Duality in stochastic processes from the viewpoint of basis expansions

Abstract

A new derivation method of duality relations in stochastic processes is proposed. The current focus is on the duality between stochastic differential equations and birth-death processes. Although previous derivation methods have been based on the viewpoint of time-evolution operators, the current derivation is based on basis expansions. In addition, only the tool needed for the derivation is the integration by parts, which is rather simple and understandable. The viewpoint of basis expansions enables us to obtain various dual stochastic processes. As a demonstration, dual processes based on Taylor-type and Hermite polynomials are derived.

Numerical methods to evaluate Koopman matrix from system equations

“…After a brief review of the stochastic differential equations, the connection with the Koopman operator is introduced. Thereafter, based on the discussion for the dual stochastic process reported in [12,13], a formal discussion for the evaluation of the Koopman matrix has been explained. Furthermore, a novel numerical technique based on extrapolation has been proposed.…”

“…is the adjoint operator of L. As discussed in [13], the adjoint operator can be easily obtained on performing integration by parts. Further, the function ϕ α (x, t) obeys the following time-evolution equation (the backward Kolmogorov equation):…”

“…Here, the simple functions {x n } are employed as the basis. Although other basis functions such as Hermite polynomials as shown in [13], may be employed, for simplicity, the following explanation has been restricted to the cases with the simple basis.…”

“…After recovering stochasticity as reported in [12,13], the derived equations correspond to chemical master equations, which can be numerically solved using conventional Runge-Kutta methods or Monte Carlo methods such as the Gillespie algorithm [17]. However, the approach based on coupled ordinary differential equations is not employed here.…”

“…In addition, a concept of a dual stochastic process has been employed to obtain the Koopman matrix. First, a basis expansion provides a dual (stochastic) process from the backward Kolmogorov equation [12,13], and the discussions based on combinatorics are available. Second, the connection between the dual process and the Koopman matrix has been clarified.…”