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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 31 (18 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 Relativistic Stable Processes in Halfspacelike Open Sets
, 2012
"... 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. Th ..."
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Cited by 11 (4 self)
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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
Heat kernel estimate for ∆ + ∆α/2 in C1,1 open sets
 J. London Math. Soc
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Green Function Estimates for Relativistic Stable Processes in Halfspacelike Open Sets
, 2010
"... In this paper, we establish sharp twosided estimates for the Green functions of relativistic stable processes (i.e. Green functions for nonlocal operators m − (m 2/α − ∆) α/2) in halfspacelike C 1,1 open sets. The estimates are uniform in m ∈ (0, M] for each fixed M ∈ (0, ∞). When m ↓ 0, our esti ..."
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Cited by 5 (4 self)
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In this paper, we establish sharp twosided estimates for the Green functions of relativistic stable processes (i.e. Green functions for nonlocal operators m − (m 2/α − ∆) α/2) in halfspacelike C 1,1 open sets. The estimates are uniform in m ∈ (0, M] for each fixed M ∈ (0, ∞). When m ↓ 0, our estimates reduce to the sharp Green function estimates for −(−∆) α/2 in such kind of open sets that were obtained recently in [13]. As a tool for proving our Green function estimates, we show that a boundary Harnack principle for X m, which is uniform for all m ∈ (0, ∞), holds for a large class of nonsmooth open sets.
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 (5 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.
Exact asymptotic for distribution densities of Lévy functionals
 ELECTRON. J. PROBAB
, 2011
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Perturbation by NonLocal Operators
, 2012
"... Suppose that d ≥ 1 and 0 < β < α < 2. We establish the existence and uniqueness of the fundamental solution q b (t, x, y) to a class of (possibly nonsymmetric) nonlocal operators L b = ∆ α/2 + S b, where S b ∫ f(x): = A(d, −β) R d f(x + z) − f(x) − ∇f(x) · z1{z≤1}) b(x, z) dz z d+β ..."
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Cited by 4 (3 self)
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Suppose that d ≥ 1 and 0 < β < α < 2. We establish the existence and uniqueness of the fundamental solution q b (t, x, y) to a class of (possibly nonsymmetric) nonlocal operators L b = ∆ α/2 + S b, where S b ∫ f(x): = A(d, −β) R d f(x + z) − f(x) − ∇f(x) · z1{z≤1}) b(x, z) dz z d+β and b(x, z) is a bounded measurable function on R d ×R d with b(x, z) = b(x, −z) for x, z ∈ R d. Here A(d, −β) is a normalizing constant so that S b = ∆ β /2 when b(x, z) ≡ 1. We show that if b(x, z) ≥ − A(d,−α) A(d,−β) zβ−α, then q b (t, x, y) is a strictly positive continuous function and it uniquely determines a conservative Feller process X b, which has strong Feller property. The Feller process X b is the unique solution to the martingale problem of (L b, S(R d)), where S(R d) denotes the space of tempered functions on R d. Furthermore, sharp twosided estimates on q b (t, x, y) are derived. In stark contrast with the gradient perturbations, these estimates exhibit different behaviors for different types of b(x, z). The model considered in this paper contains the following as a special case. Let Y and Z be (rotationally) symmetric αstable process and symmetric βstable processes on R d, respectively, that are independent to each other. Solution to stochastic differential equations dXt = dYt + c(Xt−)dZt has infinitesimal generator L b with b(x, z) = c(x)  β.
Dirichlet Heat Kernel Estimates for Subordinate Brownian Motions with Gaussian Components
, 1303
"... In this paper, we derive explicit sharp twosided estimates for the Dirichlet heat kernels, in C 1,1 open sets D in R d, of a large class of subordinate Brownian motions with Gaussian components. When D is bounded, our sharp twosided Dirichlet heat kernel estimates hold for all t> 0. Integrating ..."
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Cited by 3 (2 self)
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In this paper, we derive explicit sharp twosided estimates for the Dirichlet heat kernels, in C 1,1 open sets D in R d, of a large class of subordinate Brownian motions with Gaussian components. When D is bounded, our sharp twosided Dirichlet heat kernel estimates hold for all t> 0. Integrating the heat kernel estimates with respect to the time variable t, we obtain sharp twosided estimates for the Green functions, in bounded C 1,1 open sets, of such subordinate Brownian motions with Gaussian components.
HEAT KERNELS AND ANALYTICITY OF NONSYMMETRIC JUMP DIFFUSION SEMIGROUPS
"... Abstract. Let d � 1 and α ∈ (0, 2). Consider the following nonlocal and nonsymmetric Lévytype operator on Rd: L κ ∫ α f (x): = p.v. Rd κ(x, z) ( f (x + z) − f (x)) dz, ..."
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Abstract. Let d � 1 and α ∈ (0, 2). Consider the following nonlocal and nonsymmetric Lévytype operator on Rd: L κ ∫ α f (x): = p.v. Rd κ(x, z) ( f (x + z) − f (x)) dz,