Improvement of system identification of stochastic systems via Koopman generator and locally weighted expectation

The paper explicitly derives b_i(x) = ℓ_{i+1}^⊤ ψ(x) and a_{ij}(x) ≃ ℓ_k^⊤ ψ(x) − x_j ℓ_{i+1}^⊤ ψ(x) − x_i ℓ_{j+1}^⊤ ψ(x) by applying the Koopman generator L to monomials f=x_i and f=x_i x_j, using A=ΣΣ^⊤ and the matrix representation of L in a monomial dictionary; see the generator definition and A(x)=Σ(x)Σ(x)^⊤, the dictionary setup, and the identification formulas in Sect. III.C (equations corresponding to (19), (22), (26), (27)) . The candidate solution follows the same steps and arrives at the same formulas. The only minor nuance is that the paper marks the projection to the finite dictionary with ≃, while the candidate solution writes equalities; otherwise the reasoning and formulas are the same.