2018/05/16 by Sergei V. Krivov, Krivov, Sergei V.
Physics and Astronomy · #Chemical Physics (physics.chem-ph) #FOS: Physical sciences #Quantum Physics (quant-ph) #Statistical Mechanics (cond-mat.stat-mech) #cond-mat.stat-mech #physics.chem-ph #quant-ph
paper · pdf · doi:10.48550/arxiv.1805.06455
fixed typos, theory of quasi-stationary distributions was used to derive the equations. moved part of the material to Appendix
arxiv created 2018/09/25 · arxiv updated 2018/09/26
The standard approach in solving stochastic equations is eigenvector decomposition. Using separation ansatz P(i,t)=u(i)eμt one obtains standard equation for eigenvectors Ku=μu, 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)-νt 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.