Convergence of invariant densities in the small-noise limit

Let ρ₀ be an invariant probability density of a deterministic dynamical system f and ρ the invariant probability density of a random perturbation of f by additive noise of amplitude . Suppose ρ₀ is stochastically stable in the sense that ρ → ρ₀ as → 0. Through a systematic numerical study of concrete examples, I show that:

1. The rate of convergence of ρ to ρ₀ as → 0 is frequently governed by power laws: ||ρ − ρ₀||₁ ∼ ^γ for some γ > 0.

2. When the deterministic system f exhibits exponential decay of correlations, a simple heuristic can correctly predict the exponent γ based on the structure of ρ₀.

3. 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 ρ to ρ₀ as → 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 ||ρ − ρ₀||₁ for 1.5–2.5 decades of and provides an opportunity to discuss and compare standard numerical methods for computing invariant probability densities in some depth.