Local time penalizations with various clocks for one-dimensional diffusions

TL;DR

This paper investigates the limiting behavior of one-dimensional diffusions under various local time-based penalizations, establishing convergence results and path interpretations without restrictive resolvent assumptions.

Contribution

It introduces a general framework for local time penalizations with random clocks, extending previous deterministic clock results and providing universal path interpretations.

Findings

Convergence of penalized diffusions as the clock tends to infinity

No resolvent assumptions needed for convergence with random clocks

Path representations via universal sigma-finite measures

Abstract

We study some limit theorems for the law of a generalized one-dimensional diffusion weighted and normalized by a non-negative function of the local time evaluated at a parametrized family of random times (which we will call a clock). As the clock tends to infinity, we show that the initial process converges towards a new penalized process, which generally depends on the chosen clock. However, unlike with deterministic clocks, no specific assumptions are needed on the resolvent of the diffusion. We then give a path interpretation of these penalized processes via some universal $ \sigma $-finite measures.