Strong Lyapunov functions for rough systems

The paper proves that if a strong Lyapunov function V (assumption (HV)) exists for the drift f in a rough differential equation dy = f(y) dt + g(y) dx driven by a stationary-increment rough path, then there is a pullback absorbing ball and a compact tempered global pullback attractor A for the induced RDS; the key pullback estimate is V(φ(t,θ^{-t}ω)y0) ≤ e^{-δ t}V(y0) + C_λ/δ + R̄_λ(ω) with R̄_λ(ω) = Σ_{k≥0} e^{-kδ} H(|||x(θ^{-k}ω)|||_{p-var,[-1,0]}), where H(ξ) = L_V(16C_pC_g)^p λ^{1−p} ξ^p + 8 L_V C_p C_g ξ (Theorem 8 and Proposition 10) . The model’s solution reproduces the same strategy: Doss–Sussmann transformation on greedy partitions to make the perturbations (ψ,η) small, derive the rough Lyapunov increment and the past-weighted sum R̄_λ, build the pullback absorbing ball B_α via α^{-1}(C_λ/δ + R̄_λ + ε), and then construct the attractor as A(ω) = ⋂_{n≥0} φ(n,θ^{-n}ω) B̄_ε(θ^{-n}ω) . Both arguments rely on the same core estimates under (HV) and (HX) and the same RDS framework for cocycle and continuity . A minor discrepancy: the paper states B̄_ε is forward invariant with equality φ(t,ω)B̄_ε(ω) = B̄_ε(θ^tω) (Theorem 14), but its proof shows inclusion; equality generally requires additional surjectivity assumptions. The model correctly flags this subtlety and uses inclusion, which suffices for the attractor construction . Integrability and temperedness claims for R̄_λ and the attractor radius under K_poly assumptions also align with the paper’s Lemma/Proposition-level results .