Random dynamical systems for McKean–Vlasov SDEs via rough path theory

The paper’s Theorem 3.10 constructs the RDS φ(t, ω, (y, μ)) = (ϕt(ω, y, μ), St μ) on Ep and proves measurability, the perfect cocycle identity, and continuity in time under Assumptions 3.1, 3.7, 3.9, exactly the object and properties the candidate establishes. The proof strategy in the paper proceeds via (i) well-posedness and Lipschitz-in-time continuity for the nonlinear Fokker–Planck equation (Theorem 3.4) and stability in the bespoke Dp metric, (ii) a time-inhomogeneous rough path well-posedness and stability theorem for the RDE with drift (Theorem 3.8), and (iii) an explicit measurability argument via dyadic rough path approximations and a direct proof of the cocycle property (Section 4.3). These match the model’s Steps 1–5 at the level of ideas: build St, use the rough-path Itô map for ϕt with time-dependent, measure-parametrized coefficients, then verify measurability and cocycle via the Brownian rough path shift. Differences are present in technical details (the paper employs a Doss–Sussmann transform for unbounded drift and an explicit measurability-perfection setup), but the logical structure is the same. Therefore, both are correct and substantially the same proof at the level of construction and verification of the RDS. Key corroborating loci: the definition and proof of the RDS (Theorem 3.10) and its measurability/cocycle proof (Section 4.3) align with the candidate’s Steps 3–4; the PDE continuity and semigroup properties (Theorem 3.4) align with Step 1; the RDE well-posedness and continuity of the Itô map (Theorem 3.8) align with Step 2; Brownian rough path cocycle/shift structure is built in Section 2.