (III), generalising previous findings of part of the authors. This is based on joint work with G. Akemann, T. Checinski and E. Strahov On stiff problems via Dirichlet forms 李利平 中国科学院 Abstract: The stiff problem is concerned with the thermal conduction model with a very small barrier, which is treated as a singular material of zero volume. In this talk, we shall build a phase transition for the stiff problem in one-dimensional space and that related to the Walsh's Brownian motion. It turns out that the phase transition fairly depends on the total thermal resistance of the barrier, and the three phases corresponds to the so-called impermeable pattern, semi-permeable pattern and permeable pattern of thermal conduction respectively. For each pattern, the related boundary condition at the barrier of the flux will be also derived. Mathematically, we shall introduce and explore the so-called snapping out Markov process, which is the probabilistic counterpart of semi-permeable pattern for the stiff problem. We refer to arXiv: 1804.02634 for this recent work. Long-term behaviors of stochastic flows of continuous-state branching processes and extremal processes 马春华 南开大学 Abstract: A class of stochastic flows of continuous state branching process were constructed by Legall and Bertoin (2000) and Dawson and Li (2012) through subordinators and SDEs respectively. In this work, we study the long-term behavior of flows of continuous-state branching processes through extremal processes, which arise in the case of supercritical processes with infinite mean and of subcritical processes with infinite variation. The jumps of these extremal processes are interpreted as specific initial individuals whose progenies overwhelm the population. These individuals, which correspond to the records of a certain Poisson point process embedded in the flow, are called super-individuals. They radically increase the growth rate to +∞ in the supercritical case, and slow down the rate of extinction in the subcritical one. Uniqueness in law for stable-like processes of variable order 金鹏 University of Wuppertal White noise solution to 2D stochastic Euler equations 罗德军 中国科学院 Abstract: We consider the weak vorticity formulation of the 2D Euler equations perturbed by transport type multiplicative noises. It is known that the white noise measure is the weak limit of point vortices when the number of vortex points goes to infinity. Based on this fact, we show that, under suitable conditions, the 2D stochastic Euler equation has a white noise solution, namely, the law of the solution is absolutely continuous with respect to the white noise measure at any time. Moreover, the density function satisfies a gradient estimate and the Fokker-Planck equation. This is a joint work with Franco Flandoli. Heavy tail and light tail of Cox-Ingersoll-Ross processes with regime-switching 邵井海 天津大学 Abstract: In this talk, we shall introduce some results on the tail behavior of the Cox-Ingersoll-Ross (CIR) processes with regime-switching. One essential difference between CIR process with regime-switching and without regime-switching is: the stationary distribution for CIR process with regime-switching could be heavy-tailed, however, without switching, the corresponding stationary distribution must be light-tailed. Our results provide a theoretical evidence of the existence of regime-switching for interest rates model based on its heavy-tailed empirical evidence. In this work, we first provide sharp criteria to justify the existence of stationary distribution for the CIR process with regime-switching, which is applied to study the long term returns of interest rates. Then under the existence of the stationary distribution, we provide a criterion to justify whether its stationary distribution is heavy-tailed or not. Stabilization of regime-switching processes by feedback control based on discrete time observations II: state-dependent case 席福宝 北京理工大学 Abstract: This work investigates the almost sure stabilization of a class of regime-switching systems based on discrete-time observations of both continuous and discrete components. It develops Shao's work [SIAM J. Control Optim., 55(2017), pp. 724--740] in two aspects: first, to provide sufficient conditions for almost sure stability in lieu of moment stability; second, to investigate a class of state-dependent regime-switching processes instead of state-independent ones. To realize these developments, we establish an estimation of the exponential functional of Markov chains based on the spectral theory of linear operator. Moreover, through constructing suitable coupling processes based on Skorokhod's representation of jumping process, we realize the control from up and below of the evolution of state-dependent switching process by state-independent Markov chains. In addition, we also append an explicit construction of the general processes of regime-switching systems based on discrete-time observations. (Joint work with Jinghai Shao.) Ergodicity for SDEs Driven byα-Stable processes with Markov switching 张振中 东华大学 Abstract: In this talk , we consider the ergodicity for stochastic differential equations driven by symmetric α-stable processes with Markovian switching inWasserstein distances. Some sufficient conditions for the exponential ergodicity are presented by using the theory of M-matrix, coupling method and Lyapunov function method. As applications, the Ornstein-Uhlenbeck type process and some other processes driven by symmetric α-stable processes with Markovian switching are presented to illustrate our results. In addition, under some conditions, an explicit expression of the invariant measure for Ornstein-Uhlenbeck process is given. Functional inequalities on loop space over a general non-compact manifold 陈昕 上海交通大学 Abstract: We will prove several asymptotic gradient estimates for heat kernel on a general non-compact manifold. Based on these estimates, we could construct an O-U Dirichlet form on loop space over a general non-compact manifold which is only complete and stochastically complete. Moreover, a local log-Sobolev inequality and a log-Sobolev inequality with potential term will also be established. This talk is based on a joint work with Xue-Mei Li and Bo Wu. Dirichlet heat kernel estimates for a vector of multiple independent one-dimensional stable processes 胡二彦 天津大学 AbstractConsider, in d-dimensional Euclidean space, a vector$X$of d independent one-dimensional symmetric stable processes of order$\alpha$. We are concerned with two-sided Dirichlet heat kernel estimates of$X$on$C^{1,1}$domains in$\mathbb{R}^d\$. Our results are sharp in some domains satisfying certain conditions. This is a joint work with Professor Zhen-Qing Chen and Guohuan Zhao.

