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Overdamped ULA and underdamped BAOAB have different harmonic variance errors

Example

For overdamped ULA targeting $N (0, 1)$ , take step $η = 1$ . For underdamped BAOAB, take the harmonic oscillator $m = k = β = γ = 1$ and step $h = 1$ . The two invariant laws concern different state variables and transition kernels; their variance formulas are juxtaposed only to make that distinction explicit.

Facts & Assumptions

Given: The ULA Gaussian theorem, the harmonic BAOAB theorem, $η = 1$ , and $m = k = β = γ = 1$ , $h = 1$ .

[F1]

The Gaussian ULA invariant law has variance $2 / (2 - γ)$ : for the standard normal target and $0 < η < 2$ , ULA has invariant variance $2 / (2 - η)$ .

[F2]

Exact invariant covariance and stability threshold for scalar harmonic BAOAB: for $0 < h < 2 \sqrt{m / k}$ , BAOAB has $Var (Q) = 1 / (β k)$ and $Var (P) = (m / β) (1 - h^{2} k / (4 m))$ .

[A1]

The Axiom of Choice: AC is inherited through [F1] and [F2]; the side-by-side substitutions use no additional choice.

Verification

Given: The ULA Gaussian theorem, the harmonic BAOAB theorem, $η = 1$ , and $m = k = β = γ = 1$ , $h = 1$ .

[F1] The Gaussian ULA invariant law has variance $2 / (2 - γ)$ : for the standard normal target and $0 < η < 2$ , ULA has invariant variance $2 / (2 - η)$ .

[F2] Exact invariant covariance and stability threshold for scalar harmonic BAOAB: for $0 < h < 2 \sqrt{m / k}$ , BAOAB has $Var (Q) = 1 / (β k)$ and $Var (P) = (m / β) (1 - h^{2} k / (4 m))$ .

[A1] The Axiom of Choice: AC is inherited through [F1] and [F2]; the side-by-side substitutions use no additional choice.

Given: The parameters in the Example.

1.1F1givenalgebra

Since $0 < η = 1 < 2$ , [F1] gives the ULA invariant variance $2 / (2 - 1) = 2$ , compared with the target variance $1$ .

2.1A1F2givenalgebra∎

For BAOAB, $h = 1 < 2 \sqrt{m / k} = 2$ and $r = h^{2} k / (4 m) = 1 / 4$ . Thus [F2] gives position variance $1$ and momentum variance $1 (1 - 1 / 4) = 3 / 4$ , compared with the canonical momentum variance $1$ . The ULA variance describes a single overdamped coordinate; the BAOAB pair describes both position and momentum. Neither result supplies the other's invariant-law claim.

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Sources

Alain Durmus and Éric Moulines, Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithm, §3.3 (standard reference, not scraped)
Benedict Leimkuhler and Charles Matthews, Rational Construction of Stochastic Numerical Methods for Molecular Sampling, §2 and Appendix (standard reference, not scraped)