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An unstable harmonic BAOAB step has no invariant probability

Example

For the one-dimensional harmonic oscillator with $m = k = β = γ = 1$ , choose the BAOAB step $h = 5 / 2$ . The method's affine Gaussian transition is unstable, so this discrete chain has no invariant probability.

Facts & Assumptions

Given: $m = k = β = γ = 1$ , $h = 5 / 2$ , and the harmonic BAOAB theorem.

[F1]

Exact invariant covariance and stability threshold for scalar harmonic BAOAB: for the scalar harmonic chain, no invariant probability exists when $h > 2 \sqrt{m / k}$ .

[A1]

The Axiom of Choice: AC is inherited through [F1]; the threshold comparison uses no additional choice.

Verification

Given: $m = k = β = γ = 1$ , $h = 5 / 2$ , and the harmonic BAOAB theorem.

[F1] Exact invariant covariance and stability threshold for scalar harmonic BAOAB: for the scalar harmonic chain, no invariant probability exists when $h > 2 \sqrt{m / k}$ .

[A1] The Axiom of Choice: AC is inherited through [F1]; the threshold comparison uses no additional choice.

Given: The parameters in the Example.

1.1F1givenalgebra

The threshold is $2 \sqrt{m / k} = 2$ , and $5 / 2 > 2$ . Thus the strict unstable case of [F1] applies.

2.1A1F1step 1.1∎

Therefore the BAOAB transition at this step size has no invariant probability law. This example does not infer failure at the boundary $h = 2$ .

Depends on

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Sources

Benedict Leimkuhler and Charles Matthews, Rational Construction of Stochastic Numerical Methods for Molecular Sampling, §2 and Appendix (standard reference, not scraped)
Jean-Noël Roux, Stéphane Rodts, and Gabriel Stoltz, Introduction à la physique statistique et à la physique quantique, §5.2.2 (standard reference, not scraped)