Limiting laws associated with Brownian motion perturbed by its maximum, minmum and local time II

TL;DR

This paper develops probability measures that modify Wiener measure based on the process's maximum, minimum, and local time, and studies the resulting laws of the canonical process.

Contribution

It introduces new probability measures penalizing Wiener measure by functionals of maximum, minimum, and local time, analyzing their impact on the process law.

Findings

Derived laws of the process under penalized measures

Extended understanding of Brownian motion with boundary constraints

Provided mathematical framework for penalization by extrema and local time

Abstract

We obtain probability measures on the canonical space penalizing the Wiener measure by a function of its maximum (resp. minimum, local time). We study the law of the canonical process under these new probability measures.