Results 1  10
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24
Heat kernel estimates for Dirichlet fractional Laplacian
 J. European Math. Soc
"... In this paper, we consider the fractional Laplacian −(−∆) α/2 on an open subset in R d with zero exterior condition. We establish sharp twosided estimates for the heat kernel of such Dirichlet fractional Laplacian in C 1,1 open sets. This heat kernel is also the transition density of a rotationally ..."
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Cited by 35 (19 self)
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In this paper, we consider the fractional Laplacian −(−∆) α/2 on an open subset in R d with zero exterior condition. We establish sharp twosided estimates for the heat kernel of such Dirichlet fractional Laplacian in C 1,1 open sets. This heat kernel is also the transition density of a rotationally symmetric stable process killed upon leaving a C 1,1 open set. Our results are the first sharp twosided estimates for the Dirichlet heat kernel of a nonlocal operator on open sets.
Sharp heat kernel estimates for relativistic stable processes in open sets
"... In this paper, we establish sharp twosided estimates for the transition densities of relativistic stable processes (or equivalently, for the heat kernels of the operators m − (m 2/α − ∆) α/2) in C 1,1 open sets. The estimates are uniform in m ∈ (0,M] for each fixed M> 0. Letting m ↓ 0, the estimate ..."
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Cited by 17 (14 self)
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In this paper, we establish sharp twosided estimates for the transition densities of relativistic stable processes (or equivalently, for the heat kernels of the operators m − (m 2/α − ∆) α/2) in C 1,1 open sets. The estimates are uniform in m ∈ (0,M] for each fixed M> 0. Letting m ↓ 0, the estimates given in this paper recover the Dirichlet heat kernel estimates for −(−∆) α/2 in C 1,1open sets obtained in [9]. Sharp twosided estimates are also obtained for Green functions of relativistic stable processes in halfspacelike C 1,1 open sets and bounded C 1,1 open sets.
Global heat kernel estimates for fractional Laplacians in unbounded open sets
 Probab. Theory Relat. Fields, DOI 10.1007/s0044000902560 (online first
"... In this paper, we derive global sharp heat kernel estimates for symmetric αstable processes (or equivalently, for the fractional Laplacian with zero exterior condition) in two classes of unbounded C 1,1 open sets in R d: halfspacelike open sets and exterior open sets. These open sets can be disco ..."
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Cited by 17 (12 self)
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In this paper, we derive global sharp heat kernel estimates for symmetric αstable processes (or equivalently, for the fractional Laplacian with zero exterior condition) in two classes of unbounded C 1,1 open sets in R d: halfspacelike open sets and exterior open sets. These open sets can be disconnected. We focus in particular on explicit estimates for pD(t, x, y) for all t> 0 and x, y ∈ D. Our approach is based on the idea that for x and y in D far from the boundary and t sufficiently large, we can compare pD(t, x, y) to the heat kernel in a well understood open set: either a halfspace or R d; while for the general case we can reduce them to the above case by pushing x and y inside away from the boundary. As a consequence, sharp Green functions estimates are obtained for the Dirichlet fractional Laplacian in these two types of open sets. Global sharp heat kernel estimates and Green function estimates are also obtained for censored stable processes (or equivalently, for regional fractional Laplacian) in exterior open sets.
Boundary Harnack inequality for regional fractional Laplacian
"... We consider boundary Harnack inequalities for regional fractional Laplacian which are generators of censored stablelike processes on G taking κ(x, y)/x − y  n+α dxdy, x, y ∈ G, as the jumping measure. When G is a C 1,β−1 open set, 1 < α < β ≤ 2, and κ ∈ C 1 (G × G) bounded between two positive nu ..."
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Cited by 11 (0 self)
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We consider boundary Harnack inequalities for regional fractional Laplacian which are generators of censored stablelike processes on G taking κ(x, y)/x − y  n+α dxdy, x, y ∈ G, as the jumping measure. When G is a C 1,β−1 open set, 1 < α < β ≤ 2, and κ ∈ C 1 (G × G) bounded between two positive numbers, we prove a boundary Harnack inequality giving dist(x, ∂G) α−1 order decay for harmonic functions near the boundary. For a C 1,β−1 open set D ⊂ D ⊂ G, 0 < α ≤ (1 ∨ α) < β ≤ 2, we prove a boundary Harnack inequality giving dist(x, ∂D) α/2 order decay for harmonic functions near the boundary. These results are generalizations of the previous results for the homogeneous case on C 1,1 open sets. The method in this paper can be applied to study more general Markov operators. Key words fractional Laplacian, regional fractional Laplacian, symmetric αstable processes, censored stablelike processes, (super) subharmonic function, Carleson estimate, Harnack inequality, boundary Harnack inequality MR(2000) Subject Classification: Primary 60G52, Secondary 60J45,47G20 1
Symmetric jump processes and their heat kernel estimates
 Sci. China Ser. A
"... We survey the recent development of the DeGiorgiNashMoserAronson type theory for a class of symmetric jump processes (or equivalently, a class of symmetric integrodifferential operators). We focus on the sharp twosided estimates for the transition density functions (or heat kernels) of the proc ..."
