Analytical Mechanics in Stochastic Dynamics: Most Probable Path, Large-Deviation Rate Function and Hamilton-Jacobi Equation

Abstract

Analytical (rational) mechanics is the mathematical structure of Newtonian deterministic dynamics developed by D'Alembert, Langrange, 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 $\dot{x}=b(x)$ are all rare events. However, conditioned on an occurence of such an event, the most probable trajectory of the stochastic motion is the solution to Lagrangian mechanics with $\mathcal{L}=\|\dot{q}-b(q)\|^2/4$ and Hamiltonian equations with $H(p,q)=\|p\|^2+b(q)\cdot p$. 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 $\nabla\times b\neq 0$ corresponds to a Newtonian system with a Lorentz force $\ddot{q}=(\nabla\times b)\times \dot{q}+1/2\nabla\|b\|^2$. 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.