Critical Transitions in Interacting Particle Systems: An Onsager-Machlup Action Functional Framework
This paper establishes an indirect approximation theorem for the most probable transition pathway of a stochastic interacting particle system in the mean-field framework. This paper studied the problem of indirect approximation of the most probable transition pathway of an interacting particle system (i.e., a high-dimensional stochastic dynamical system) and its mean field limit equation (McKean-Vlasov stochastic differential equation). This study is based on the Onsager-Machlup action functional, reformulated the problem as an optimal control problem. This paper completes the derivation using the stochastic Pontryagin's Maximum Principle. This paper proves the existence and uniqueness theorem for the solution to the mean-field optimal control problem of McKean-Vlasov stochastic differential equations and establishes a system of equations that determine the control parameters $θ^{*}$ and $θ^{N}$ respectively. There are few studies on the most probable transition pathways of stochastic interacting particle systems, it is still a great challenge to solve the most probable transition pathways directly or to approximate it with the mean field limit system. Therefore, this paper first gave the proof of correspondence between the core equation of Pontryagin's Maximum Principle, that is, Hamiltonian extreme condition equation. In other words, this relationship indirectly illustrates the correspondence between the most probable transition pathways of stochastic interacting particle systems and those of mean-field systems.