Exponential stability of stationary distributions for some nonlocal transport equations: application to neural dynamic

Audric Drogoul 1 Romain Veltz 1
1 MATHNEURO - Mathématiques pour les Neurosciences
CRISAM - Inria Sophia Antipolis - Méditerranée
Abstract : In this work, we study the exponential stability of the stationary distribution for a McKean–Vlasov equation recently derived in [FL14]. We complete the convergence result proved in their paper. Our proof relies on two principal arguments. First, the linearized semigroup is positive which allows to precisely pinpoint the spectrum of the infinitesimal generator ; this is a remarkable property. Second, we use a time rescaling argument to transform the original quasilinear equation into a semilinear one. Interestingly, this convergence result can be interpreted as the existence of a locally exponentially attracting center manifold.
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Audric Drogoul, Romain Veltz. Exponential stability of stationary distributions for some nonlocal transport equations: application to neural dynamic. [Research Report] RR-8899, INRIA Sophia Antipolis - Méditerranée. 2016. 〈hal-01290264〉

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