Potentials of Continuous Markov Process and Random Perturbations

TL;DR

This paper explores the decomposition of vector fields in diffusion processes using scalar and bivector potentials, providing probabilistic interpretations and linking to thermodynamics and Lyapunov functions.

Contribution

It introduces a novel decomposition of diffusion drift fields with probabilistic meaning and demonstrates the existence of generalized gradient forms in deterministic systems via random perturbations.

Findings

Decomposition of vector fields into gradient, perpendicular, and divergence-free components.

Probabilistic interpretation of cycle velocity in thermodynamics.

Deterministic systems can have generalized gradient forms with Lyapunov functions.

Abstract

With a scalar potential and a bivector potential, the vector field associated with the drift of a diffusion is decomposed into a generalized gradient field, a field perpendicular to the gradient, and a divergence-free field. We give such decomposition a probabilistic interpretation by introducing cycle velocity from a bivectorial formalism of nonequilibrium thermodynamics. New understandings on the mean rates of thermodynamic quantities are presented. Deterministic dynamical system is further proven to admit a generalized gradient form with the emerged potential as the Lyapunov function by the method of random perturbations.