SEMINARS
Boundary theory of subordinate killed Levy processes

2018-01-10　16:00 — 17:00

1106, Math Building

Panki Kim

Abstract: Let $Z$ be a subordinate Brownian motion in $\mathbf{R}^d$, $d\ge 3$, via a subordinator with Laplace exponent $\phi$.  We kill the process $Z$ upon exiting a bounded open set $D\subset \R^d$ to obtain the killed process $Z^D$, and then we subordinate the process $Z^D$ by a subordinator with Laplace exponent $\psi$. The resulting process is denoted by $Y^D$. Both $\phi$ and $\psi$ are assumed to satisfy certain weak scaling conditions at infinity.

In this talk, I will present some recent results on the potential theory, in particular the boundary theory, of $Y^D$.  First, in case that $D$ is a $\kappa$-fat bounded open set, we show that the Harnack inequality holds. If, in addition, $D$ satisfies the local exterior volume condition, then we prove the Carleson estimate.  In case $D$ is a smooth open set and the lower weak scaling index of $\psi$ is strictly larger than $1/2$, we establish the boundary Harnack principle with explicit decay rate near the boundary of $D$. On the other hand, when $\psi(\lambda)=\lambda^{\gamma}$ with $\gamma\in (0,1/2]$, we show that the boundary Harnack principle near the boundary of $D$ fails for any bounded $C^{1,1}$ open set $D$. Our results give the first example where the Carleson estimate holds true, but the boundary Harnack principle does not.

We also prove a boundary Harnack principle for non-negative functions harmonic in  a smooth open set $E$ strictly contained in $D$, showing that the behavior of $Y^D$ in the interior of $D$ is determined by the composition $\psi\circ \phi$.

This talk is based a joint paper with Renming Song and Zoran Vondracek.