Electronic Journal of Probability

Local stability of Kolmogorov forward equations for finite state nonlinear Markov processes

Amarjit Budhiraja, Paul Dupuis, Markus Fischer, and Kavita Ramanan

Full-text: Open access

Abstract

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.

Article information

Source
Electron. J. Probab., Volume 20 (2015), paper no. 81, 30 pp.

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465067187

Digital Object Identifier
doi:10.1214/EJP.v20-4004

Mathematical Reviews number (MathSciNet)
MR3383565

Zentralblatt MATH identifier
1337.60240

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 93D30: Scalar and vector Lyapunov functions 34D20: Stability 60F10: Large deviations 60K25: Queueing theory [See also 68M20, 90B22]

Keywords
Nonlinear Markov processes weakly interact- ing particle systems interacting Markov chains mean field limit stability metastability Lyapunov functions relative entropy large deviations

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Budhiraja, Amarjit; Dupuis, Paul; Fischer, Markus; Ramanan, Kavita. Local stability of Kolmogorov forward equations for finite state nonlinear Markov processes. Electron. J. Probab. 20 (2015), paper no. 81, 30 pp. doi:10.1214/EJP.v20-4004. https://projecteuclid.org/euclid.ejp/1465067187


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