Local Stability of McKean–Vlasov Equations Arising from Heterogeneous Gibbs Systems Using Limit of Relative Entropies

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 emp...

Descripción completa

Guardado en:

Detalles Bibliográficos
Autores principales:	Donald A. Dawson, Ahmed Sid-Ali, Yiqiang Q. Zhao
Formato:	article
Lenguaje:	EN
Publicado:	MDPI AG 2021
Materias:	McKean–Vlasov Gibbs measure relative entropy Lyapunov function jump processes interacting particle systems Science Q Astrophysics QB460-466 Physics QC1-999
Acceso en línea:	https://doaj.org/article/624dda75c0f24d8a836f48fd90874496
Etiquetas:	Agregar Etiqueta Sin Etiquetas, Sea el primero en etiquetar este registro!

id	oai:doaj.org-article:624dda75c0f24d8a836f48fd90874496
record_format	dspace
spelling	oai:doaj.org-article:624dda75c0f24d8a836f48fd908744962021-11-25T17:29:25ZLocal Stability of McKean–Vlasov Equations Arising from Heterogeneous Gibbs Systems Using Limit of Relative Entropies10.3390/e231114071099-4300https://doaj.org/article/624dda75c0f24d8a836f48fd908744962021-10-01T00:00:00Zhttps://www.mdpi.com/1099-4300/23/11/1407https://doaj.org/toc/1099-4300A 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 <i>N</i>-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.Donald A. DawsonAhmed Sid-AliYiqiang Q. ZhaoMDPI AGarticleMcKean–VlasovGibbs measurerelative entropyLyapunov functionjump processesinteracting particle systemsScienceQAstrophysicsQB460-466PhysicsQC1-999ENEntropy, Vol 23, Iss 1407, p 1407 (2021)
institution	DOAJ
collection	DOAJ
language	EN
topic	McKean–Vlasov Gibbs measure relative entropy Lyapunov function jump processes interacting particle systems Science Q Astrophysics QB460-466 Physics QC1-999
spellingShingle	McKean–Vlasov Gibbs measure relative entropy Lyapunov function jump processes interacting particle systems Science Q Astrophysics QB460-466 Physics QC1-999 Donald A. Dawson Ahmed Sid-Ali Yiqiang Q. Zhao Local Stability of McKean–Vlasov Equations Arising from Heterogeneous Gibbs Systems Using Limit of Relative Entropies
description	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 <i>N</i>-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.
format	article
author	Donald A. Dawson Ahmed Sid-Ali Yiqiang Q. Zhao
author_facet	Donald A. Dawson Ahmed Sid-Ali Yiqiang Q. Zhao
author_sort	Donald A. Dawson
title	Local Stability of McKean–Vlasov Equations Arising from Heterogeneous Gibbs Systems Using Limit of Relative Entropies
title_short	Local Stability of McKean–Vlasov Equations Arising from Heterogeneous Gibbs Systems Using Limit of Relative Entropies
title_full	Local Stability of McKean–Vlasov Equations Arising from Heterogeneous Gibbs Systems Using Limit of Relative Entropies
title_fullStr	Local Stability of McKean–Vlasov Equations Arising from Heterogeneous Gibbs Systems Using Limit of Relative Entropies
title_full_unstemmed	Local Stability of McKean–Vlasov Equations Arising from Heterogeneous Gibbs Systems Using Limit of Relative Entropies
title_sort	local stability of mckean–vlasov equations arising from heterogeneous gibbs systems using limit of relative entropies
publisher	MDPI AG
publishDate	2021
url	https://doaj.org/article/624dda75c0f24d8a836f48fd90874496
work_keys_str_mv	AT donaldadawson localstabilityofmckeanvlasovequationsarisingfromheterogeneousgibbssystemsusinglimitofrelativeentropies AT ahmedsidali localstabilityofmckeanvlasovequationsarisingfromheterogeneousgibbssystemsusinglimitofrelativeentropies AT yiqiangqzhao localstabilityofmckeanvlasovequationsarisingfromheterogeneousgibbssystemsusinglimitofrelativeentropies
_version_	1718412288595066880

Local Stability of McKean–Vlasov Equations Arising from Heterogeneous Gibbs Systems Using Limit of Relative Entropies

Ejemplares similares