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We consider the fractional Laplacian -(-Δ) 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.

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.

In this paper, we consider a large class of purely discontinuous rotationally symmetric Lévy processes. We establish sharp two-sided estimates for the transition densities of such processes killed upon leaving an open set D. When D is a κ-fat open set, the sharp two-sided estimates are given in terms of surviving probabilities and the global transition density of the Lévy process. When D is a C 1 , 1 open set and the Lévy exponent of the process is given by Ψ ( ξ ) = ϕ ( | ξ | 2 ) with ϕ being a

Let μ = μ 1 ⋯ μ d be such that each μ i is a signed measure on \R d belonging to the Kato class \K d , 1 . The existence and uniqueness of a continuous Markov process X on \R d , called a Brownian motion with drift μ , was recently established by Bass and Chen. In this paper we study the potential theory of X . We show that X has a continuous density q μ and that there exist positive constants c i , i = 1 , ⋯ , 9 , such that c 1 e - c 2 t t - d 2 e - c 3 x - y 2 2 t ≤ q μ t x y ≤ c 4 e c 5 t t -

A subordinate Brownian motion is a Lévy process that can be obtained by replacing the time of the Brownian motion by an independent subordinator. The infinitesimal generator of a subordinate Brownian motion is−ϕ(−Δ), where ϕ is the Laplace exponent of the subordinator. In this paper, we consider a large class of subordinate Brownian motions without diffusion component and with ϕ comparable to a regularly varying function at infinity. This class of processes includes symmetric stable processes, r

A subordinate Brownian motion <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a Lévy process which can be obtained by replacing the time of the Brownian motion by an independent subordinator. In this paper, when the Laplace exponent <inline-formula content-type="math/mathml"> <m

Abstract In this paper we prove the uniform boundary Harnack principle in general open sets for harmonic functions with respect to a large class of rotationally symmetric purely discontinuous Lévy processes.

We extend the concept of intrinsic ultracontractivity to non-symmetric semigroups and prove the intrinsic ultracontractivity of the Dirichlet semigroups of non-symmetric second order elliptic operators in bounded Lipschitz domains.