Dynamics and kinematics of reciprocal diffusions

Abstract

This paper examines the dynamics and kinematics of reciprocal diffusions. Reciprocal processes were introduced by Bernstein in 1932, and were later studied in detail by Jamison. The reciprocal diffusions are constructed here by specifying their finite joint densities in terms of the Green’s function of a general heat operator, and an end-point density. A path integral interpretation of the heat operator Green’s function is provided, which is used to derive a stochastic form of Newton’s law, as well as a conditional distribution for the velocity of a diffusing particle given its position. These results are then employed to derive two conservation laws expressing the conservation of mass and momentum. The conservation laws do not form a closed system of equations, in general, except for two subclasses of reciprocal diffusions, the Markov and quantum diffusions.