Optimal Coordinate as a General Method in Stochastic Dynamics

Abstract: A general method to describe stochastic dynamics of Markov processes is suggested. The method aims to solve three related problems. The determination of an optimal coordinate for the description of stochastic dynamics. The reconstruction of time from an ensemble of stochastic trajectories. The decomposition of stationary stochastic dynamics on eigen-modes which do not decay exponentially with time. The problems are solved by introducing additive eigenvectors which are transformed by a stochastic matrix in a simple way - every component is translated on a constant distance. Such solutions have peculiar properties. For example, an optimal coordinate for stochastic dynamics with detailed balance is a multi-valued function. An optimal coordinate for a random walk on the line corresponds to the conventional eigenvector of the one dimensional Dirac equation. The equation for the optimal coordinate in a slow varying potential reduces to the Hamilton-Jacobi equation for the action function.