2024/11/21 by Stein Andreas Bethuelsen, Bethuelsen, Stein Andreas, Florian Völlering +1
Biochemistry, Genetics and Molecular Biology · Mathematics · Physics and Astronomy · #Diffusion and Search Dynamics #FOS: Mathematics #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.2411.13926
openalex publication_date 2024/11/21 · openalex created_date 2024/11/24 · openalex updated_date 2026/07/28
Consider the dynamic environment governed by a Poissonian field of independent particles evolving as simple random walks on ℤd. The random walk on random walks model refers to a particular stochastic process on ℤd whose evolution at time t depends on the number of such particles at its location. We derive classical limit theorems for this instrumental model of a random walk in a dynamic random environment, applicable in sufficiently high dimensions. More precisely, for d ≥ 5, we prove a strong law of large numbers and large deviation estimates. Further, for d≥ 9, we obtain a functional central limit theorem under the annealed law. These results are non-perturbative in the sense that they hold for any positive density of the Poissonian field. Under the aforementioned assumptions on the dimension they therefore improve on previous work on the model. Moreover, they stand in contrast to the anomalous behaviour predicted in low dimensions.