Intrinsic ultracontractivity for non-symmetric Lévy processes
Panki Kim, Renming Song · Forum Mathematicum · 2009
Recently in [Kim and Song, Intrinsic ultracontractivity of non-symmetric diffusion semigroups in bounded domains: 2006, Kim and Song, Intrinsic ultracontractivity of non-symmetric diffusion with measure-valued drifts and potentials: 2006], we extended the concept of intrinsic ultracontractivity to non-symmetric semigroups and proved that for a large class of non-symmetric diffusions Z with measure-valued drift and potential, the semigroup of Z D (the process obtained by killing Z upon exiting D ) in a bounded domain is intrinsic ultracontractive under very mild assumptions. In this paper, we study the intrinsic ultracontractivity for non-symmetric discontinuous Lévy processes. We prove that, for a large class of non-symmetric discontinuous Lévy processes X such that the Lebesgue measure is absolutely continuous with respect to the Lévy measure of X , the semigroup of X D in any bounded open set D is intrinsic ultracontractive. In particular, for the non-symmetric stable process X discussed in [Vondraček, Glas. Mat. Ser. 37: 211–233, 2002], the semigroup of X D is intrinsic ultracontractive for any bounded set D . Using the intrinsic ultracontractivity, we show that the parabolic boundary Harnack principle is true for those processes. Moreover, we get that the supremum of the expected conditional lifetimes in a bounded open set is finite. We also have results of the same nature when the Lévy measure is compactly supported.