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Section: New Results

Numerical Probability

Regularity of probability laws using an interpolation method

This work was motivated by previous studies by N. Fournier, J. Printemps, E. Clément, A. Debusche and V. Bally, on the regularity of the law of the solutions of stochastic differential equations with low regularity coefficients - such as diffusion processes with Hölder coefficients or many other examples including jump type equations, Boltzmann equation or Stochastic PDE's. Since we do not have sufficient regularity, the usual approach by Malliavin calculus fails in this framework. We use the following alternative idea: We approximate the law of the random variable $X$ (the solution of the equation at hand) by a sequence $X\left(n\right)$ of random variables which are smooth. Consequently we are able to establish integration by parts formulas for $X\left(n\right)$, to obtain the absolutely continuity of the law of $X\left(n\right)$, and to establish estimates for the density of the law of $X\left(n\right)$ and its derivatives. Note that the derivatives of the densities of $X\left(n\right)$ 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 $X\left(n\right)$ to $X$ is faster 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 knows how 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 [48] by V. Bally and Lucia Caramellino. As an application we discussed in [50] the regularity of the law of a Wiener functional under a Hörmander type non degeneracy condition.

A stochastic parametrix representation for the density of a Markov process.

Classical results of PDE theory (due to A. Friedmann) assert that, under uniform ellipticity conditions, the law of a diffusion process has a continuous density (the approach of A. Friedmann is analytical and concerns PDE's instead of the corresponding diffusion process). The method developed by A. Friedmann is known as the "parametrix method". V. Bally In collaboration with A. Kohatzu Higa gave a probabilistic approach which represents the probabilistic counterpart of the parametrix method [33] . They obtained a probabilistic representation for the density of the law of the solution of a SDE and more generally, for a class of Markov processes including solutions of jump type SDE's. This representation may be considered as a perfect simulation scheme and so represents a starting point for Monte Carlo simulation. However the random variable which appears in the stochastic representation has infinite variance, so direct simulation gives unstable results (as some preliminary tests have proved). In order to obtain an efficient simulation scheme some more work on the reduction of variance has to be done - and this does not seem trivial.

The distance between two density functions and convergence in total variation.

V. Bally and L. Caramellino have obtained estimates of the distance between the densities of the law of two random variables using an abstract variant of Malliavin calculus. They used these estimates in order to study the convergence in total variation of a sequence of random variables. This has been done in [49] . They are now working on more specific examples concerning the Central Limit Theorem [32] . In the last years the convergence in entropy distance and in total variation distance for several variants of the CLT has been considered in papers by S. Bobkov, F. Götze, G. Peccati, Y. Nourdin, D. Nualart and G. Poly. This is a very active research. Moreover, in an working paper in collaboration with his Phd student R. Clement, V. Bally uses similar methods in order to study the total variation distance between two Markov semigroups and for approximation schemes purposes. A special interest is devoted to higher order schemes such as the Victoir Nyomia scheme.

An invariance principle for stochastic series (U- Statistics).

Vlad Bally and Lucia Caramellino are working on invariance principles for stochastic series of polynomial type. In the case of polynomials of degree one we must have the classical Central Limit Theorem (for random variables which are not identically distributed). For polynomials of higher order we are in the framework of the so called U statistics which have been introduced by Hoffdings in t 1948 and which play an important role in modern statistics. Our contribution in this topic concerns convergence in total variation distance for this type of objects. We use abstract Malliavin calculus and more generally, the methods mentioned in the above paragraph.