: 11, 167-188, LNM 850 (1981)BOULEAU, Nicolas
Propriétés d'invariance du domaine du générateur infinitésimal étendu d'un processus de Markov
The main result of the paper of Kunita (Nagoya Math. J.
, 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 generatorsNature: Original Retrieve article from Numdam