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39
Heat kernel estimates for jump processes of mixed types on metric measure spaces
 FIELDS
"... In this paper, we investigate symmetric jumptype processes on a class of metric measure spaces with jumping intensities comparable to radially symmetric functions on the spaces. The class of metric measure spaces includes the Alfors dregular sets, which is a class of fractal sets that contains ge ..."
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Cited by 51 (30 self)
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In this paper, we investigate symmetric jumptype processes on a class of metric measure spaces with jumping intensities comparable to radially symmetric functions on the spaces. The class of metric measure spaces includes the Alfors dregular sets, which is a class of fractal sets that contains geometrically selfsimilar sets. A typical example of our jumptype processes is the symmetric jump process with jumping intensity e −c0(x,y)x−y � α2 α1 c(α, x, y) ν(dα) x − y  d+α where ν is a probability measure on [α1, α2] ⊂ (0, 2), c(α, x, y) is a jointly measurable function that is symmetric in (x, y) and is bounded between two positive constants, and c0(x, y) is a
Nonlocal Dirichlet forms and symmetric jump processes
 Transactions of the American Mathematical Society
, 1999
"... We consider the symmetric nonlocal Dirichlet form (E, F) given by E(f, f) = (f(y) − f(x)) 2 J(x, y)dxdy Rd Rd with F the closure of the set of C 1 functions on R d with compact support with respect to E1, where E1(f, f): = E(f, f) + ∫ R d f(x) 2 dx, and where the jump kernel J satisfies κ1y − x ..."
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Cited by 30 (16 self)
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We consider the symmetric nonlocal Dirichlet form (E, F) given by E(f, f) = (f(y) − f(x)) 2 J(x, y)dxdy Rd Rd with F the closure of the set of C 1 functions on R d with compact support with respect to E1, where E1(f, f): = E(f, f) + ∫ R d f(x) 2 dx, and where the jump kernel J satisfies κ1y − x  −d−α ≤ J(x, y) ≤ κ2y − x  −d−β for 0 < α < β < 2, x − y  < 1. This assumption allows the corresponding jump process to have jump intensities whose size depends on the position of the process and the direction of the jump. We prove upper and lower estimates on the heat kernel. We construct a strong Markov process corresponding to (E, F). We prove a parabolic Harnack inequality for nonnegative functions that solve the heat equation with respect to E. Finally we construct an example where the corresponding harmonic functions need not be continuous.
Harnack inequality for some classes of Markov processes
"... In this paper we establish a Harnack inequality for nonnegative harmonic functions of some classes of Markov processes with jumps. ..."
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Cited by 25 (13 self)
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In this paper we establish a Harnack inequality for nonnegative harmonic functions of some classes of Markov processes with jumps.
Potential theory of truncated stable processes
 MATHEMATISCHE ZEITSCHRIFT
, 2007
"... For any α ∈ (0, 2), a truncated symmetric αstable process is a symmetric Lévy process in R d with a Lévy density given by cx  −d−α 1{x<1} for some constant c. In this paper we study the potential theory of truncated symmetric stable processes in detail. We prove a Harnack inequality for non ..."
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Cited by 22 (18 self)
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For any α ∈ (0, 2), a truncated symmetric αstable process is a symmetric Lévy process in R d with a Lévy density given by cx  −d−α 1{x<1} for some constant c. In this paper we study the potential theory of truncated symmetric stable processes in detail. We prove a Harnack inequality for nonnegative harmonic functions of these processes. We also establish a boundary Harnack principle for nonnegative functions which are harmonic with respect to these processes in bounded convex domains. We give an example of a nonconvex domain for which the boundary Harnack principle fails.
Stochastic differential equations with jumps
 TRANS. AMER. MATH. SOC
, 2004
"... This paper is a survey of uniqueness results for stochastic differential equations with jumps and regularity results for the corresponding harmonic functions. ..."
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Cited by 18 (2 self)
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This paper is a survey of uniqueness results for stochastic differential equations with jumps and regularity results for the corresponding harmonic functions.
Harnack inequality for some discontinuous Markov processes with a diffusion part
 GLASNIK MATEMATIČKI
, 2005
"... In this paper we establish a Harnack inequality for nonnegative harmonic functions of some discontinuous Markov processes with a diffusion part. ..."
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Cited by 16 (11 self)
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In this paper we establish a Harnack inequality for nonnegative harmonic functions of some discontinuous Markov processes with a diffusion part.
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.
Parabolic Harnack inequality for the mixture of Brownian motion and stable process
 TOHOKU MATH. J
"... Let X be a mixture of independent Brownian motion and symmetric stable process. In this paper we establish sharp bounds for transition density of X, and prove a parabolic Harnack inequality for nonnegative parabolic functions of X. ..."
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Cited by 13 (9 self)
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Let X be a mixture of independent Brownian motion and symmetric stable process. In this paper we establish sharp bounds for transition density of X, and prove a parabolic Harnack inequality for nonnegative parabolic functions of X.
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 posit ..."
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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.