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4 matches found
XV: 11, 167-188, LNM 850 (1981)
BOULEAU, Nicolas
Propriétés d'invariance du domaine du générateur infinitésimal étendu d'un processus de Markov (Markov processes)
The main result of the paper of Kunita (Nagoya Math. J., 36, 1969) showed that the domain of the extended generator $A$ of a right Markov semigroup is an algebra if and only if the angle brackets of all martingales are absolutely continuous with respect to the measure $dt$. See also 1010. Such semigroups are called here semigroups of Lebesgue type''. Kunita's result is sharpened here: it is proved in particular that if some non-affine convex function $f$ operates on the domain, then the semigroup is of Lebesgue type (Kunita's result corresponds to $f(x)=x^2$) and if the second derivative of $f$ is not absolutely continuous, then the semigroup has no diffusion part (i.e., all martingales are purely discontinuous). The second part of the paper is devoted to the behaviour of the extended domain under an absolutely continuous change of probability (arising from a multiplicative functional)
Keywords: Semigroup theory, Carré du champ, Infinitesimal generators
Nature: 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 laws
Nature: Original
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XX: 13, 162-185, LNM 1204 (1986)
BOULEAU, Nicolas; LAMBERTON, Damien
Théorie de Littlewood-Paley et processus stables (Applications of martingale theory, Markov processes)
Meyer' probabilistic approach to Littlewood-Paley inequalities (1010, 1510) is extended by replacing the underlying Brownian motion with a stable process. The following spectral multiplicator theorem is obtained: If $(P_t)_{t\geq 0}$ is a symmetric Markov semigroup with spectral representation $P_t=\int_{[0,\infty)}e^{-t\lambda} dE_{\lambda}$, and if $M$ is a function on $R_+$ defined by $M(\lambda)=\lambda\int_0^\infty r(y)e^{-y\lambda}dy,$ where $r(y)$ is bounded and Borel on $R_+$, then the operator $T_M=\int_{[0,\infty)}M(\lambda)dE_{\lambda},$ which is obviously bounded on $L^2$, is actually bounded on all $L^p$ spaces of the invariant measure, $1<p<\infty$. The method also leads to new Littlewood-Paley inequalities for semigroups admitting a carré du champ operator
Keywords: Littlewood-Paley theory, Semigroup theory, Riesz transforms, Stable processes, Inequalities, Singular integrals, Carré du champ
Nature: Original
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XLIII: 14, 341-349, LNM 2006 (2011)
BOULEAU, Nicolas
The Lent Particle Method for Marked Point Processes (General theory of processes, Point processes)
Nature: Original