Open Access
2023 Gibbsian dynamics and the generalized Langevin equation
David P. Herzog, Jonathan C. Mattingly, Hung D. Nguyen
Author Affiliations +
Electron. J. Probab. 28: 1-29 (2023). DOI: 10.1214/23-EJP904


We study the statistically invariant structures of the nonlinear generalized Langevin equation (GLE) with a power-law memory kernel. For a broad class of memory kernels, including those in the subdiffusive regime, we construct solutions of the GLE using a Gibbsian framework, which does not rely on existing Markovian approximations. Moreover, we provide conditions on the decay of the memory to ensure uniqueness of statistically steady states, generalizing previous known results for the GLE under particular kernels as a sum of exponentials.

Funding Statement

The authors graciously acknowledge support from the Department of Mathematics at Duke University and the Department of Mathematics at Iowa State University. DPH was supported in part by NSF Grants DMS-1612898 and DMS-1855504. JCM thanks the NSF for its partial support through the grant DMS-1613337.


The authors would like to thank Gustavo Didier for helpful discussions in the development of this work. The authors also would like to thank the anonymous reviewer for their valuable comments and suggestions.


Download Citation

David P. Herzog. Jonathan C. Mattingly. Hung D. Nguyen. "Gibbsian dynamics and the generalized Langevin equation." Electron. J. Probab. 28 1 - 29, 2023.


Received: 8 November 2021; Accepted: 12 January 2023; Published: 2023
First available in Project Euclid: 20 January 2023

Digital Object Identifier: 10.1214/23-EJP904

Primary: 60H10

Keywords: Gibbsian dynamics , Invariant measures , Langevin equation with memory

Vol.28 • 2023
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