Exponential stability of the stationary distribution of a mean field of spiking neural network

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:270

Article

Drogoul, Audric¹  Veltz, Romain¹ 

[1] Univ Cote Azut, INRIA, 2004 Route Lucioles,BP 93, F-06902 Sophia Antipolis, France

关键词: McKean-Vlasov equations;    Nonlocal nonlinear transport equation;    Boundary condition;    Stationary distribution;    Nonlinear stability;    Center manifold;   

DOI  :  10.1016/j.jde.2020.08.001

来源: Elsevier

【 摘 要 】

In this work, we study the exponential stability of the stationary distribution of a McKean-Vlasov equation, of nonlinear hyperbolic type which was recently derived in [1,2]. We complement the convergence result proved in [2] using tools from dynamical systems theory. Our proof relies on two principal arguments in addition to a Picard-like iteration method. First, the linearized semigroup is positive which allows to precisely pinpoint the spectrum of the infinitesimal generator. Second, we use a time rescaling argument to transform the original quasilinear equation into another one for which the nonlinear flow is differentiable. Interestingly, this convergence result can be interpreted as the existence of a locally exponentially attracting center manifold for a hyperbolic equation. (C) 2020 Published by Elsevier Inc.

【 授权许可】

Free