Transience and Recurrence of Markov Processes with Constrained Local Time

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 Benjamini and Berestycki (2011); Kolb and Savov (2016), we study transience and recurrence for a broad class of Markov processes. In order to understand the local time, we determine the distribution of a nondecreasing Lévy 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. Denisov and Wachtel, 2015; Garbit, 2009), 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.

Brownian Motion Conditioned to Spend Limited Time Below a Barrier

“…Proceeding from Doob's work, similar problems have been considered for more complicated processes and more complicated or time-dependent sets to be avoided. Many examples are referred to in the introduction of [Bar20]. In the present paper, we advance in a different direction: We allow a Brownian motion with an arbitrary starting point to spend limited time in the negative half-line.…”