Section: New Results
Stochastic Analysis and Malliavin calculus

Invariance principles for stochastic polynomials [40].
With L. Caramellino (Roma), V. Bally has studied invariance principles for stochastic polynomials. This is a generalization of the classical invariance principle from the Central Limit Theorem, of interest in $U$statistics. The main contribution concerns convergence in total variation distance, using an abstract variant of Malliavin calculus for general random variables which verify a Doeblin type condition.

Convergence in distribution norms in the Central Limit Theorem and Edgworth expansions [39] (V. Bally, L. Caramellino and G. Poly).
The convergence in "distribution norms" represents an extension of the convergence in total variation distance which permits to take into account some singular phenomenons. The main tool is the abstract Malliavin calculus mentioned above. Several examples are given in the paper and an outstanding application concerns the estimates of the number of roots of trigonometric polynomials. considered in a second paper; see [40].

Bolzmann equation and Piecewise Deterministic Markov Processes. (see [41], [37]). In collaboration with D. Goreac and V. Rabiet, V. Bally has studied the regularity of the semigroup of $PDM{P}^{\text{'}}s$ and, as an application estimates of the distance between two such semigroups. An interesting example is given by the two dimensional homogeneous Bolzmann equation. Furthermore, V. Bally obtained some exponential estimates for the function solution of this equation.