We prove a quenched invariance principle for continuous-time random walks in a dynamically averaging environment on . In the beginning, the conductances may fluctuate substantially, but we assume that as time proceeds, the fluctuations decrease according to a typical diffusive scaling and eventually approach constant unit conductances. The proof relies on a coupling with the standard continuous time simple random walk.
The authors acknowledge support from the DFG through Scientific Network Stochastic Processes on Evolving Networks”. The research of CM is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)–443916008. The authors would like to acknowledge the financial support of the CogniGron research center and the Ubbo Emmius Funds (Univ. of Groningen).
The authors thank the anonymous referee for reading through the entire manuscript thoroughly and for providing us with high-quality and constructive feedback. In particular, the referee suggested the extension to the quenched invariance principle. The authors thank H. Pitters for inspiring discussions in the early phase of this project.
"Quenched invariance principle for random walks on dynamically averaging random conductances." Electron. Commun. Probab. 26 1 - 13, 2021. https://doi.org/10.1214/21-ECP440