Path integral and instanton calculus for stochastic jump processes with a finite number of states

Abstract

We develop a method to analyze stochastic jump processes with a finite number of states ${0,1,...,N}$. It is known that the system can be analyzed using su(2) Lie algebra and the spin coherent state path integral. First, we analyze a linear susceptible-infected-susceptible epidemic model by using the Lie algebraic method and path integral. Then, we apply the instanton approximation to more general models where stochastic noise can induce bimodal distributions. By using density variables, we show that instanton solution can be interpreted as a solution of the equation of motion of the deterministic system corresponding to the stochastic system. It is also shown that instanton approximation correctly reproduces the noise-induced bimodality of the system.