Wiener integral for the coordinate process under the $ \sigma $-finite measure unifying Brownian penalisations

TL;DR

This paper defines a Wiener integral under a specific sigma-finite measure that unifies Brownian penalizations, analyzes its decomposition around the last exit time from zero, and lays groundwork for a Cameron--Martin formula in this context.

Contribution

It introduces a Wiener integral under a sigma-finite measure unifying Brownian penalizations and studies its decomposition, advancing the theoretical framework for Cameron--Martin formulas in this setting.

Findings

Decomposition of Wiener integral before and after last exit time from 0

Foundation for Cameron--Martin formula under sigma-finite measure

Unification of Brownian penalizations through the measure

Abstract

Wiener integral for the coordinate process is defined under the $ \sigma $-finite measure unifying Brownian penalisations, which has been introduced by Najnudel, Roynette and Yor. Its decomposition before and after last exit time from 0 is studied. This study prepares for the author's recent study of Cameron--Martin formula for the $ \sigma $-finite measure.