# Stochastic Deformation

The program of Stochastic Deformation was started in 1984-5 and initially motivated by the puzzling probabilistic content of Quantum Theory. Part of the results obtained there provided a mathematical re-interpretation of Feynman Path Integral approach, in terms of two adjoint parabolic equations. The method, however, is considerably more general and can be regarded as a systematic way to deform classical structures into probabilistic ones, along the sample paths of appropriate stochastic processes. Such processes are often referred to as Bernstein reciprocal, variational, or even local Markov or two-sided Markov.

Feynman considered informally only diffusion and jump processes but our Stochastic deformation is, in principle, applicable to any kind of processes. All the associated probability measures are intrinsically invariant under time reversal in a more general sense than the one traditionally considered by probabilists.

This program can also be regarded in the perspective of Statistical Mechanics, more precisely Quantum Statistical Physics.

The community of Mass transportation has recently re-discovered the "Schroedinger problem" (1931) which is, in fact, at the origin of our Stochastic Deformation. It is now in fast development in various foreign academic institutions.