Abstract
In this paper we study harmonic functions of subordinate killed Brownian motion in a domain D. We first prove that, when the killed Brownian semigroup in D is intrinsic ultracontractive, all nonnegative harmonic functions of the subordinate killed Brownian motion in D are continuous and then we establish a Harnack inequality for these harmonic functions. We then show that, when D is a bounded Lipschitz domain, both the Martin boundary and the minimal Martin boundary of the subordinate killed Brownian motion in D coincide with the Euclidean boundary ∂D. We also show that, when D is a bounded Lipschitz domain, a boundary Harnack principle holds for positive harmonic functions of the subordinate killed Brownian motion in D.
Original language | English (US) |
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Pages (from-to) | 399-426 |
Number of pages | 28 |
Journal | Journal of Functional Analysis |
Volume | 215 |
Issue number | 2 |
DOIs | |
State | Published - Oct 15 2004 |
Keywords
- Boundary Harnack principle
- Fractional Laplacian
- Green function
- Harmonic functions
- Harnack inequality
- Intrinsic ultracontractivity
- Killed Brownian motions
- Martin boundary
- Martin kernel
- Subordination
ASJC Scopus subject areas
- Analysis