2006.06808
LIMIT BEHAVIOR OF THE INVARIANT MEASURE FOR LANGEVIN DYNAMICS
Gerardo Barrera
correcthigh confidence
- Category
- Not specified
- Journal tier
- Note/Short/Other
- Processed
- Sep 28, 2025, 12:55 AM
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Audit review
The paper proves that the scaled invariant law ε^{d/2} μ_ε(√ε dx) converges in W2 to N(0, Σ) by a synchronous-coupling/disintegration argument with an explicit Gaussian comparison and a suitable time choice t_ε, under (H) and (G) (Theorem 1.1 and Section 2) . The candidate solution proves the same limit by rescaling the dynamics, generator convergence L_ε → L_0 on compacts, uniqueness of the OU invariant law, and uniform moment bounds, then upgrades weak convergence to W2 using second moments. Both arguments are logically sound and reach the same conclusion; they differ mainly in technique (coupling vs. generator/semigroup).
Referee report (LaTeX)
\textbf{Recommendation:} minor revisions
\textbf{Journal Tier:} note/short/other
\textbf{Justification:}
The manuscript gives a clean, coupling-based proof of a Gaussian limit for the small-noise invariant measure under a strong monotonicity assumption. The result is classical in spirit but the argument is concise and pedagogical, avoiding explicit Gibbs-form computations. Minor clarifications (explicit time choice, brief discussion of the role of (G) and its possible relaxation) would strengthen the presentation.