On weighted pseudo almost automorphic mild solutions for some mean field stochastic evolution equations.

Moustapha Dieye, Amadou Diop, Mamadou Moustapha Mbaye, Mark A. McKibben

correctmedium confidence

Category: Not specified
Journal tier: Specialist/Solid
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper’s main theorems (existence/uniqueness of an L2-bounded mild solution; almost automorphy in distribution; and weighted pseudo almost automorphy) are proven via a two-level fixed-point scheme: (i) a contraction on Cb(R;L2(P,H)) for frozen law, and (ii) a contraction on Cb(R;P(H)) for the law map, plus a sequential argument for almost automorphy. The candidate solution reproduces the same two-level fixed-point structure and smallness thresholds, and then gives an alternate (but compatible) route to almost automorphy by showing the frozen-law map is a contraction on the SAA subspace under the paper’s numerical bound. Constants, spaces, and hypotheses match, and the minor differences (e.g., how the contraction of the law map is algebraically expressed; the approach to almost automorphy) are equivalent under the same smallness conditions.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

The manuscript establishes well-posedness and regularity-in-distribution (almost automorphy and weighted pseudo almost automorphy) for a broad class of mean-field nonautonomous stochastic evolution equations driven jointly by Wiener and fractional Brownian noises, under AT conditions and exponential stability. The fixed-point strategy for the state and law maps is sound, constants are tracked, and examples are provided. Minor clarifications of notation and a brief explicit note on the deterministic nature of the fBm integrand would improve clarity.