Université Paris-Est Université Paris-Est - Marne-la-Vallée Université Paris-Est - Créteil Val-de-Marne Centre National de la Recherche Scientifique

Régularité des lois de probabilité par une méthode d'interpolation

23/05/2013 - 13:30 - 14:30

In the last 30 years Malliavin calculus became the main instrument in the study of the regularity of probability laws: this is an infinite dimensional differential calculus which permits to built integration by parts formulas. Once we have such a formula the regularity of the law follows using Fourier analysis or various alternative arguments. But if one deals with solutions of stochastic equations with coefficients which are not sufficiently regular then this functional is not in the domain of the differential operators from Malliavin calculus and so this method does not apply. Recently, initiated by N. Fournier, an alternative approach appeared: instead of using one "perfect" integration by parts formula one constructs a sequence of integration by parts formulas based on a sequence of functionals which approximates the basic functional. The approximation error tends to zero and the weights in the integration by parts formulas blow up. If one is able to find a good equilibrium between these two quantities then one obtains regularity properties. It turns out that this procedure fits in the classical framework of the real interpolation method. In my talk I will explain this procedure and, based on it, give a general criterion of regularity for probability laws.