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On the reconstruction of a killed Markov process

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On the reconstruction of a killed Markov process SÉMINAIRE DE PROBABILITÉS (STRASBOURG) SADAO SATO On the reconstruction of a killed Markov process Séminaire de probabilités (Strasbourg), tome 26 (1992), p. 540-559. <http://www.numdam.org/item?id=SPS_1992__26__540_0> © Springer-Verlag, Berlin Heidelberg New York, 1992, tous droits réservés. L’accès aux archives du séminaire de probabilités (Strasbourg) (http://www-irma. u-strasbg.fr/irma/semproba/index.shtml), implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commer- ciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ On the reconstruction of a killed Markov process By Sadao Sato 1. Introduction Let ( Yt) be a right continuous strong Markov process and ( St) be its semigroup. Let (ifft) be a multiplicative functional satisfying 0~Mt~ l. Then the formula (1.1) Ttf(x)=Ex[f(Yt)Mt] defines a subordinated semigroup. P. A. Meyer obtained the existence and the uniqueness (up to the life time of Yt) of such an Mt for every positive semigroup subordinated to ( St) (see [DM3]). A Markov process with semigroup (Tt) is called (according to Dynkin) a subprocess of (Yt). It is a strong Markov process if the functional (M t) is exact, and in particular if a.s.. Conversely, let (Xt) be a right continuous strong Markov process with semigroup ( Tt). The problem in this paper is to construct in a natural way a strong Markov process (Yt) with semigroup (St) such that 1) (Xt) is subordinated to (Yt) 2) (Yt) is as close as possible to being conservative. This problem was studied by several authors. In Ikeda, Nagasawa and Watanabe [INW], it is constructed using the "piecing out method". They also proved that it is conservative under the condit

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