Convergence of invariant densities in the small-noise limit

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 - ρ₀∥₁ ∼ 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.