INTRINSIC ULTRACONTRACTIVITY, CONDITIONAL LIFETIMES AND CONDITIONAL GAUGE FOR SYMMETRIC

STABLE PROCESSES ON ROUGH DOMAINS

ZHEN-QING CHEN AND RENMING SONG

Abstract. For a symmetric α-stable process X on R^{n} with 0 < α < 2, n ≥ 2 and a domain D ⊂ 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 Hölder 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 Hölder domain of order 0 are uniformly bounded.