Tables of integrals

Stochastic differential equations and diffusion processes

Special Functions and Their Applications. By N. N. Lebedev, translated by R. A. Silverman. Pp. xii, 308. 96s. 1965. (Prentice-Hall)

Diffusion processes and their sample paths

This article reviews several definitions of the fractional Laplace operator (-Delta)^{alpha/2} (0 < alpha < 2) in R^d, also known as the Riesz fractional derivative operator, as an operator on Lebesgue spaces L^p, on the space C_0 of continuous functions vanishing at infinity and on the space C_{bu} of bounded uniformly continuous functions. Among these definitions are ones involving singular integrals, semigroups of operators, Bochner's subordination and harmonic extensions. We collect and extend known results in order to prove that all these definitions agree: on each of the function spaces considered, the corresponding operators have common domain and they coincide on that common domain.

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Ten equivalent definitions of the fractional laplace operator

Boundary Potential Theory for Schr#x00F6 dinger Operators Based on Fractional Laplacian.- Nontangential Convergence for #x03B1 -harmonic Functions.- Eigenvalues and Eigenfunctions for Stable Processes.- Potential Theory of Subordinate Brownian Motion.

Potential Analysis of Stable Processes and its Extensions

Potential theory for the α-stable Schrödinger operator on bounded Lipschitz domains

The purpose of the paper is to find explicit formulas for basic objects pertaining to the potential theory of the operator (I- Δ) α/2 , which is based on Bessel potentials J α = (I- Δ) -α/2 , 0 < α < 2. We compute the harmonic measure of the half-space and obtain a concise form for the corresponding Green function of the operator (I- Δ) α/2 . As an application we provide sharp estimates for the Green function of the half-space for the relativistic process.

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Bessel potentials, hitting distributions and Green functions

Let \(T_1^{(\mu)}\) be the first hitting time of the point 1 by the Bessel process with index μ ∈ ℝ starting from x > 1. Using an integral formula for the density \(q_x^{(\mu)}(t)\) of \(T_1^{(\mu)}\), obtained in Byczkowski and Ryznar (Stud Math 173(1):19–38, 2006), we prove sharp estimates of the density of \(T_1^{(\mu)}\) which exhibit the dependence both on time and space variables. Our result provides optimal uniform estimates for the density of the hitting time of the unit ball by the Brownian motion in ℝn, which improve existing bounds. Another application is to provide sharp estimates for the Poisson kernel for half-spaces for hyperbolic Brownian motion in real hyperbolic spaces.

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Hitting Times of Bessel Processes

The boundary Harnack principle for fractional Laplacian is now known to hold in Lipschitz domains [5]. It states that if two nonnegative functions, harmonic with respect to a symmetric stable Levy process vanish continuously outside a Lipschitz domain, near a part of its boundary, then the ratio of the functions is bounded inside the domain, near this part of the boundary. We give a probabilistic proof of the assertion using elementary properties of the stable process.

Tomasz Byczkowski

Papers

Potential Analysis of Stable Processes and its Extensions

Potential theory for the α-stable Schrödinger operator on bounded Lipschitz domains

Bessel potentials, hitting distributions and Green functions

Hitting Times of Bessel Processes

Probabilistic Proof of Boundary Harnack Principle for α-Harmonic Functions