Periodic-orbit determination of dynamical correlations in stochastic processes

TL;DR

This paper demonstrates that large deviation statistics and time correlation functions of finite state Markov processes can be effectively analyzed using the periodic orbits of their associated Kalman maps, linking stochastic and deterministic dynamics.

Contribution

It establishes a connection between Markov processes and deterministic maps, enabling analysis of stochastic correlations via periodic orbits in the Kalman map.

Findings

Large deviation quantities match between Markov processes and Kalman maps.

Time correlations in Markov processes can be described by unstable periodic orbits.

The approach provides a new perspective on stochastic-deterministic correspondence.

Abstract

It is shown that large deviation statistical quantities of the discrete time, finite state Markov process $P_{n+1}^{(j)}=\sum_{k=1}^NH_{jk}P_n^{(k)}$, where P_n^{(j)} is the probability for the j-state at the time step n and H_{jk} is the transition probability, completely coincides with those from the Kalman map corresponding to the above Markov process. Furthermore, it is demonstrated that by using simple examples, time correlation functions in finite state Markov processes can be well described in terms of unstable periodic orbits embedded in the equivalent Kalman maps.