Inverting Ray-Knight identities on trees

TL;DR

This paper introduces and inverts Ray-Knight identities on trees, expressing the inversions via repelling jump processes, and connects these to self-repelling diffusions in the limit, extending previous work to a broader setting.

Contribution

It presents new inversion formulas for Ray-Knight identities on trees using repelling jump processes, generalizing prior results and linking loop soups with self-repelling diffusions.

Findings

Inversion formulas expressed through repelling jump processes.

Connection between fine mesh limits and self-repelling diffusions.

Extension of previous results to a more general setting on trees.

Abstract

In this paper, we first introduce the Ray-Knight identity and percolation Ray-Knight identity related to loop soup with intensity $\alpha (\ge 0)$ on trees. Then we present the inversions of the above identities, which are expressed in terms of repelling jump processes. In particular, the inversion in the case of $\alpha=0$ gives the conditional law of a Markov jump process given its local time field. We further show that the fine mesh limits of these repelling jump processes are the self-repelling diffusions \cite{Aidekon} involved in the inversion of the Ray-Knight identity on the corresponding metric graph. This is a generalization of results in \cite{2016Inverting,lupu2019inverting,LupuEJP657}, where the authors explore the case of $\alpha=1/2$ on a general graph. Our construction is different from \cite{2016Inverting,lupu2019inverting} and based on the link between random networks and loop soups.