Statistics and Data Science Seminar
Renming Song
UIUC
Factorizations and estimates of Dirichlet heat kernels for non-local operators with critical killings
Abstract: 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.
Wednesday November 14, 2018 at 4:00 PM in 636 SEO