Reciprocal diffusions in flat space

Abstract

We describe the theory of reciprocal diffusions in flat space. A reciprocal process is a Markov random field on a one dimensional parameter space. Every Markov process is reciprocal but not vice versa. We descibe the first and second order mean differential characteristics of reciprocal diffusions. This includes a new definition of stochastic acceleration. We show that reciprocal diffusions satisfy stochastic differential equations of second order. Associated to a reciprocal diffusion is a sequence of conservation laws, the first two of which are the familiar continuity and Euler equations. There are two cases where these laws can be closed after the first two. They are the mutually exclusive subclasses of Markov and quantum diffusions. The latter corresponds to solutions of the Schrodinger equation and may be part of a stochastic description of quantum mechanics.