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, 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. 
Hao Ge and Hong Qian
math-ph cond-mat.stat-mech math.DS math.MP math.PR