Phase-Amplitude Description of Stochastic Oscillators: A Parameterization Method Approach

The paper’s Theorem 5.1 states precisely that if Θ and Σ are C^1 on D′, satisfy L†Θ=2π/T and L†Σ=λ_Floq Σ, and have linearly independent gradients, then there exists a unique vector field F with flow φ_t such that E[Θ(X(t))]=Θ(φ_t(x0)) and E[Σ(X(t))]=Σ(φ_t(x0)); differentiating Θ(φ_t) and Σ(φ_t) yields the constraints ∇Θ·F=2π/T and ∇Σ·F=λ_Floq Σ, and inverting the 2×2 matrix of gradients gives the explicit formula F=([∇Θ^T;∇Σ^T]^{-1}[2π/T;λ_Floq Σ]) (Theorem 5.1, Eqs. (5.5)–(5.6) ; see also the same displayed equations ). The paper justifies the mean-evolution identities via Dynkin’s formula (Eqs. (4.2)–(4.5) ). The candidate solution mirrors this line-by-line: identical pointwise linear-algebraic construction of F, the same uniqueness argument from invertibility, and the same expectation identities via Itô/Dynkin. The model adds mild regularity remarks (Peano/Picard–Lindelöf), which are consistent but not essential for the paper’s claims.