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35
Twosided heat kernel estimates for censored stablelike processes
, 2008
"... In this paper we study the precise behavior of the transition density functions of censored (resurrected) αstablelike processes in C 1,1 open sets in R d, where d ≥ 1 and α ∈ (1, 2). We first show that the semigroup of the censored αstablelike process in any bounded Lipschitz open set is intrins ..."
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Cited by 25 (17 self)
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In this paper we study the precise behavior of the transition density functions of censored (resurrected) αstablelike processes in C 1,1 open sets in R d, where d ≥ 1 and α ∈ (1, 2). We first show that the semigroup of the censored αstablelike process in any bounded Lipschitz open set is intrinsically ultracontractive. We then establish sharp twosided estimates for the transition density functions of a large class of censored αstablelike processes in C 1,1 open sets. We further obtain sharp twosided estimates for the Green functions of these censored αstablelike processes in 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 19 (13 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.
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.
Heat kernel estimates for the fractional Laplacian
, 2009
"... We give sharp estimates for the heat kernel of the fractional Laplacian with Dirichlet condition for a general class of domains including Lipschitz domains. ..."
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Cited by 16 (1 self)
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We give sharp estimates for the heat kernel of the fractional Laplacian with Dirichlet condition for a general class of domains including Lipschitz domains.
Potential theory of subordinate Brownian motions revisited’, Stochastic analysis and applications to finance–essays
 in honour of Jiaan Yan, (eds
, 2012
"... The paper discusses and surveys some aspects of the potential theory of subordinate Brownian motion under the assumption that the Laplace exponent of the corresponding subordinator is comparable to a regularly varying function at infinity. This extends some results previously obtained under stronger ..."
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Cited by 14 (11 self)
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The paper discusses and surveys some aspects of the potential theory of subordinate Brownian motion under the assumption that the Laplace exponent of the corresponding subordinator is comparable to a regularly varying function at infinity. This extends some results previously obtained under stronger conditions.
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
On the potential theory of onedimensional subordinate Brownian motions with continuous components
, 2008
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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 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.