Journal of the
Korean Mathematical Society JKMS

This is an introduction to a stochastic version of E. Cartan's symplectic mechanics. A class of time-symmetric (``Bernstein") diffusion processes is used to deform stochastically the exterior derivative of the Poincar\'e--Cartan one-form on the extended phase space. The resulting symplectic two-form is shown to contain the (a.e.) dynamical laws of the diffusions. This can be regarded as a geometrization of Feynman's path integral approach to quantum theory; when Planck's constant reduce to zero, we recover Cartan's mechanics. The underlying strategy is the one of ``Euclidean Quantum Mechanics".

Keywords: Feynman integrals, diffusions, symplectic mechanics

MSC numbers: Primary:81S40, 58D30, Secondary:60H07, 60J70