Open Access

# Weak-coupling limit for ergodic environments

### Abstract

The main aim of this work is to establish an averaging principle for a wide class of interacting particle systems in the continuum. This principle is an important step in the analysis of Markov evolutions and is usually applied for the associated semigroups related to backward Kolmogorov equations, c.f. [27]. Our approach is based on the study of forward Kolmogorov equations (a.k.a. Fokker-Planck equations). We describe a system evolving as a Markov process on the space of finite configurations, whereas its rates depend on the actual state of another (equilibrium) process on the space of locally finite configurations. We will show that ergodicity of the environment process implies the averaging principle for the solutions of the coupled Fokker-Planck equations.

Key words: Averaging principle, Fokker-Planck equation, interacting particle systems, weak-coupling, random evolution.

### Article Information

 Title Weak-coupling limit for ergodic environments Source Methods Funct. Anal. Topology, Vol. 25 (2019), no. 2, 118-133 MathSciNet MR3978676 Milestones Received 31/07/2018; Revised 22/01/2019 Copyright The Author(s) 2019 (CC BY-SA)

### Authors Information

Martin Friesen
Fakultät für Mathematik und Naturwissenschaften, Bergische Universität Wuppertal, Gaußstraße 20, 42119 Wuppertal, Germany

Yuri Kondratiev
Department of Mathematics, Bielefeld University, Germany

### Citation Example

Martin Friesen and Yuri Kondratiev, Weak-coupling limit for ergodic environments, Methods Funct. Anal. Topology 25 (2019), no. 2, 118-133.

### BibTex

@article {MFAT1167,
AUTHOR = {Martin Friesen and Yuri Kondratiev},
TITLE = {Weak-coupling limit for ergodic environments},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {25},
YEAR = {2019},
NUMBER = {2},
PAGES = {118-133},
ISSN = {1029-3531},
MRNUMBER = {MR3978676},
URL = {http://mfat.imath.kiev.ua/article/?id=1167},
}

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