Inverting Ray-Knight identities on trees

Abstract

In this paper, we first introduce the Ray-Knight identity and percolation Ray-Knight identity related to loop soup with intensity $\mathit{\alpha }(\ge 0)$ on trees. Then we present the inversions of the above identities, which are expressed in terms of repelling jump processes. The inversion in the case of $\mathit{\alpha }=0$ gives the conditional law of a continuous-time Markov chain given its local time field; while the inversion in the case of $\mathit{\alpha }>0$ gives the conditional law of a Markovian loop soup given its local time field. We further show that the fine mesh limits of these repelling jump processes are the self-repelling diffusions involved in the inversion of the Ray-Knight identity on the corresponding metric graph.