Exact Asymptotic Formulas for the Heat Kernels of Space and Time-Fractional Equations

Abstract: We study the asymptotic behaviour of the fundamental solutions (heat kernels) of non-local (partial and pseudo differential) equations with fractional operators in time and space. In particular, we obtain exact asymptotic formulas for the heat kernels of time-changed Brownian motions and Cauchy processes. This talk is based on a joint work with Prof. Rene Schilling (TU Dresden, Germany).

Abstract:我们将介绍一类带Levy跳的中立随机微分方程的Euler近似解。利用Gronwall不等式、Holder不等式及BDG不等式，在局部Lipschitz条件和线性增长下，证明了近似解在均方意义下收敛于真实解。还将介绍一类带有限延迟的随机泛函微分方程的EM逼近，定义了该方程的带随机步长的EM算法，得到了随机步长的两个特点：有限个步长求和是停时；可列无限多个步长求和是发散的。最终由离散形式的非负半鞅收敛定理得到了在系数满足局部Lipschitz条件和单调条件下，带随机步长的EM数值解几乎处处收敛到零。

### 参会人员

 姓名 职称 E-mail 单位 毕秀春 副研究员 中国科技大学 陈传钟 教授 chk2006@hainnu.edu.cn 海南师范大学 陈昕 特别研究员 Chenxin217@sjtu.edu.cn 上海交通大学 陈晔 讲师 chenyexfw@163.com 湖南文理学院 陈勇 副教授 zhishi@pku.org.cn 湖南科技大学 邓昌松 副教授 dengcs@whu.edu.cn 武汉大学 董昭 研究员 zdong@amt.ac.cn 中国科学院 高武军 讲师 gaowj@sustc.edu.cn 南方科技大学 韩新方 副教授 xfanghan@163.com 海南师范大学 郝子墨 研究生 武汉大学 胡二彦 讲师 eryan.hu@tju.edu.cn 天津大学 胡泽春 教授 zchu@scu.edu.cn 四川大学 金鹏 讲师 pjin1982@googlemail.com University of Wuppertal 黎怀谦 副教授 huaiqianlee@gmail.com 天津大学应用数学中心 李金芸 实验员 lijinyun1991@163.com 海南师范大学 李利平 助理研究员 liliping@amss.ac.cn 中国科学院 李霓 副教授 nl_hainnu@163.com 海南师范大学 李彤 研究生 tongli0510@126.com 中山大学 李英 讲师 527651108@qq.com 湘潭大学 李应求 教授 liyq-2001@163.com 长沙理工大学 廖波 教授 dragonbw@163.com 海南师范大学 刘党政 副教授 dzliu@ustc.edu.cn 中国科学技术大学 刘伟 教授 weiliu@jsnu.edu.cn 江苏师范大学 罗德军 副研究员 luodj@amss.ac.cn 中国科学院 吕广迎 副教授 gylvmaths@126.com 河南大学 马春华 副教授 mach@nankai.edu.cn 南开大学 马丽 副教授 malihnsd@163.com 海南师范大学 马志明 院士 mazm@amt.ac.cn 中国科学院 彭雪 讲师 pengxuemath@scu.edu.cn 四川大学 任永 教授 renyong@126.com 安徽师范大学 邵井海 教授 shaojh@bnu.edu.cn 天津大学 申广君 教授 gjshen@163.com 安徽师范大学 宋将 研究生 360048910@qq.com 海南师范大学 宋永生 副研究员 yssong@amss.ac.cn 中国科学院数学数学与系统科学研究院 王立飞 讲师 flywit1986@163.com 河北师范大学 王荔丹 讲师 lidanw@nankai.edu.cn 南开大学 王珍 研究生 武汉大学 韦东 研究生 coconall@qq.com 海南师范大学 魏茸 研究生 Wrbeauty@163.com 海南师范大学 温馨 研究生 244172482@qq.com 中山大学 巫静 副教授 wjjosie@hotmail.com 中山大学 吴明燕 研究生 武汉大学 武伟娜 讲师 wuweinaforever@163.com 南京财经大学 席福宝 教授 xifb@bit.edu.cn 北京理工大学 夏鹏程 研究生 武汉大学 向开南 教授 kainanxiang@nankai.edu.cn 南开大学 闫理坦 教授 东华大学 杨明 研究生 18289640711@163.com 海南师范大学 杨赛赛 博士 中国科技大学 翟建梁 副教授 Zhaijl@ustc.edu.cn 中国科学技术大学 张 登 副教授 dzhang@sjtu.edu.cn 上海交通大学 张希承 教授 XichengZhang@gmail.com 武汉大学 张裕华 研究生 505857044@qq.com 海南师范大学 张振中 副教授 zzzhang@dhu.edu.cn 东华大学 赵国焕 博士后 zhaoguohuan@gmail.com 中国科学院 周国立 副教授 zhouguoli736@126.com 重庆大学 祖力 副教授 zulihsd2014@163.com 海南师范大学

（点击下载）