Darning and gluing of diffusions

TL;DR

This paper introduces a method for 'darning' and 'gluing' compact sets within diffusion processes, enabling iterative modifications in potential theory and applications to heat kernel studies across various geometries.

Contribution

It develops a potential-theoretic framework for darning and gluing of compact sets in diffusion processes, applicable to Euclidean spaces and manifolds of different dimensions.

Findings

Framework allows iterative darning and gluing of compact sets.

Applicable to Euclidean spaces and manifolds of various dimensions.

Relevance to recent heat kernel research.

Abstract

We introduce darning of compact sets (darning and gluing of finite unions of compact sets), which are not thin at any of their points, in a potential-theoretic framework which may be described, analytically, in terms of harmonic kernels/harmonic functions or, probabilistically, in terms of a diffusion. This is accomplished without leaving our kind of setting so that the procedure can be iterated without any problem. It applies to darning and gluing of compacts in Euclidean spaces (manifolds) of different dimensions, which is of interest pertaining to recent studies on heat kernels.