Convergence of invariant densities in the small-noise limit

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8 Scopus citations

Abstract

Let ρ0 be an invariant probability density of a deterministic dynamical system f and ρε the invariant probability density of a random perturbation of f by additive noise of amplitude ε. Suppose ρ0 is stochastically stable in the sense that ρε → 0 as ε → 0. Through a systematic numerical study of concrete examples, I show that: The rate of convergence of ρε to 7rho;ε0 as ε → 0 is frequently governed by power laws: ∥ρε - ρ01 ∼ ε γ for some γ > 0. When the deterministic system / exhibits exponential decay of correlations, a simple heuristic can correctly predict the exponent γ based on the structure of ρ0. The heuristic fails for systems with some 'intermittency', i.e. systems which do not exhibit exponential decay of correlations. For these examples, the convergence of ρε to ρ0 as ε → 0 continues to be governed by power laws but the heuristic provides only an upper bound on the power law exponent γ. Furthermore, this numerical study requires the computation of ∥ρε - ρ0∥ 1 for 1.5-2.5 decades of ε and provides an opportunity to discuss and compare standard numerical methods for computing invariant probability densities in some depth.

Original languageEnglish (US)
Pages (from-to)659-683
Number of pages25
JournalNonlinearity
Volume18
Issue number2
DOIs
StatePublished - Mar 2005

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Physics and Astronomy(all)
  • Applied Mathematics

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