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Geometric methods in probability

A general line of research of GFMUL is the study of geometry of the various path spaces arising in Stochastic Analysis.

Another one could be called a program of stochastic deformation, in the same sense as Quantum Mechanics can be regarded as deformation of Classical Mechanics. In particular, geometrical structures designed originally for underlying smooth trajectories are deformed as little as possible so as to become compatible with the very irregular realizations of some suited Markovian stochastic processes. For instance, symplectic geometry was designed for smooth solutions of Hamiltonian equations of motion. In recent years, we have deformed the classical tools of this geometry for the need of diffusion processes. New, unexpected, relations between them arise in this way, and are relevant even in stochastic finance applications.

Another example of our deformation strategy is classical Hydrodynamics. One can regard Navier-Stokes equation as a deformation of Euler equation. The geometry of Eulerian flows, studied by V. Arnold and others, can also be deformed in the same perspective. (Cf. "Stochastic analysis")

The method advocated seems to be of interest much more generally, for the geometrical study of a number of PDE.