Transience and Recurrence of Markov Processes with Constrained Local Time

TL;DR

This paper investigates the transience and recurrence of Markov processes with constrained local time, providing explicit distributions and conditions for their long-term behavior, extending previous Brownian motion studies to broader Markov classes.

Contribution

It introduces a framework for analyzing Markov processes with time-varying local time constraints and derives explicit distributions and criteria for transience and recurrence.

Findings

Necessary and sufficient conditions for transience or recurrence.

Explicit distribution of the inverse local time for the conditioned process.

Characterization of the entropic repulsion envelope in the recurrent case.

Abstract

We study Markov processes conditioned so that their local time must grow slower than a prescribed function. Building upon recent work on Brownian motion with constrained local time in [5] and [33], we study transience and recurrence for a broad class of Markov processes. In order to understand the distribution of the local time, we determine the distribution of a non-decreasing L\'evy process (the inverse local time) conditioned to remain above a given level which varies in time. We study a time-dependent region, in contrast to previous works in which a process is conditioned to remain in a fixed region (e.g. [21,27]), so we must study boundary crossing probabilities for a family of curves, and thus obtain uniform asymptotics for such a family. Main results include necessary and sufficient conditions for transience or recurrence of the conditioned Markov process. We will explicitly determine the distribution of the inverse local time for the conditioned process, and in the transient case, we explicitly determine the law of the conditioned Markov process. In the recurrent case, we characterise the "entropic repulsion envelope" via necessary and sufficient conditions.