On développe une approche générale du théorème limite centrale presque-sûre pour les martingales vectorielles quasi-continues à gauche convenablement normalisées dont on dégage une extension quadratique et un nouveau théorème de la limite centrale. L'application de ce résultat à l'estimation de la variance d'un processus à accroissements indépendants et stationnaires illustre l'usage qu'on peut en faire en statistique.

We investigate the dissipativity properties of a class of scalar secondorder parabolic partial differential equations with time-dependentcoefficients. We provide explicit condition on the drift term which ensurethat the relative entropy of one particular orbit with respect to some otherone decreases to zero. The decay rate is obtained explicitly by the use ofa Sobolev logarithmic inequality for the associated semigroup, which is derived by an adaptation of Bakry's Γ-calculus.As a byproduct, the systematic method for constructing entropieswhich we propose here also yields the well-known intermediate asymptotics for the heat equation in a very quick way, and without having to rescale the original equation.