Method to describe stochastic dynamics using an optimal coordinate

A general method to describe the 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; and the decomposition of stationary stochastic dynamics into eigenmodes 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 by a constant distance. Such solutions have peculiar properties. For example, an optimal coordinate for stochastic dynamics with detailed balance is a multivalued function. An optimal coordinate for a random walk on a line corresponds to the conventional eigenvector of the one-dimensional Dirac equation. The equation for the optimal coordinate in a slowly varying potential reduces to the Hamilton-Jacobi equation for the action function.