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We develop a variational scheme in a field theoretic approach to a stochasticprocess. While various stochastic processes can be expressed using masterequations, in general it is difficult to solve the master equations exactly,and it is also hard to solve the master equations numerically because of thecurse of dimensionality. The field theoretic approach has been used in order tostudy such complicated master equations, and the variational scheme achievestremendous reduction in the dimensionality of master equations. For thevariational method, only the Poisson ansatz has been used, in which onerestricts the variational function to a Poisson distribution. Hence, one hasdealt with only restricted fluctuation effects. We develop the variationalmethod further, which enables us to treat an arbitrary variational function. Itis shown that the variational scheme developed gives a quantitatively goodapproximation for master equations which describe a stochastic gene regulatorynetwork.

A field theoretic approach to master equations and a variational method beyond the Poisson ansatz