Intrinsic Ultracontractivity, Conditional Lifetimes and Conditional Gauge for Symmetric Stable Processes on Rough Domains

Details Intrinsic Ultracontractivity, Conditional Lifetimes and Conditional Gauge for Symmetric Stable Processes on Rough Domains Intrinsic Ultracontractivity, Conditional Lifetimes and Conditional Gauge for Symmetric Stable Processes on Rough Domains

1998-09-29

arxiv.org/abs/math/9809180v1

Mathematics

Probability

Scientific paper

For a symmetric $\alpha$-stable process $X$ on $\RR^n$ with $0<\alpha <2$, $n\geq 2$ and a domain $D \subset \RR^n$, let $L^D$ be the infinitesimal generator of the subprocess of $X$ killed upon leaving $D$. For a Kato class function $q$, it is shown that $L^D+q$ is intrinsic ultracontractive on a H\"older domain $D$ of order 0. This is then used to establish the conditional gauge theorem for $X$ on bounded Lipschitz domains in $\RR^n$. It is also shown that the conditional lifetimes for symmetric stable process in a H\"older domain of order 0 are uniformly bounded.

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