ANALYTICAL MECHANICS IN STOCHASTIC DYNAMICS: MOST PROBABLE PATH, LARGE-DEVIATION RATE FUNCTION AND HAMILTON-JACOBI EQUATION

Analytical (rational) mechanics is the mathematical structure of Newtonian deterministic dynamics developed by D'Alembert, Lagrange, Hamilton, Jacobi, and many other luminaries of applied mathematics. Diffusion as a stochastic process of an overdamped individual particle immersed in a fluid, initiated by Einstein, Smoluchowski, Langevin and Wiener, has no momentum since its path is nowhere differentiable. In this exposition, we illustrate how analytical mechanics arises in stochastic dynamics from a randomly perturbed ordinary differential equation dX(t) = b(X-t) dt + epsilon dW(t), where W-t is a Brownian motion. In the limit of vanishingly small epsilon, the solution to the stochastic differential equation other than (x) over dot = b(x) are all rare events. However, conditioned on an occurrence of such an event, the most probable trajectory of the stochastic motion is the solution to Lagrangian mechanics with L = parallel to(q) over dot - b(q)parallel to(2)/4 and Hamiltonian equations with H(p, q) = parallel to p parallel to(2) + b(q).p. Hamiltonian conservation law implies that the most probable trajectory for a rare event has a uniform excess kinetic energy along its path. Rare events can also be characterized by the principle of large deviations which expresses the probability density function for X-t as f(x, t) = e(-u(x,t)/epsilon), where u(x,t) is called a large-deviation rate function which satisfies the corresponding Hamilton-Jacobi equation. An irreversible diffusion process with del x b not equal 0 corresponds to a Newtonian system with a Lorentz force (sic) = (del x b) x (q) over dot + (1/2)del parallel to b parallel to(2). The connection between stochastic motion and analytical mechanics can be explored in terms of various techniques of applied mathematics, for example, singular perturbations, viscosity solutions and integrable systems.