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.

Wiener integral for the coordinate process under theσ-finite measure unifying Brownian penalisations

“…There is another way of constructing Wiener integral for the 3-dimensional Bessel process, which is due to Funaki-Hariya-Yor ( [4]; see also [5], [2] and [3]). Note that this method will play a crucial role in [12].…”

“…As an application of the Wiener integral under W , the author studies, in a separate paper [12], Cameron-Martin formula for W , i.e., quasi-invariance of W under deterministic translations.…”

“…Its decomposition before and after last exit time from 0 is studied. This study prepares for the author's recent study [12] of Cameron-Martin formula for the σ-finite measure.…”