Local time of infinite time horizon Brownian bridge

TL;DR

This paper introduces an infinite time horizon Brownian bridge driven by a Langevin equation with time-dependent drift, analyzing its local time, asymptotic behavior, and regularity despite non-stationarity.

Contribution

It develops a framework for studying non-stationary infinite horizon Brownian bridges, including existence, asymptotics, and regularity of local time.

Findings

The process converges to zero almost surely as time approaches infinity.

The local time exists and its asymptotic behavior is characterized.

Hölder continuity in time and space variables is established.

Abstract

We introduce an infinite time horizon Brownian bridge which is determined by a stochastic Langevin equation with time dependent drift coefficient. We show that this process goes to zero almost surely when the time goes to infinity and study the existence and asymptotic behavior of its local time as well as its H\"older continuity in time variable and in location variable. The main difficulty is the lack of stationarity of the process so that the powerful tools for stationary (Gaussian) processes are not applicable. We employ the Garsia-Rodemich-Rumsey inequality to get around this type of difficulty.