Geometric Mechanics is a field applying geometric methods to various mechanical systems , from Mechanics of particles to Fluid dynamics.In recent years it became necessary to "add noise" to such systems. The more systematic way known to do this is Quantum Mechanics, but generally there are no well defined associated probability measures.Our Stochastic Deformation method, however, allows to do that rigorously along the paths of diffusion processes, for instance.

We develop a new Stochastic Geometric Mechanics theory, extending Geometric Mechanics of classical (deterministic) dynamical systems to the case of systems with randomness, using diffusion processes with values in Lie groups. Randomness may be explained by some noisy perturbation of the system or, in a more fundamental way, as being at the heart of the phenomena usually described by deterministic equations.

For infinite dimensional mechanical systems, like hydrodynamic ones, Navier-Stokes equation, for instance, can be regarded as a stochastic deformation of Euler equation and be studied in this intrinsic probabilistic perspective.