Local Stability of McKean–Vlasov Equations Arising from Heterogeneous Gibbs Systems Using Limit of Relative Entropies |
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Authors: | Donald A. Dawson Ahmed Sid-Ali Yiqiang Q. Zhao |
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Affiliation: | School of Mathematics and Statistics, Carleton University, 1125 Colonel by Drive, Ottawa, ON K1S 5B6, Canada; (D.A.D.); (Y.Q.Z.) |
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Abstract: | A family of heterogeneous mean-field systems with jumps is analyzed. These systems are constructed as a Gibbs measure on block graphs. When the total number of particles goes to infinity, the law of large numbers is shown to hold in a multi-class context, resulting in the weak convergence of the empirical vector towards the solution of a McKean–Vlasov system of equations. We then investigate the local stability of the limiting McKean–Vlasov system through the construction of a local Lyapunov function. We first compute the limit of adequately scaled relative entropy functions associated with the explicit stationary distribution of the N-particles system. Using a Laplace principle for empirical vectors, we show that the limit takes an explicit form. Then we demonstrate that this limit satisfies a descent property, which, combined with some mild assumptions shows that it is indeed a local Lyapunov function. |
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Keywords: | McKean– Vlasov, Gibbs measure, relative entropy, Lyapunov function, jump processes, interacting particle systems, differential equations, nonlinearity |
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