Equivalent Markov processes under gauge group

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 to resolve equations, included the case in a previous work in which the sample space is time-dependent [Phys. Rev. E 90, 022125 (2014)]. We found a general solution through dilation of the state space, although the prior probability distribution of the states defined in this new space takes smaller values with respect to that 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 the Markov process in general could be linked via gauge (local) transformations, and we present some illustrative examples for this result.

Figure 1

Diagram associated with the finite pure birth process with the infinitesimal generator ${\mathtt{Q}}_{+}$.

Figure 2

Diagram associated with the finite pure death process with the infinitesimal generator ${\mathtt{Q}}_{-}$.

Figure 3

This diagram shows the composed process on the state space $\mathbf{S}\approx {\mathcal{S}}^{\prime }$, in the sense that $\left|\mathbf{S}\right|=|{\mathcal{S}}^{\prime }|$, with the infinitesimal generator $\mathbf{Q}$.