Schrödinger conditional brownian motion and stochastic calculus of variations

Abstract

We present a new concept of Schrodinger conditional Brownian motion, which may be considered as a generalization of the h-process given by Doob in the case that the harmonic function h is replaced by a positive solution u of Schrodinger equation M = 0.The potential term V is in the basic class K which includes unbounded functions such as the Coulomb potential.We prove that the change from the Brownian motion to the “Schrodinger process” corresponds to the change from zero drift to the drift V(lnu). Moreover, we prove that the Schrodinger process and drift V(ln u)provide the solution of the stochastic calculus of variations corresponding to a problem of least average action. This is the same type as the stochastic optimal control problem originated by Fleming.