CUTOFF STABILITY OF MULTIVARIATE GEOMETRIC BROWNIAN MOTION

G. Barrera, M.A. Högele, J.C. Pardo

incompletemedium confidence

Category: math.DS
Journal tier: Specialist/Solid
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The commuting case (Theorem 2.1) in the paper matches the candidate’s derivation and is correct. In the first-order non-commutative case (Theorem 3.1), both the paper and the model hinge on the hypothesis Γ := ([A,B]+[A,B]∗)^2/6 < 0 to get a dominant cubic decay. But since Ĉ := [A,B]+[A,B]∗ is self-adjoint, Ĉ^2 is positive semidefinite, so Γ < 0 cannot hold as stated. The paper’s proof thereafter uses Γ < 0 critically (e.g., shifting A by pΓΓ), so that part is not currently sound. The model mirrors this assumption and also has a sign error in the t^2 cross-term of H(t).

Referee report (LaTeX)

\textbf{Recommendation:} major revisions

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

The manuscript makes a valuable contribution by connecting cutoff phenomena to algebraic structure (commutators) in linear SDEs and by deriving explicit time scales and windows. The commuting case is clean and correct. However, the main non-commutative theorem relies critically on an assumption Γ := ([A,B]+[A,B]∗)\^2/6<0 that cannot hold as stated because [A,B]+[A,B]∗ is self-adjoint, so its square is positive semidefinite. This sign inconsistency affects the heart of the exponential asymptotics and the shift argument for A. Clarifying the intended definition or correcting the sign is essential. With these issues addressed, the results could be solid and of interest.