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

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