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X: 05, 44-77, LNM 511 (1976)
KUNITA, Hiroshi
Absolute continuity for Markov processes (Markov processes)
This paper is devoted to a progressive'' Lebesgue decomposition of the laws of a Markov process with respect to a second one in the same filtration, and the structure of the corresponding density. The two processes are assumed to be Hunt processes, and for part of the paper satisfy Hunt's hypothesis (K) (all excessive functions are regular, or semi-polar sets are polar). The topics discussed are the following: Lévy systems and the relation between the Lévy systems of a process and of its transform by a multiplicative functional; structure of exact perfect terminal times, which are shown to be hitting times of sets in space-time, by the process $(X_{t-},X_t)$ (a version of a result of Walsh-Weil, Ann. Sci. ENS, 5, 1972); the Lebesgue decomposition'' of a Markov process with respect to another, and the fact that if absolute continuity holds on the germ field it also holds up to some maximal terminal time; a condition for this terminal time to be equal to the lifetime, under hypothesis (K)
Comment: The pasting together of the Lebesgue decompositions of a probability measure with respect to another one, on the $\sigma$-fields of a given filtration, is called the Kunita decomposition, and is not restricted to Markov processes. For the general case, see Yoeurp, in LN 1118, Grossissements de filtrations, 1985
Keywords: Absolute continuity of laws, Hunt processes, Terminal times, Kunita decomposition
Nature: Original
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