Drifts versus forces: the Ehrenfest theorem for Markovian diffusions

Abstract

Following Stratonovich, we make a general analysis of the external force manifestations in the dynamics of Markov diffusion processes. The transformation connecting transition densities of the process with the respective (unique) Feynman-Kac kernels induces the local field of accelerations, which equals the gradient of the Feynman-Kac potential and enters the straightforward analog of the Ehrenfest theorem. The latter encompasses not only Nelson's or Zambrini's diffusions but also the familiar non-equilibrium statistical physics processes, like the standard Brownian motion in the external force field (Smoluchowski diffusions).