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Section: New Results

Regularity of probability laws using an interpolation method

Participant : Vlad Bally.

This work was motivated by previous papers of Nicolas Fournier, J. Printemps, E. Clément, A. Debusche and of myself, concerning the regularity of the law of the solutions of some equations with coefficients with little regularity - for example diffusion processes with Hölder coefficients (but also many other examples including jump type equations, Bolzmann equation or Stochastic PDE's). Since we do not have sufficient regularity the usual approach by Malliavin calculus fails in this framework. Then one may use an alternative idea which roughly speaking is the following: We approximate the law of the random variable X (the solution of the equation at hand) by a sequence X(n) of random variables which are smooth and consequently we are able to establish integration by parts formulas for X(n) and we are able to obtain the absolutely continuity of the law of X(n) and to establish estimates for the density of the law of X(n) and for its derivatives. Notice that the derivatives of the densities of X(n) generally blow up - so we can not derive directly results concerning the density of the law of X. But, if the speed of convergence of X(n) to X is stronger than the blow up, then we may obtain results concerning the density of the law of X. It turns out that this approach fits in the framework of interpolation spaces and that the criterion of regularity for the law of X amounts to the characterization of an interpolation space between a space of distributions and a space of smooth functions. Although the theory of interpolation spaces is very well developed and one already know to characterize the interpolation spaces for Sobolev spaces of positive and negative indices, we have not found in the (huge) literature a result which covers the problem we are concerned with. So, although our result may be viewed as an interpolation result, it is a new one. The above work is treated in the paper [62] (in collaboration with Lucia Caramellino). As an application we discussed in [48] the regularity of the law of a Wiener functional under a Hörmander type non degeneracy condition.