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

报告人：Prof. Renming Song  University of Illinois at Urbana-Champaign

时间：2018年12月13日 上午9:00-10:00

地点：二教 2606

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 ${/mathbf P}_x(/zeta >t)/widehat{/mathbf P}_y(/widehat{/zeta}>t)/widetilde{q}(t,x,y)$. This is joint work with Soobin Cho, Panki Kim and Zoran Vondracek.