An exact reduction of the master equation to a strictly stable system with an explicit expression for the stationary distribution

TL;DR

This paper presents a method to reduce the dimensionality of the Master equation for finite-state Markov processes, resulting in a stable system with an explicit stationary distribution, simplifying analysis and computation.

Contribution

The authors introduce a novel reduction technique that transforms the Master equation into a stable, lower-dimensional system with an explicit stationary distribution expression.

Findings

Reduces the Master equation by one dimension using probability normalization.

Produces a stable, affine differential equation with a non-singular generator matrix.

Provides an explicit formula for the stationary distribution.

Abstract

The evolution of a continuous time Markov process with a finite number of states is usually calculated by the Master equation - a linear differential equations with a singular generator matrix. We derive a general method for reducing the dimensionality of the Master equation by one by using the probability normalization constraint, thus obtaining a affine differential equation with a (non-singular) stable generator matrix. Additionally, the reduced form yields a simple explicit expression for the stationary probability distribution, which is usually derived implicitly. Finally, we discuss the application of this method to stochastic differential equations.