Equivalent Markov processes under gauge group

TL;DR

This paper demonstrates that all Markov processes on countable state spaces can be connected through gauge transformations, providing a new perspective and tools for analyzing and solving these processes.

Contribution

It establishes a general framework linking Markov processes via gauge transformations and illustrates how this approach can be used to analyze and solve such processes.

Findings

All Markov processes are connected via gauge transformations.

Gauge transformations can be used to solve equations related to Markov processes.

Examples illustrate the application of gauge transformations to Markov processes.

Abstract

We have studied Markov processes on denumerable state space and continuous time. We found that all these processes are connected via gauge transformations. We have used this result before as a method for resolution of equations, included the case where the sample space is time dependent in a previous work \textit{Phys. Rev. E} \textbf{90}, 022125 (2014). We found a general solution through a dilation of the state space, although the prior probability distribution of the states defined in this new space takes smaller values with respect to the one in the initial problem. The gauge (local) group of dilations modifies the distribution on the dilated space to restore the original process. In this work we show how Markov process in general could be linked via gauge (local) transformations and we present some illustrative examples for this results.