In this paper we study the precise behavior of the transition density functions of censored (resurrected) α-stable-like 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 α-stable-like process in any bounded Lipschitz open set is intrinsically ultracontractive. We then establish sharp two-sided estimates for the transition density functions of a large class of censored α-stable-like processes in C 1,1 open sets. We further obtain sharp two-sided estimates for the Green functions of these censored α-stable-like processes in bounded C 1,1 open sets.

Abstract: We give sharp estimates for the heat kernel of the fractional Laplacian with Dirichlet condition for a general class of domains including Lipschitz domains.

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

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.

Abstract not found

Suppose that d ≥ 1 and α ∈ (0, 2). In this paper, we establish by using probabilistic methods sharp two-sided pointwise estimates for the Dirichlet heat kernels of { ∆ + a α ∆ α/2; a ∈ (0, 1]} on half-space-like C1,1 domains for all time t> 0. The large time estimates for half-spacelike domains are very different from those for bounded domains. Our estimates are uniform in a ∈ (0, 1] in the sense that the constants in the estimates are independent of a ∈ (0, 1]. Thus they yield the Dirichlet heat kernel estimates for Brownian motion in half-space-like domains by taking a → 0. Integrating the heat kernel estimates with respect to the time variable t, we obtain uniform sharp two-sided estimates for the Green functions of { ∆ + aα∆α/2; a ∈ (0, 1]} in half-space-like C1,1 domains in Rd.

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