The program of Stochastic Deformation was started in 1984-5 and initially motivated by the puzzling probabilistic content of Quantum Theory. But 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 models into probabilistic ones, along the sample paths of appropriate stochastic processes. Such processes are often referred to as reciprocal, Bernstein, 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.
One way to interpret this program is as a (stochastic) deformation of Geometric Mechanics. But it can also be regarded in the perspective of Statistical Mechanics.
For instance the original motivation of this program is known as the "Schrödinger problem" in the Mass Transportation community.