ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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At the critical quadratic ULA step size, no invariant law exists

Example

Let $U (x) = x^{2} / 2$ on $R$ and set $γ = 2$ in ULA. Then the update is $X^{'} = - X + 2 Z$ for a fresh independent $Z \sim N (0, 1)$ . This Markov kernel has no invariant probability law.

Facts & Assumptions

Given: AC; the ULA update for $U (x) = x^{2} / 2$ at $γ = 2$ .

[A1]

AC is assumed in the standard-normal construction. The Axiom of Choice Standard normal and normal laws

[F1]

For $U (x) = x^{2} / 2$ , $\nabla U (x) = x$ , so the ULA update at $γ = 2$ is $X^{'} = - X + 2 Z$ , with fresh independent $Z \sim N (0, 1)$ . Unadjusted Langevin algorithm (ULA) Standard normal and normal laws

[F2]

Affine maps and sums of independent variables transform characteristic functions by $φ_{a X + b} (t) = e^{i t b} φ_{X} (a t)$ and by multiplication. Characteristic functions under affine maps and independent sums

[F3]

Every characteristic function has $φ (0) = 1$ , is continuous, and satisfies $φ (- t) = \overline{φ (t)}$ . Characteristic function of a real random variable Basic properties of characteristic functions

[F4]

The standard normal characteristic function is $e^{- t^{2} / 2}$ . Characteristic function of a normal law

Verification

Given: The update and facts above.

1.1A1F1F2F4given

Suppose for contradiction that an invariant probability $μ$ exists and write $φ$ for its characteristic function. Invariance of the kernel in [F1], together with the affine characteristic-function identity [F2] and the standard-normal characteristic function [F4], gives for every real $t$ , $φ (t) = φ (- t) e^{- 2 t^{2}} .$

2.1F3step 1.1algebra∎

By [F3], $∣ φ (- t) ∣ = ∣ φ (t) ∣$ . For each $t \neq 0$ , the displayed identity implies $∣ φ (t) ∣ = ∣ φ (t) ∣ e^{- 2 t^{2}}$ , and $e^{- 2 t^{2}} < 1$ forces $φ (t) = 0$ . Thus $φ$ vanishes at every nonzero $t$ , contradicting continuity at zero and $φ (0) = 1$ from [F3]. No invariant probability can exist.

Depends on

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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)