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Factorizations and estimates of Dirichlet heat kernels for non-local operators with critical killings
Factorizations and estimates of Dirichlet heat kernels for non-local operators with critical killings
教师介绍

本讲教师:Renming Song
所属学科:工科
人  气:476

课程介绍
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 $kappa(x)=Cdelta_D(x)^{-alpha}$ 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 $pin [alpha-1, alpha)$ 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 ${mathbf R}^dsetminus {0}$ 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.

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