Towards a Theory of Additive Eigenvectors

TL;DR

This paper introduces a new theoretical framework for additive eigenvectors in stochastic processes, offering an alternative to the standard eigenvector approach and linking it to conditioned Markov processes.

Contribution

It develops a formal theory of additive eigenvectors, deriving their equations and demonstrating their application to one-dimensional stochastic processes.

Findings

Additive eigenvectors describe conditioned Markov processes.

Derived differential equations for additive eigenvectors.

Applied theory to the telegraph equation, revealing unique properties.

Abstract

The standard approach in solving stochastic equations is eigenvector decomposition. Using separation ansatz $P(i,t)=u(i)e^{\mu t}$ one obtains standard equation for eigenvectors $Ku=\mu u$, where $K$ is the rate matrix of the master equation. While universally accepted, the standard approach is not the only possibility. Using additive separation ansatz $S(i,t)=W(i)-\nu t$ one arrives at additive eigenvectors. Here we suggest a theory of such eigenvectors. We argue that additive eigenvectors describe conditioned Markov processes and derive corresponding equations. The formalism is applied to one-dimensional stochastic process corresponding to the telegraph equation. We derive differential equations for additive eigenvectors and explore their properties. The proposed theory of additive eigenvectors provides a new description of stochastic processes with peculiar properties.