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2015 Local stability of Kolmogorov forward equations for finite state nonlinear Markov processes
Amarjit Budhiraja, Paul Dupuis, Markus Fischer, Kavita Ramanan
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Electron. J. Probab. 20: 1-30 (2015). DOI: 10.1214/EJP.v20-4004


The focus of this work is on local stability of a class of nonlinear ordinary differential equations (ODE) that describe limits of empirical measures associated with finite-state exchangeable weakly interacting N -particle systems. Local Lyapunov functions are identified for several classes of such ODE, including those associated with systems with slow adaptation and Gibbs systems. Using previous results and large deviations heuristics, a partial differential equation (PDE) associated with the nonlinear ODE is introduced and it is shown that positive definite subsolutions of this PDE serve as local Lyapunov functions for the ODE. This PDE characterization is used to construct explicit Lyapunov functions for a broad class of models called locally Gibbs systems. This class of models is significantly larger than the family of Gibbs systems and several examples of such systems are presented, including models with nearest neighbor jumps and models with simultaneous jumps that arise in applications.


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Amarjit Budhiraja. Paul Dupuis. Markus Fischer. Kavita Ramanan. "Local stability of Kolmogorov forward equations for finite state nonlinear Markov processes." Electron. J. Probab. 20 1 - 30, 2015.


Accepted: 4 August 2015; Published: 2015
First available in Project Euclid: 4 June 2016

zbMATH: 1337.60240
MathSciNet: MR3383565
Digital Object Identifier: 10.1214/EJP.v20-4004

Primary: 60K35
Secondary: 34D20, 60F10, 60K25, 93D30


Vol.20 • 2015
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