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2015 Phase transitions in nonlinear filtering
Patrick Rebeschini, Ramon van Handel
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Electron. J. Probab. 20: 1-46 (2015). DOI: 10.1214/EJP.v20-3281


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.


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Patrick Rebeschini. Ramon van Handel. "Phase transitions in nonlinear filtering." Electron. J. Probab. 20 1 - 46, 2015.


Accepted: 1 February 2015; Published: 2015
First available in Project Euclid: 4 June 2016

zbMATH: 1351.37043
MathSciNet: MR3311220
Digital Object Identifier: 10.1214/EJP.v20-3281

Primary: 37A50
Secondary: 60G35 , 60K35 , 82B26 , 82B44

Keywords: conditional ergodicity and mixing , filtering in infinite dimension , Phase transitions


Vol.20 • 2015
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