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29
Boundary behavior of harmonic functions for truncated stable processes
 J. THEORET. PROBAB
, 2008
"... For any α ∈ (0, 2), a truncated symmetric αstable process in R d is a symmetric Lévy process in R d with no diffusion part and with a Lévy density given by cx  −d−α 1{x<1} for some constant c. In (Kim and Song in Math. Z. 256(1): 139–173, 2007) we have studied the potential theory of trunca ..."
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Cited by 26 (16 self)
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For any α ∈ (0, 2), a truncated symmetric αstable process in R d is a symmetric Lévy process in R d with no diffusion part and with a Lévy density given by cx  −d−α 1{x<1} for some constant c. In (Kim and Song in Math. Z. 256(1): 139–173, 2007) we have studied the potential theory of truncated symmetric stable processes. Among other things, we proved that the boundary Harnack principle is valid for the positive harmonic functions of this process in any bounded convex domain and showed that the Martin boundary of any bounded convex domain with respect to this process is the same as the Euclidean boundary. However, for truncated symmetric stable processes, the boundary Harnack principle is not valid in nonconvex domains. In this paper, we show that, for a large class of not necessarily convex bounded open sets in R d called bounded roughly connected κfat open sets (including bounded nonconvex κfat domains), the Martin boundary with respect to any truncated symmetric stable process is still the same as the Euclidean boundary. We also show that, for truncated symmetric stable processes a relative Fatou type theorem is true in bounded roughly connected κfat open sets.
Dirichlet Heat Kernel Estimates for ∆ α/2 + ∆ β/2
, 2009
"... For d ≥ 1 and 0 < β < α < 2, consider a family of pseudo differential operators { ∆ α + a β ∆ β/2; a ∈ [0, 1]} on R d that evolves continuously from ∆ α/2 to ∆ α/2 + ∆ β/2. It gives arise to a family of Lévy processes {X a, a ∈ [0, 1]} on R d, where each X a is the independent sum of a symm ..."
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Cited by 15 (9 self)
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For d ≥ 1 and 0 < β < α < 2, consider a family of pseudo differential operators { ∆ α + a β ∆ β/2; a ∈ [0, 1]} on R d that evolves continuously from ∆ α/2 to ∆ α/2 + ∆ β/2. It gives arise to a family of Lévy processes {X a, a ∈ [0, 1]} on R d, where each X a is the independent sum of a symmetric αstable process and a symmetric βstable process with weight a. For any C 1,1 open set D ⊂ R d, we establish explicit sharp twosided estimates (uniform in a ∈ [0, 1]) for the transition density function of the subprocess X a,D of X a killed upon leaving the open set D. The infinitesimal generator of X a,D is the nonlocal operator ∆ α + a β ∆ β/2 with zero exterior condition on D c. As consequences of these sharp heat kernel estimates, we obtain uniform sharp Green function estimates for X a,D and uniform boundary Harnack principle for X a in D with explicit decay rate.
Boundary Harnack principle for Brownian motions with measurevalued drifts in bounded Lipschitz domains
 MATHEMATISCHE ANNALEN
, 2007
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Estimates on Green functions and Schrödingertype equations for nonsymmetric diffusions with measurevalued drifts
 J. MATH. ANAL. APPL.
, 2007
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Central limit theorems for supercritical branching Markov processes
, 2013
"... In this paper we establish spatial central limit theorems for a large class of supercritical branching Markov processes with general spatialdependent branching mechanisms. These are generalizations of the spatial central limit theorems proved in [1] for branching OU processes with binary branching ..."
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Cited by 7 (6 self)
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In this paper we establish spatial central limit theorems for a large class of supercritical branching Markov processes with general spatialdependent branching mechanisms. These are generalizations of the spatial central limit theorems proved in [1] for branching OU processes with binary branching mechanisms. Compared with the results of [1], our central limit theorems are more satisfactory in the sense that the normal random variables in our theorems are nondegenerate.
Limit Theorems for Some Critical Superprocesses. Available on ArXiv: http://arxiv.org/abs/1403.1342
, 2014
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L log L condition for supercritical branching Hunt processes
"... In this paper we use the spine decomposition and martingale change of measure to establish a KestenStigum L log L theorem for branching Hunt processes. This result is a generalization of the results in [1] and [9] for branching diffusions. ..."
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Cited by 3 (2 self)
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In this paper we use the spine decomposition and martingale change of measure to establish a KestenStigum L log L theorem for branching Hunt processes. This result is a generalization of the results in [1] and [9] for branching diffusions.
Intrinsic Ultracontractivity on Riemannian Manifolds with Infinite Volume Measures
, 2008
"... By establishing the intrinsic superPoincaré inequality, some explicit conditions are presented for diffusion semigroups on a noncompact complete Riemannian manifold to be intrinsically ultracontractive. These conditions, as well as the resulting uniform upper bounds on the intrinsic heat kernels, ..."
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Cited by 2 (1 self)
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By establishing the intrinsic superPoincaré inequality, some explicit conditions are presented for diffusion semigroups on a noncompact complete Riemannian manifold to be intrinsically ultracontractive. These conditions, as well as the resulting uniform upper bounds on the intrinsic heat kernels, are sharp for some concrete examples. AMS subject Classification: 58G32, 60J60