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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 truncated ..."
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Cited by 13 (11 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.
Estimates on Green functions and Schrödingertype equations for nonsymmetric diffusions with measurevalued drifts
 J. MATH. ANAL. APPL.
, 2007
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Boundary Harnack principle for Brownian motions with measurevalued drifts in bounded Lipschitz domains
 MATHEMATISCHE ANNALEN
, 2007
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R.: L log L condition for supercritical branching Hunt processes
 J. Theor. Probab
, 2011
"... Abstract In this paper we use the spine decomposition and martingale change of measure to establish a Kesten–Stigum L log L theorem for branching Hunt processes. This result is a generalization of the results in Asmussen and Hering (Z. Wahrscheinlichkeitstheor. ..."
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Cited by 2 (2 self)
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Abstract In this paper we use the spine decomposition and martingale change of measure to establish a Kesten–Stigum L log L theorem for branching Hunt processes. This result is a generalization of the results in Asmussen and Hering (Z. Wahrscheinlichkeitstheor.
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 symmetric αs ..."
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Cited by 1 (1 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.
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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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
Sharp Green Function Estimates for ∆ + ∆ α/2 in C 1,1 Open Sets and Their Applications
, 2009
"... We consider a family of pseudo differential operators { ∆ + a α ∆ α/2; a ∈ [0, 1]} on R d that evolves continuously from ∆ to ∆+ ∆ α/2, where d ≥ 1 and α ∈ (0, 2). It gives rise to a family of Lévy processes {X a, a ∈ [0, 1]}, where X a is the sum of a Brownian motion and an independent symmetric α ..."
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We consider a family of pseudo differential operators { ∆ + a α ∆ α/2; a ∈ [0, 1]} on R d that evolves continuously from ∆ to ∆+ ∆ α/2, where d ≥ 1 and α ∈ (0, 2). It gives rise to a family of Lévy processes {X a, a ∈ [0, 1]}, where X a is the sum of a Brownian motion and an independent symmetric αstable process with weight a. Using a recently obtained uniform boundary Harnack principle with explicit decay rate, we establish sharp bounds for the Green function of the process X a killed upon exiting a bounded C 1,1 open set D ⊂ R d. As a consequence, we identify the Martin boundary of D with respect to X a with its Euclidean boundary. Finally, sharp Green function estimates are derived for certain Lévy processes which can be obtained as perturbations of X a.