On planar Brownian motion singularly tilted through a point potential

TL;DR

This paper introduces a family of two-dimensional diffusions with singular drifts influenced by a point potential, enabling visits to the origin, and analyzes their local time processes and inverses.

Contribution

It characterizes a local time process at the origin for these diffusions and studies the law of its inverse, extending understanding of singularly tilted planar Brownian motions.

Findings

Diffusions can visit the origin due to the singular drift.

Local time process at the origin is characterized.

Law of the inverse local time process is analyzed.

Abstract

We discuss a family of time-inhomogeneous two-dimensional diffusions, defined over a finite time interval $[0,T]$, having transition density functions that are expressible in terms of the integral kernels for negative exponentials of the two-dimensional Schr\"odinger operator with a point potential at the origin. These diffusions have a singular drift pointing in the direction of the origin that is strong enough to enable the possibly of visiting there, in contrast to a two-dimensional Brownian motion. Our main focus is on characterizing a local time process at the origin analogous to that for a one-dimensional Brownian motion and on studying the law of its process inverse.