Wiener integral for the coordinate process
is defined under the σ-finite measure unifying Brownian penalisations,
which has been introduced by [Najnudel et al., C. R. Math. Acad. Sci. Paris 345 (2007) 459–466] and [Najnudel et al., MSJ Memoirs 19. Mathematical Society of Japan, Tokyo (2009)].
Its decomposition before and after last exit time from 0
This study prepares for the author's recent study [K. Yano,
J. Funct. Anal. 258 (2010) 3492–3516] of Cameron-Martin formula
for the σ-finite measure.