2014/06/05 by Benjamin Fehrman, Benjamin J. Fehrman, Fehrman, Benjamin J.
Computer Science · Mathematics · #35B27 #35B53 #60J60 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Probability (math.PR) #Stochastic processes and statistical mechanics #advanced mathematical theories #math.AP #math.PR #msc:35B27 #msc:35B53 #msc:60J60
paper · pdf · doi:10.48550/arxiv.1406.1549
28 pages. arXiv admin note: substantial text overlap with arXiv:1404.5274
arxiv created 2014/06/05 · openalex publication_date 2014/06/05 · arxiv updated 2014/06/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We obtain a Liouville property for stationary diffusions in random environment which are small, isotropic perturbations of Brownian motion in spacial dimension greater than two. Precisely, we prove that, on a subset of full probability, the constant functions are the only strictly sub-linear maps which are invariant with respect to the evolution of the diffusion. And, we prove that the constant functions are the only bounded, ancient maps which are invariant under the evolution. These results depend upon the previous work of Fehrman [3] and Sznitman and Zeitouni [7] and, in the first case, our methods are motivated by the work, in the discrete setting, of Benjamini, Duminil-Copin, Kozma and Yadin [1].