日  期:2019年5月10日

时  间:14:00

地  点:理科楼 LA106

第一个报告14:00-15:00，提问5分钟，休息10分钟。

报告题目: Factorizations and estimates of Dirichlet heat kernels for non-local operators with critical killings

摘  要: In this talk I will discuss heat kernel estimates for critical perturbations of non-local operators. To be more precise, let $X$ be the reflected $\alpha$-stable process in the closure of a smooth open set $D$, and $X^D$ the process killed upon exiting $D$. We consider potentials of the form $\kappa(x)=C\delta_D(x)^{-\alpha}$ with positive $C$ and the corresponding Feynman-Kac semigroups. Such potentials do not belong to the Kato class. We obtain sharp two-sided estimates for the heat kernel of the perturbed semigroups. The interior estimates of the heat kernels have the usual $\alpha$-stable form, while the boundary decay is of the form $\delta_D(x)^p$ with non-negative $p\in [\alpha-1, \alpha)$ depending on the precise value of the constant $C$. Our result recovers the heat kernel estimates of both the censored and the killed stable process in $D$. Analogous estimates are obtained for the heat kernel of the Feynman-Kac semigroup of the $\alpha$-stable process in ${\mathbf R}^d\setminus \{0\}$ through the potential $C|x|^{-\alpha}$.

    All estimates are derived from a more general result described as follows:Let $X$ be a Hunt process on a locally compact separable metric space in a strong duality with $\widehat{X}$. Assume that transition densities of $X$ and $\widehat{X}$  are comparable to the function $\widetilde{q}(t,x,y)$ defined in terms of the volume of balls and a certain scaling function. For an open set $D$ consider the killed process $X^D$, and a critical smooth measure on $D$ with the corresponding positive additive functional $(A_t)$.  We show that the heat kernel of the the Feynman-Kac semigroup of $X^D$ through the multiplicative functional $\exp(-A_t)$ admits the factorization of the form ${\mathbf P}_x(\zeta >t)\widehat{\mathbf P}_y(\widehat{\zeta}>t)\widetilde{q}(t,x,y)$.

报告人简介:宋仁明，美国伊利诺伊大学教授, 在Ann. Probability, Probab. Theory Related Fields , Math. Ann, J. Eur. Math. Soc. , J. Funct. Anal. ,  Proc. Lond. Math. Soc. , T Trans. Amer. Math. Soc. , Stochastic Process. Appl. 等数学和概率论Top杂志上发表论文100多篇，出版专著2部。

详见个人主页 https://faculty.math.illinois.edu/~rsong/

第二个报告 15:15-16:15，提问5分钟，休息10分钟。

报告题目: Stochastic 3D Leray-α Model with Fractional Dissipation

摘  要: In this talk we study the well-posedness of stochastic 3D Leray-α model with general fractional dissipation driven by multiplicative noise. This model is the stochastic 3D Navier Stokes equation regularized through a smoothing kernel of order θ1 in the nonlinear term and a θ2-fractional Laplacian. In the case of θ1 ≥ 0, θ2 > 0 and θ1 +θ2 ≥ 5/4, we prove the global existence and uniqueness of strong solutions. We also study the Freidlin-Wentzell large deviation principle and the ergodicity for this model with degenerate stochastic force. This is a joint work with Li Shihu and Liu Wei.

报告人简介：谢颖超，江苏师范大学教授（兼南开大学教授，博士生导师），获国家人事部、教育部：全国模范教师称号。在Stochastic Process. Appl., J. Differential Equations, Potential Analysis, Discrete and Continuous Dynamic System-A等国际一流期刊发表近百篇论著。

详见个人主页：http://maths.xznu.edu.cn/5098/list.htm