## Intrinsic Ultracontractivity, Conditional Lifetimes and Conditional Gauge for Symmetric Stable Processes on Rough Domains (1998)

Citations: | 15 - 6 self |

### BibTeX

@MISC{Chen98intrinsicultracontractivity,,

author = {Zhen-qing Chen and Renming Song},

title = {Intrinsic Ultracontractivity, Conditional Lifetimes and Conditional Gauge for Symmetric Stable Processes on Rough Domains},

year = {1998}

}

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### Abstract

For a symmetric ff-stable process X on R n with 0 ! ff ! 2, n 2 and a domain D ae R 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 Holder domain D of order 0. This is then used to establish the conditional gauge theorem for X on bounded Lipschitz domains in R n . It is also shown that the conditional lifetimes for symmetric stable process in a Holder domain of order 0 are uniformly bounded. Keywords and phrases: Symmetric stable processes, Feynman-Kac semigroup, conditional gauge theorem, logarithmic Sobolev inequality, intrinsic ultracontractivity. Running Title: Conditional Gauge Theorem The research of this author is supported in part by NSA Grant MDA904-98-1-0044 1 Introduction A symmetric ff-stable process X on R n is a L'evy process whose transition density p(t; x \Gamma y) relative to Lebesgue measure is uniquely determined by ...