Cameron--Martin formula for the $ \sigma $-finite measure unifying Brownian penalisations

TL;DR

This paper proves a quasi-invariance property for a special sigma-finite measure related to Brownian penalizations, using Wiener integrals for Bessel processes to unify various approaches.

Contribution

It introduces a new proof of quasi-invariance for the measure unifying Brownian penalizations, leveraging Wiener integrals for centered Bessel processes.

Findings

Established quasi-invariance under translation for the measure

Unified different Brownian penalization approaches

Applied Wiener integrals for centered Bessel processes

Abstract

Quasi-invariance under translation is established for the $ \sigma $-finite measure unifying Brownian penalisations, which has been introduced by Najnudel, Roynette and Yor. For this purpose, the theory of Wiener integrals for centered Bessel processes, due to Funaki, Hariya and Yor, plays a key role.