Potential theory of subordinate killed Brownian motion in a domain

Abstract. Subordination of a killed Brownian motion in a bounded domain D⊂ℝd via an α/2-stable subordinator gives a process Zt whose infinitesimal generator is −(−Δ|D)α/2, the fractional power of the negative Dirichlet Laplacian. In this paper we study the properties of the process Zt in a Lipschitz domain D by comparing the process with the rotationally invariant α-stable process killed upon exiting D. We show that these processes have comparable killing measures, prove the intrinsic ultracontractivity of the generator of Zt, prove the intrinsic ultracontractivity of the semigroup of Zt, and, in the case when D is a bounded C1,1 domain, obtain bounds on the Green function and the jumping kernel of Zt.

[1]  Nobuyuki Ikeda,et al.  On some relations between the harmonic measure and the Lévy measure for a certain class of Markov processes , 1962 .

[2]  Akihiko Miyake.  The subordination of Lévy system for Markov processes , 1969 .

[3]  M. Bartlett,et al.  Markov Processes and Potential Theory , 1972, The Mathematical Gazette.

[4]  H. Triebel.  Interpolation Theory, Function Spaces, Differential Operators , 1978 .

[5]  N. Bouleau.  Quelques resultats probabilistes sur la subordination au sens de Bochner , 1984 .

[6]  E. Davies,et al.  Heat kernels and spectral theory , 1989 .

[7]  T. K. Carne.  HEAT KERNELS AND SPECTRAL THEORY: (Cambridge Tracts in Mathematics 92) , 1990 .

[8]  Burgess Davis.  Intrinsic ultracontractivity and the Dirichlet Laplacian , 1991 .

[9]  R. Bañuelos.  Intrinsic ultracontractivity and eigenfunction estimates for Schrodinger operators , 1991 .

[10]  W. Hayman,et al.  Classical and modern potential theory and applications , 1994 .

[11]  R. Schilling.  On the domain of the generator of a subordinate semigroup , 1996 .

[12]  T. Kulczycki.  Properties of Green function of symmetric stable processes , 1997 .

[13]  Renming Song,et al.  Intrinsic ultracontractivity and conditional gauge for symmetric stable processes , 1997 .

[14]  T. Kulczycki.  Intrinsic ultracontractivity for symmetric stable processes , 1998 .

[15]  Intrinsic ultracontractivity, conditional lifetimes and conditional gauge for symmetric stable processes on rough domains , 1998, math/9809180.

[16]  Renming Song,et al.  Estimates on Green functions and Poisson kernels for symmetric stable processes , 1998 .

[17]  R. Schilling,et al.  Some Dirichlet spaces obtained by subordinate reflected diffusions , 1999 .

[18]  W. Farkas,et al.  Sobolev Spaces on Non Smooth Domains and Dirichlet Forms Related to Subordinate Reflecting Diffusions , 2001 .

[19]  Renming Song,et al.  HARDY INEQUALITY FOR CENSORED STABLE PROCESSES , 2003 .

[20]  乔花玲,et al.  关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解 , 2003 .