THE PERFECTION OF LOCAL SEMI-FLOWS AND LOCAL RANDOM DYNAMICAL SYSTEMS WITH APPLICATIONS TO SDES

The paper’s Theorem 2.7 precisely addresses the task: from a local semi-flow (φ,Θ) that satisfies the shift-covariance and lifetime relations (2.4)–(2.5) almost surely for each s>0, it constructs a local RDS (θ,ϕ,τ) indistinguishable from the time-0 slice (φ0,·,Θ(0,·)), with continuity/globality inherited . The proof uses a robust perfection step (Proposition 2.5) via a coffin state and essential limits to remove the s-dependent exceptional null sets and enforce identities for all real s, t and all ω, before defining the cocycle ϕ and its lifetime τ; the key construction is sketched explicitly in the proof of Theorem 2.7 and relies on the formal definitions of local semi-flows and local RDS in Section 2 . By contrast, the model’s solution defines ϕt(x,ω):=φ0,t(x,ω) and τ(x,ω):=Θ(0,x,ω) and attempts a “perfection” by intersecting over rational s and using right-continuity to pass to all reals. This step is not justified under the paper’s assumptions: passing from the rational cocycle identity to real s requires a uniform perfection argument (as in Proposition 2.5), not merely right-continuity in t or s. In particular, the model’s limit interchange tacitly needs continuity in the initial condition x (or a topology/continuity in ω under θ) which is not assumed in Theorem 2.7; the paper’s proof avoids this by constructing a perfected semi-flow first and only then defining the cocycle . Therefore, the paper’s result/proof is correct, while the model’s argument contains a substantive gap in extending the almost-sure rational-time identities to all real times.