Panki Kim

Heat kernel estimates for the Dirichlet fractional Laplacian

In this paper, we consider the fractional Laplacian -(-?)a/2 on an open subset in Rd with zero exterior condition. We establish sharp two-sided estimates for the heat kernel of such Dirichlet fractional Laplacian in C1.1 open sets. This heat kernel is also the transition density of a rotationally symmetric a-stable process killed upon leaving a C1.1 open set. Our results are the first sharp two-sided estimates for the Dirichlet heat kernel of a non-local operator on open sets.

Global Heat Kernel Estimates for Symmetric Jump Processes

In this paper, we study sharp heat kernel estimates for a large class of symmetric jump-type processes in R d for all t > 0. A prototype of the processes under consideration are symmetric jump processes on R d with jumping intensity

Dirichlet heat kernel estimates for fractional Laplacian with gradient perturbation

Suppose that d≥2 and α∈(1,2). Let D be a bounded C1,1 open set in Rd and b an Rd-valued function on Rd whose components are in a certain Kato class of the rotationally symmetric α-stable process. In this paper, we derive sharp two-sided heat kernel estimates for Lb=Δα/2+b⋅∇ in D with zero exterior condition. We also obtain the boundary Harnack principle for Lb in D with explicit decay rate.

Potential Theory of Subordinate Brownian Motions Revisited

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.

Sharp heat kernel estimates for relativistic stable processes in open sets

In this paper, we establish sharp two-sided estimates for the transition densities of relativistic stable processes [i.e., for the heat kernels of the operators m − (m2/α − Δ)α/2] in C1,1 open sets. Here m > 0 and α ∈ (0, 2). The estimates are uniform in m ∈ (0, M] for each fixed M > 0. Letting m ↓ 0, we recover the Dirichlet heat kernel estimates for Δα/2 := −(−Δ)α/2 in C1,1 open sets obtained in [14]. Sharp two-sided estimates are also obtained for Green functions of relativistic stable processes in bounded C1,1 open sets.

Two-sided Green function estimates for killed subordinate Brownian motions

A subordinate Brownian motion is a Levy 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, relativistic stable processes, sums of independent symmetric stable processes, sums of independent relativistic stable processes and much more. We give sharp two-sided estimates on the Green functions of these subordinate Brownian motions in any bounded κ-fat open set D.W henD is a bounded C 1,1 open set, we establish an explicit form of the estimates in terms of the distance to the boundary. As a consequence of such sharp Green function estimates, we obtain a boundary Harnack principle in C 1,1 open sets with explicit rate of decay.

Uniform boundary Harnack principle for rotationally symmetric Lévy processes in general open sets

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.

Boundary Harnack principle for Δ+Δ^{/2}

For d � 1 and � 2 (0,2), consider the family of pseudo differential operators f �+b� �/2 ;b 2 (0,1)g on R d that evolves continuously fromto � + � �/2 . In this paper, we establish a uniform boundary Harnack principle (BHP) with explicit boundary decay rate for nonnegative functions which are harmonic with respect to �+b� �/2 (or equivalently, the sum of a Brownian motion and an independent symmetric�-stable process with constant multipleb 1/� ) inC 1,1 open sets. Here a uniform BHP means that the comparing constant in the BHP is independent of b 2 (0,1). Along the way, a uniform Carleson type estimate is established for nonnegative functions which are harmonic with respect to � +b� �/2 in Lipschitz open sets. Our method employs a combination of probabilistic and analytic techniques.

Dirichlet heat kernel estimates for rotationally symmetric Lévy processes

In this paper, we consider a large class of purely discontinuous rotationally symmetric Levy processes. We establish sharp two-sided estimates for the transition densities of such processes killed upon leaving an open set D.W henD 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 Levy process. When D is a C 1,1 open set and the Levy exponent of the process is given by Ψ(ξ )= φ(|ξ| 2 ) with φ being a complete Bernstein function satisfying a mild growth condition at infinity, our two-sided estimates are explicit in terms of Ψ, the distance function to the boundary of D and the Levy density of X. This gives an affirmative answer to the conjecture posted in Chen, Kim and Song (Global heat kernel estimates for relativistic stable processes in half-space-like open sets. Potential Anal. 36 (2012) 235-261). Our results are the first sharp two-sided Dirichlet heat kernel estimates for a large class of symmetric Levy processes with general Levy exponents. We also derive an explicit lower bound estimate for symmetric Levy processes on R d in terms of their Levy exponents.

Green function estimates for subordinate Brownian motions: Stable and beyond

A subordinate Brownian motion