Eder, O.J., Lackner, T., & Posch, M. ORCID: https://orcid.org/0000000186499129 (1985). Green’sfunction solution for a special class of master equations. Physical Review A 31 (1) 366377. 10.1103/PhysRevA.31.366.

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Abstract
We consider a onedimensional stochastic process described by a master equation and calculate the timedependent distribution function. The time evolution of the system is given by the conditional probability h(x,t‖x0), where x and x0 are continuous variables. Assuming a quite general dependence of the transition probability WΩ(x→x’) on a parameter Ω, we show that the backward form of the master equation can be used to calculate arbitrary conditional averages 〈f(x)‖x0⟩t up to any given power in Ω−1. This general expansion procedure will be used to construct h(x,t‖x0) itself. We show that—introducing a new stochastic variable y—the conditional probability h(x,t‖x0) can be expanded into a series of Hermite functions. The coefficients of this expansion, bn(t), which depend on the expansion parameter Ω, can be uniquely determined via a recursion relation. We show that in the limit Ω→∞ all coefficients bn(t) vanish, except b0, which is time independent. In this limit a Gaussian distribution for the conditional probability is obtained, which is in agreement with the socalled linear noise approximation.
Item Type:  Article 

Research Programs:  Acid Rain Program (ACI) 
Depositing User:  Romeo Molina 
Date Deposited:  15 Apr 2016 11:33 
Last Modified:  27 Aug 2021 17:26 
URI:  http://pure.iiasa.ac.at/12722 
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