On Concentration Properties of Partially Observed Chaotic Systems

Daniel Paulin, Ajay Jasra, Dan Crisan, Alexandros Beskos

Research output: Contribution to journalArticlepeer-review

Abstract

In this paper we present results on the concentration properties of the smoothing and filtering distributions of some partially observed chaotic dynamical systems. We show that, rather surprisingly, for the geometric model of the Lorenz equations, as well as some other chaotic dynamical systems, the smoothing and filtering distributions do not concentrate around the true position of the signal, as the number of observations tends to ∞. Instead, under various assumptions on the observation noise, we show that the expected value of the diameter of the support of the smoothing and filtering distributions remains lower bounded by a constant multiplied by the standard deviation of the noise, independently of the number of observations. Conversely, under rather general conditions, the diameter of the support of the smoothing and filtering distributions are upper bounded by a constant multiplied by the standard deviation of the noise. To some extent, applications to the three-dimensional Lorenz 63 model and to the Lorenz 96 model of arbitrarily large dimension are considered.
Original languageEnglish
Pages (from-to)440-479
Number of pages40
JournalAdvances in Applied Probability
Volume50
Issue number2
DOIs
Publication statusPublished - 1 Jun 2018
Externally publishedYes

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