Zhen-Qing Chen

Heat kernel estimates for stable-like processes on d-sets

The notion of d-set arises in the theory of function spaces and in fractal geometry. Geometrically self-similar sets are typical examples of d-sets. In this paper stable-like processes on d-sets are investigated, which include reflected stable processes in Euclidean domains as a special case. More precisely, we establish parabolic Harnack principle and derive sharp two-sided heat kernel estimate for such stable-like processes. Results on the exact Hausdorff dimensions for the range of stable-like processes are also obtained.

Non-local Dirichlet forms and symmetric jump processes

We consider the non-local symmetric Dirichlet form (E,F) given by with F the closure with respect to E 1 of the set of C 1 functions on R d with compact support, where E 1 (f, f):= E(f, f) + f Rd f(x) 2 dx, and where the jump kernel J satisfies for 0 < α < β < 2, |x - y| < 1. This assumption allows the corresponding jump process to have jump intensities whose sizes depend on the position of the process and the direction of the jump. We prove upper and lower estimates on the heat kernel. We construct a strong Markov process corresponding to (E,F). We prove a parabolic Harnack inequality for non-negative functions that solve the heat equation with respect to E. Finally we construct an example where the corresponding harmonic functions need not be continuous.

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.

Gaugeability and conditional gaugeability

New Kato classes are introduced for general transient Borel right processes, for which gauge and conditional gauge theorems hold. These new classes are the genuine extensions of the Green-tight measures in the classical Brownian motion case. However, the main focus of this paper is on establishing various equivalent conditions and consequences of gaugeability and conditional gaugeability. We show that gaugeability, conditional gaugeability and the subcriticality for the associated Schrodinger operators are equivalent for transient Borel right processes with strong duals. Analytic characterizations of gaugeability and conditional gaugeability are given for general symmetric Markov processes. These analytic characterizations are very useful in determining whether a process perturbed by a potential is gaugeable or conditionally gaugeable in concrete cases. Connections with the positivity of the spectral radii of the associated Schrodinger operators are also established.

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

Drift transforms and Green function estimates for discontinuous processes

In this paper, we consider Girsanov transforms of pure jump type for discontinuous Markov processes. We show that, under some quite natural conditions, the Green functions of the Girsanov transformed process are comparable to those of the original process. As an application of the general results, the drift transform of symmetric stable processes is studied in detail. In particular, we show that the relativistic a-stable process in a bounded C 1;1 -smooth open set D can be obtained from symmetric a-stable process in D through a combination of a pure jump Girsanov transform and a Feynman–Kac transform. From this, we deduce that the Green functions for these two processes in D are comparable.

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.

Space-time fractional diffusion on bounded domains

Abstract Fractional diffusion equations replace the integer-order derivatives in space and time by their fractional-order analogues. They are used in physics to model anomalous diffusion. This paper develops strong solutions of space–time fractional diffusion equations on bounded domains, as well as probabilistic representations of these solutions, which are useful for particle tracking codes.

On reflecting diffusion processes and Skorokhod decompositions

SummaryLetG be ad-dimensional bounded Euclidean domain, H1 (G) the set off in L2(G) such that ∇f (defined in the distribution sense) is in L2(G). Reflecting diffusion processes associated with the Dirichlet spaces (H1(G), ℰ) on L2(G, σdx) are considered in this paper, where A=(aij is a symmetric, bounded, uniformly ellipticd×d matrix-valued function such thataij∈H1(G) for eachi,j, and σ∈H1(G) is a positive bounded function onG which is bounded away from zero. A Skorokhod decomposition is derived for the continuous reflecting Markov processes associated with (H1(G), ℰ) having starting points inG under a mild condition which is satisfied when ϖG has finite (d−1)-dimensional lower Minkowski content.

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.