Inverting the Ray-Knight identity on the line

TL;DR

This paper constructs a self-repelling diffusion on the line using a Bass-Burdzy flow, demonstrating it inverts the second Ray-Knight identity, with proofs via approximation by self-repelling jump processes.

Contribution

It introduces a novel self-repelling diffusion that inverts the Ray-Knight identity on the line, extending previous discrete results to continuous settings.

Findings

Constructed a self-repelling diffusion using Bass-Burdzy flow

Proved the diffusion inverts the second Ray-Knight identity

Established the connection through approximation by jump processes

Abstract

Using a divergent Bass-Burdzy flow we construct a self-repelling one-dimensional diffusion. Heuristically, it can be interpreted as a solution to an SDE with a singular drift involving a derivative of the local time. We show that this self-repelling diffusion inverts the second Ray-Knight identity on the line. The proof goes through an approximation by a self-repelling jump processes that has been previously shown by the authors to invert the Ray-Knight identity in the discrete.