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 decompositionNature: Original
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XX: 12, 131-161, LNM 1204 (1986)
BOULEAU, Nicolas;
HIRSCH, Francis
Propriété d'absolue continuité dans les espaces de Dirichlet et applications aux équations différentielles stochastiques (
Dirichlet forms,
Malliavin's calculus)
This is the main result of the ``Bouleau-Hirsch approach'' to absolute continuity in Malliavin calculus (see
The Malliavin calculus and related topics by D. Nualart, Springer1995). In the framework of Dirichlet spaces, a general criterion for absolute continuity of random vectors is established; it involves the image of the energy measure. This leads to a Lipschitzian functional calculus for the Ornstein-Uhlenbeck Dirichlet form on Wiener space, and gives absolute continuity of the laws of the solutions to some SDE's with coefficients that can be uniformly degenerate
Comment: These results are extended by the same authors in their book
Dirichlet Forms and Analysis on Wiener Space, De Gruyter 1991
Keywords: Dirichlet forms,
Carré du champ,
Absolute continuity of lawsNature: Original Retrieve article from Numdam
