Critical Ornstein-Uhlenbeck processes

The Ornstein-Uhlenbeck position process with the invariant measure is shown to satisfy a variational principle quite analogous to Hamilton's least action principle of classical mechanics. To prove this, a stochastic calculus of variations is developed for processes with differentiable sample paths, and which form a diffusion together with their derivative. The key tool in the derivation of stochastic Euler-Lagrange-type equations is a symmetric variant of Nelson's integration by parts formula for semimartingales simultaneously adapted to an increasing and a decreasing family of σ-algebras. An energy conservation theorem is also proved.