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Cited by 9 (6 self)
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We survey the recent development of the DeGiorgiNashMoserAronson type theory for a class of symmetric jump processes (or equivalently, a class of symmetric integrodifferential operators). We focus on the sharp twosided estimates for the transition density functions (or heat kernels) of the processes, a priori Hölder estimate and parabolic Harnack inequalities for their parabolic functions. In contrast to the second order elliptic differential operator case, the methods to establish these properties for symmetric integrodifferential operators are mainly probabilistic.
Heat kernel estimates and Harnack inequalities for some Dirichlet forms with nonlocal part
, 2009
"... We consider the Dirichlet ..."
Heat kernel of fractional Laplacian in cones
 Colloq. Math
"... This paper is devoted to the memory of Professor Andrzej Hulanicki. We give sharp estimates for the transition density of the isotropic stable Lévy process killed when leaving a right circular cone. 1 ..."
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Cited by 5 (1 self)
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This paper is devoted to the memory of Professor Andrzej Hulanicki. We give sharp estimates for the transition density of the isotropic stable Lévy process killed when leaving a right circular cone. 1
Heat Kernel Estimate for ∆ + ∆ α/2 in C 1,1 Open Sets
, 2010
"... We consider a family of pseudo differential operators { ∆ + a α ∆ α/2; a ∈ (0, 1]} on R d for every d ≥ 1 that evolves continuously from ∆ to ∆ + ∆ α/2, where α ∈ (0, 2). It gives rise to a family of Lévy processes {X a, a ∈ (0, 1]} in R d, where X a is the sum of a Brownian motion and an independen ..."
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Cited by 5 (4 self)
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We consider a family of pseudo differential operators { ∆ + a α ∆ α/2; a ∈ (0, 1]} on R d for every d ≥ 1 that evolves continuously from ∆ to ∆ + ∆ α/2, where α ∈ (0, 2). It gives rise to a family of Lévy processes {X a, a ∈ (0, 1]} in R d, where X a is the sum of a Brownian motion and an independent symmetric αstable process with weight a. We establish sharp twosided estimates for the heat kernel of ∆ + a α ∆ α/2 with zero exterior condition in a family of open subsets, including bounded C 1,1 (possibly disconnected) open sets. This heat kernel is also the transition density of the sum of a Brownian motion and an independent symmetric αstable process with weight a in such open sets. Our result is the first sharp twosided estimates for the transition density of a Markov process with both diffusion and jump components in open sets. Moreover, our result is uniform in a in the sense that the constants in the estimates are independent of a ∈ (0, 1] so that it recovers the Dirichlet heat kernel estimates for Brownian motion by taking a → 0. Integrating the heat kernel estimates in time t, we recover the twosided sharp uniform Green function estimates of X a in bounded C 1,1 open sets in R d, which were recently established in [14] by using a completely different approach.
Global Heat Kernel Estimates for Relativistic Stable Processes in Halfspacelike Open Sets
"... Abstract In this paper, by using probabilistic methods, we establish sharp twosided large time estimates for the transition densities of relativistic αstable processes with mass m ∈ (0, 1] (i.e., for the Dirichlet heat kernels of m − (m 2/α − �) α/2 with m ∈ (0, 1]) in halfspacelike C 1,1 open s ..."
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Cited by 4 (2 self)
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Abstract In this paper, by using probabilistic methods, we establish sharp twosided large time estimates for the transition densities of relativistic αstable processes with mass m ∈ (0, 1] (i.e., for the Dirichlet heat kernels of m − (m 2/α − �) α/2 with m ∈ (0, 1]) in halfspacelike C 1,1 open sets. The estimates are uniform in m in the sense that the constants are independent of m ∈ (0, 1]. Combining with the sharp twosided small time estimates, established in Chen et al. (Ann Probab, 2011), valid for all C 1,1 open sets, we have now sharp twosided estimates for the transition densities of relativistic αstable processes with mass m ∈ (0, 1] in halfspacelike C 1,1 open sets for all times. Integrating the heat kernel estimates with respect to the time variable, one can recover the sharp twosided Green function estimates for relativistic αstable processes with mass m ∈ (0, 1] in halfspacelike C 1,1 open sets established recently
Regularity results for stablelike operators
, 2008
"... For α ∈ [1,2) we consider operators of the form Lf(x) = R d [f(x + h) − f(x) − 1 (h≤1)∇f(x) · h] A(x,h) h  d+α and for α ∈ (0,1) we consider the same operator but where the ∇f term is omitted. We prove, under appropriate conditions on A(x,h), that the solution u to Lu = f will be in C α+β if ..."
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Cited by 2 (0 self)
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For α ∈ [1,2) we consider operators of the form Lf(x) = R d [f(x + h) − f(x) − 1 (h≤1)∇f(x) · h] A(x,h) h  d+α and for α ∈ (0,1) we consider the same operator but where the ∇f term is omitted. We prove, under appropriate conditions on A(x,h), that the solution u to Lu = f will be in C α+β if f ∈ C β. Subject Classification: Primary 45K05; Secondary 35B65, 60J75 1