Generalized axially symmetric potentials with distributional boundary values

TL;DR

This paper extends classical potential theory to weighted Laplace equations, characterizing distributions that generate solutions relevant to hyperbolic Brownian motion with drift, and solving related boundary value problems.

Contribution

It identifies the optimal class of tempered distributions for generalized axially symmetric potentials via $ ext{S}'$-convolution and solves the associated Dirichlet boundary value problem.

Findings

Characterization of the distributional boundary values for these potentials.

Solution of the Dirichlet boundary value problem in this context.

Establishment of sharp asymptotic growth relations for solutions.

Abstract

We study a counterpart of the classical Poisson integral for a family of weighted Laplace differential equations in Euclidean half space, solutions of which are known as generalized axially symmetric potentials. These potentials appear naturally in the study of hyperbolic Brownian motion with drift. We determine the optimal class of tempered distributions which by means of the so-called $\mathscr{S}'$-convolution can be extended to generalized axially symmetric potentials. In the process, the associated Dirichlet boundary value problem is solved, and we obtain sharp order relations for the asymptotic growth of these extensions.