Stochastic Analysis is the analytical study of phenomena with a random component. It is based on works of Wiener, Itô, Cameron-Martin, Girsanov and ˆothers, and can be also interpreted as the mathematical framework underlying Feynman path integral approach to Quantum Physics. We have been using especially Malliavin calculus of variations, a theory of differentiation on the path space equipped with the Wiener measure, which is adapted to very irregular functionals like the ones generated by Itô calculus.
Recently a research line has been developed using geometric as well as stochastic methods in classical Hydrodynamics. The starting point is Arnold's approach: in a famous paper, published in 1966, Arnold has shown that the Euler flow coincides with a geodesic in the group of volume preserving diffeomorphisms of the underlying manifold with respect to the L2 metric. Since then many works have been published on, in particular, the geometry of this (infinite dimensional) group and its consequences, for example, for the stability of the fluids motion. Our work proceeds mainly along two lines:
- The geometry of the volume preserving diffeomorphisms group:
The classical approach of Banach-modeled charts for the study of infinite dimensional Riemannian geometry is systematically replaced by the use of Malliavin-Itô charts . The geometry is developed as a stochastic Cartan–type geometry using a frame bundle approach. In fluid dynamics the escape of the energy from low to high modes induces a lack of compactness. The key point of our approach is the control of this energy transfer. It allows to study some random perturbations of the hydrodynamical equations and also to derive (by stochastic methods) results on the deterministic motions, in particular concerning their ergodicity.
- Generalization of Arnold's approach to Navier-Stokes equations
(see also "Geometric
Methods in Probability"):
We have proved a stochastic variational characterization of Navier-Stokes equation, regarded as a random perturbation of Euler flow treated by Arnold (when the underlying manifold is the two-dimensional torus): This variational principle can be reinterpreted through a Bismut-type integration by parts formula on the space of Brownian motion on the Lie group of measure preserving diffeomorphisms. Among possible consequences of our approach is the study of stability of the Navier-Stokes flow.