2014/01/31 by Patrick Rebeschini, Ramon van Handel · 6 citations
Mathematics · Physics and Astronomy · #Applied mathematics #Calculus (dental) #Computer science #Epistemology #Ergodic theory #Ergodicity #Filter (signal processing) #Markov Chains and Monte Carlo Methods #Markov chain #Markov process #Mathematical Dynamics and Fractals #Mathematical economics #Mathematics #Nonlinear system #Phase transition #Physics #Probability theory #Pure mathematics #Scope (computer science) #Simple (philosophy) #Stationary ergodic process #Statistical mechanics #Statistical physics #Statistics #Stochastic processes and statistical mechanics #math-ph #math.MP #math.PR #msc:37A50 #msc:60G35 #msc:60K35 #msc:82B26 #msc:82B44
paper · pdf · doi:10.1214/ejp.v20-3281
published in Electronic Journal of Probability 20(none) (Institute of Mathematical Statistics) · 51 pages
arxiv created 2014/12/08 · openalex publication_date 2015/01/01 · arxiv updated 2015/02/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
It has been established under very general conditions that the ergodic properties of Markov processes are inherited by their conditional distributions given partial information. While the existing theory provides a rather complete picture of classical filtering models, many infinite dimensional problems are outside its scope. Far from being a technical issue, the infinite dimensional setting gives rise to surprising phenomena and new questions in filtering theory. The aim of this paper is to discuss some elementary examples, conjectures, and general theory that arise in this setting, and to highlight connections with problems in statistical mechanics and ergodic theory. In particular, we exhibit a simple example of a uniformly ergodic model in which ergodicity of the filter undergoes a phase transition, and we develop some qualitative understanding as to when such phenomena can and cannot occur.We also discuss closely related problems in the setting of conditional Markov random fields.