Planar diffusions with a point interaction on a finite time horizon

TL;DR

This paper develops an axiomatic framework for planar diffusions with point interactions at the origin, generalizing known examples like skew-product diffusion and tilted Brownian motion, and explores their construction and properties.

Contribution

It introduces a general axiomatic approach to construct and analyze planar diffusions with point interactions, extending existing models and providing new insights into their structure.

Findings

Established admissibility and regularity conditions for such diffusions.

Provided a heuristic construction for the skew-product diffusion.

Applied the framework to measure-generated diffusions, including tilted Brownian motion.

Abstract

The skew-product diffusion [Ann. Appl. Probab. 35, 3150--3214 (2025)] and exponentially tilted planar Brownian motion [Electron. J. Probab. 30, 1--97 (2025)] are canonical examples of planar diffusions with a point interaction at the origin in the sense that their drifts are singular only at the origin and allow visits there with positive probability. However, in this article we propose an axiomatic framework for such diffusions on a finite time horizon. We isolate admissibility conditions and additional regularity hypotheses on a general driving family under which the associated diffusion, constructed as a Doob transform of point-interaction Schr\"odinger semigroup kernels, exhibits the same point interaction structure. In particular, for the ground-state driving family, we obtain a heuristic alternative construction of the skew-product diffusion based on Kolmogorov continuity arguments. We also consider formal applications of this framework to some driving families generated by measures, including the Lebesgue-driven diffusion, which is the exponentially tilted planar Brownian motion.