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107
An extension problem related to the fractional Laplacian
 1 DEGENERATE PARABOLIC EQUATIONS 19
, 2007
"... The operator square root of the Laplacian (−△) 1/2 can be obtained from the harmonic extension problem to the upper half space as the operator that maps the Dirichlet boundary condition to the Neumann condition. In this paper we obtain similar characterizations for general fractional powers of the L ..."
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Cited by 182 (20 self)
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The operator square root of the Laplacian (−△) 1/2 can be obtained from the harmonic extension problem to the upper half space as the operator that maps the Dirichlet boundary condition to the Neumann condition. In this paper we obtain similar characterizations for general fractional powers of the Laplacian and other integrodifferential operators. From those characterizations we derive some properties of these integrodifferential equations from purely local arguments in the extension problems. 1
Estimates on Green functions and Poisson kernels of symmetric stable processes
, 1998
"... for symmetric stable processes ..."
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 78 (26 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.
The Dirichlet problem for the fractional Laplacian: regularity up to the boundary
 J. Math. Pures Appl
"... Abstract. We study the regularity up to the boundary of solutions to the Dirichlet problem for the fractional Laplacian. We prove that if u is a solution of (−∆)su = g in Ω, u ≡ 0 in Rn\Ω, for some s ∈ (0, 1) and g ∈ L∞(Ω), then u is Cs(Rn) and u/δsΩ is Cα up to the boundary ∂Ω for some α ∈ (0, 1) ..."
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Cited by 69 (16 self)
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Abstract. We study the regularity up to the boundary of solutions to the Dirichlet problem for the fractional Laplacian. We prove that if u is a solution of (−∆)su = g in Ω, u ≡ 0 in Rn\Ω, for some s ∈ (0, 1) and g ∈ L∞(Ω), then u is Cs(Rn) and u/δsΩ is Cα up to the boundary ∂Ω for some α ∈ (0, 1), where δ(x) = dist(x, ∂Ω). For this, we develop a fractional analog of the Krylov boundary Harnack method. Moreover, under further regularity assumptions on g we obtain higher order Hölder estimates for u and u/δs. Namely, the Cβ norms of u and u/δs in the sets {x ∈ Ω: δ(x) ≥ ρ} are controlled by Cρs−β and Cρα−β, respectively. These regularity results are crucial tools in our proof of the Pohozaev identity for the fractional Laplacian [19, 20]. 1. Introduction and
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 48 (24 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.
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 45 (18 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.
Potential theory of special subordinators and subordinate killed stable processes
 J. Theoret. Probab
, 2006
"... In this paper we introduce a large class of subordinators called special subordinators and study their potential theory. Then we study the potential theory of processes obtained by subordinating a killed symmetric stable process in a bounded open set D with special subordinators. We establish a one ..."
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Cited by 42 (21 self)
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In this paper we introduce a large class of subordinators called special subordinators and study their potential theory. Then we study the potential theory of processes obtained by subordinating a killed symmetric stable process in a bounded open set D with special subordinators. We establish a onetoone correspondence between the nonnegative harmonic functions of the killed symmetric stable process and the nonnegative harmonic functions of the subordinate killed symmetric stable process. We show that nonnegative harmonic functions of the subordinate killed symmetric stable process are continuous and satisfy a Harnack inequality. We then show that, when D is a bounded κfat set, both the Martin boundary and the minimal Martin boundary of the subordinate killed symmetric stable process in D coincide with the Euclidean boundary ∂D.
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 estim ..."
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Cited by 40 (21 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.
The Cauchy Process and the Steklov Problem
"... Let X t be a Cauchy process in R , d 1. We investigate some of the fine spectral theoretic properties of the semigroup of this process killed upon leaving a domain D. We establish a connection between the semigroup of this process and a mixed boundary value problem for the Laplacian in one dimen ..."
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Cited by 40 (16 self)
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Let X t be a Cauchy process in R , d 1. We investigate some of the fine spectral theoretic properties of the semigroup of this process killed upon leaving a domain D. We establish a connection between the semigroup of this process and a mixed boundary value problem for the Laplacian in one dimension higher, known as the "Mixed Steklov Problem." Using this we derive a variational characterization for the eigenvalues of the Cauchy process in D. This characterization leads to many detailed properties of the eigenvalues and eigenfunctions for the Cauchy process inspired by those for Brownian motion. Our results are new even in the simplest geometric setting of the interval (1, 1) where we obtain more precise information on the size of the second and third eigenvalues and on the geometry of their corresponding eigenfunctions. Such results, although trivial for the Laplacian, take considerable work to prove for the Cauchy processes and remain open for general symmetric #stable processes. Along the way we present other general properties of the eigenfunctions, such as real analyticity, which even though well known in the case of the Laplacian, are not rarely available for more general symmetric #stable processes. #
Uniform boundary Harnack principle and generalized triangle property
 J. Funct. Anal
, 2005
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