Cameron–Martin formula for the σ-finite measure unifying Brownian penalisations

Abstract

Quasi-invariance under translation is established for the σ-finite measure unifying Brownian penalisations, which has been introduced by Najnudel, Roynette and Yor [J. Najnudel, B. Roynette, M. Yor, A remarkable σ-finite measure on C ( R + , R ) related to many Brownian penalisations, C. R. Math. Acad. Sci. Paris 345 (8) (2007) 459–466]. For this purpose, the theory of Wiener integrals for centered Bessel processes, due to Funaki, Hariya and Yor [T. Funaki, Y. Hariya, M. Yor, Wiener integrals for centered Bessel and related processes. II, ALEA Lat. Am. J. Probab. Math. Stat. 1 (2006) 225–240 (electronic)], plays a key role.