Abstract
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.
Acknowledgments
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.
Citation
David P. Herzog. Jonathan C. Mattingly. Hung D. Nguyen. "Gibbsian dynamics and the generalized Langevin equation." Electron. J. Probab. 28 1 - 29, 2023. https://doi.org/10.1214/23-EJP904
Information
