In this paper, we consider the fractional Laplacian −(−∆) α/2 on an open subset in R d with zero exterior condition. We establish sharp two-sided 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 two-sided estimates for the Dirichlet heat kernel of a non-local operator on open sets.

In this paper, we establish sharp two-sided 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 1,1-open sets obtained in [9]. Sharp two-sided estimates are also obtained for Green functions of relativistic stable processes in half-space-like C 1,1 open sets and bounded C 1,1 open sets.

In this paper, we derive global sharp heat kernel estimates for symmetric α-stable processes (or equivalently, for the fractional Laplacian with zero exterior condition) in two classes of unbounded C 1,1 open sets in R d: half-space-like 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 half-space 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.

We consider boundary Harnack inequalities for regional fractional Laplacian which are generators of censored stable-like 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 α-stable processes, censored stable-like processes, (super) subharmonic function, Carleson estimate, Harnack inequality, boundary Harnack inequality MR(2000) Subject Classification: Primary 60G52, Secondary 60J45,47G20 1

We survey the recent development of the DeGiorgi-Nash-Moser-Aronson type theory for a class of symmetric jump processes (or equivalently, a class of symmetric integro-differential operators). We focus on the sharp two-sided 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 integro-differential operators are mainly probabilistic.

We consider the Dirichlet

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

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 α-stable process with weight a. We establish sharp two-sided 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 α-stable process with weight a in such open sets. Our result is the first sharp two-sided 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 two-sided 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.

Abstract In this paper, by using probabilistic methods, we establish sharp two-sided large time estimates for the transition densities of relativistic α-stable 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 half-space-like 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 two-sided small time estimates, established in Chen et al. (Ann Probab, 2011), valid for all C 1,1 open sets, we have now sharp two-sided estimates for the transition densities of relativistic α-stable processes with mass m ∈ (0, 1] in half-space-like C 1,1 open sets for all times. Integrating the heat kernel estimates with respect to the time variable, one can recover the sharp two-sided Green function estimates for relativistic α-stable processes with mass m ∈ (0, 1] in half-space-like C 1,1 open sets established recently

For d ≥ 1 and 0 < β < α < 2, consider a family of pseudo differential operators { ∆ α + a β ∆ β/2; a ∈ [0, 1]} on R d that evolves continuously from ∆ α/2 to ∆ α/2 + ∆ β/2. It gives arise to a family of Lévy processes {X a, a ∈ [0, 1]} on R d, where each X a is the independent sum of a symmetric α-stable process and a symmetric β-stable process with weight a. For any C 1,1 open set D ⊂ R d, we establish explicit sharp two-sided estimates (uniform in a ∈ [0, 1]) for the transition density function of the subprocess X a,D of X a killed upon leaving the open set D. The infinitesimal generator of X a,D is the non-local operator ∆ α + a β ∆ β/2 with zero exterior condition on D c. As consequences of these sharp heat kernel estimates, we obtain uniform sharp Green function estimates for X a,D and uniform boundary Harnack principle for X a in D with explicit decay rate.