Abstract
We prove the large deviation principle for the trajectory of a broad class of mean field interacting Markov jump processes via a general analytic approach based on viscosity solutions. Examples include generalized Ehrenfest models as well as Curie-Weiss spin flip dynamics with singular jump rates. The main step in the proof of the large deviation principle, which is of independent interest, is the proof of the comparison principle for an associated collection of Hamilton-Jacobi equations. Additionally, we show that the large deviation principle provides a general method to identify a Lyapunov function for the associated McKean-Vlasov equation.
Highlights
We consider two models of Markov jump processes with mean-field interaction
We have n particles or spins that evolve as a pure jump process, where the jump rates of the individual particles depend on the empirical distribution of all n particles
We prove the large deviation principle (LDP) for the trajectory of these empirical quantities, with Lagrangian rate function, via a proof that an associated Hamilton–Jacobi equation has a unique viscosity solution
Summary
We consider two models of Markov jump processes with mean-field interaction. In both cases, we have n particles or spins that evolve as a pure jump process, where the jump rates of the individual particles depend on the empirical distribution of all n particles. To x(t), or μ(t), the solution of a McKean–Vlasov equation, which is a generalization of the linear Kolmogorov forward equation which would appear in the case of independent particles For these sets of models, we obtain a LDP for the trajectory of these empirical measures on the space DEi (R+), i ∈ {1, 2} of càdlàg paths on Ei of the form. We verify conditions from [22] that are necessary to obtain our large deviation result with a rate function in Lagrangian form, in the case that we have uniqueness of solutions to the Hamilton–Jacobi equations.
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