Stochastic calculus of variations and mechanics

The Cauchy problem for the heat equation with a potential term (imaginary time analogue of the Schrodinger equation) is considered. After a logarithmic transformation, a positive solution to this Cauchy problem turns into a solution to the dynamic programming equation for a problem of stochastic calculus of variations. The latter problem is one of least average action. The classical principle of least action is obtained as an appropriate parameter tends to zero.