Phase transitions in nonlinear filtering

Patrick Rebeschini, Ramon van Handel

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


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.

Original languageEnglish (US)
JournalElectronic Journal of Probability
StatePublished - 2015

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


  • Conditional ergodicity and mixing
  • Filtering in infinite dimension
  • Phase transitions


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