## Electronic Journal of Probability

### Phase transitions in nonlinear filtering

#### Abstract

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.

#### Article information

Source
Electron. J. Probab., Volume 20 (2015), paper no. 7, 46 pp.

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

https://projecteuclid.org/euclid.ejp/1465067113

Digital Object Identifier
doi:10.1214/EJP.v20-3281

Mathematical Reviews number (MathSciNet)
MR3311220

Zentralblatt MATH identifier
1351.37043

Rights

#### Citation

Rebeschini, Patrick; van Handel, Ramon. Phase transitions in nonlinear filtering. Electron. J. Probab. 20 (2015), paper no. 7, 46 pp. doi:10.1214/EJP.v20-3281. https://projecteuclid.org/euclid.ejp/1465067113

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