Penalizing null recurrent diffusions

TL;DR

This paper establishes limit theorems for normalized laws of null recurrent diffusions, focusing on functionals involving last passage times, and extends previous Brownian penalization results to a broader class of diffusions.

Contribution

It introduces new limit theorems for null recurrent diffusions based on the Lévy measure of the inverse local time, generalizing known Brownian penalization results.

Findings

Limit theorems for normalized laws of null recurrent diffusions.

Extension of Brownian penalization results to broader diffusions.

Conditions based on Lévy measure of inverse local time.

Abstract

We present some limit theorems for the normalized laws (with respect to functionals involving last passage times at a given level up to time t) of a large class of null recurrent diffusions. Our results rely on hypotheses on the L\'evy measure of the diffusion inverse local time at 0. As a special case, we recover some of the penalization results obtained by Najnudel, Roynette and Yor in the (reflected) Brownian setting.