TY - JOUR
T1 - Harmonic functions of subordinate killed Brownian motion
AU - Glover, J.
AU - Pop-Stojanovic, Z.
AU - Rao, M.
AU - Šikić, H.
AU - Song, R.
AU - Vondraček, Z.
N1 - Keywords: Killed Brownian motions; Subordination; Fractional Laplacian; Harmonic functions; Green function; Martin kernel; Martin boundary; Harnack inequality; Boundary Harnack principle; Intrinsic ultracontractivity ·Corresponding author. E-mail addresses: [email protected] (J. Glover), [email protected] (Z. Pop-Stojanovic), rao@math. ufl.edu (M. Rao), hsik [email protected] (H. Sˇ ik ić), [email protected] (R. Song), [email protected] (Z. Vondracˇ ek ). 1The research is supported in part by MZT Grant 0037118 of the Republic of Croatia and in part by a joint US-Croatia Grant INT 0302167. 2The research is supported in part by a joint US-Croatia Grant INT 0302167. 3The research is Supported in part by MZT Grant 0037107 of the Republic of Croatia and in part by a joint US-Croatia Grant INT 0302167.
PY - 2004/10/15
Y1 - 2004/10/15
N2 - 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.
AB - 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.
KW - Boundary Harnack principle
KW - Fractional Laplacian
KW - Green function
KW - Harmonic functions
KW - Harnack inequality
KW - Intrinsic ultracontractivity
KW - Killed Brownian motions
KW - Martin boundary
KW - Martin kernel
KW - Subordination
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U2 - 10.1016/j.jfa.2004.01.001
DO - 10.1016/j.jfa.2004.01.001
M3 - Article
AN - SCOPUS:4344667711
SN - 0022-1236
VL - 215
SP - 399
EP - 426
JO - Journal of Functional Analysis
JF - Journal of Functional Analysis
IS - 2
ER -