Optimal linear response for Markov Hilbert-Schmidt integral operators and stochastic dynamical systems

The paper proves Theorem 5.4 via Lagrange multipliers and second-order conditions, deriving the explicit optimizer k̇(x,y) = [f0(y)/α](h(x) − avg_{F^y_l} h) 1_{Fl}(x,y), with h = (Id − L0^*)^{-1}c, uniqueness, and L∞-regularity under extra assumptions (k0 ∈ L∞ and L0 compact on L1) . The candidate solution reaches the same formula using a Hilbert-space projection argument: it rewrites the objective as ⟨k̇, f0(y)h(x)1Fl⟩ and identifies the optimizer as the normalized orthogonal projection of g(x,y)=f0(y)h(x)1Fl onto S=Vker∩Sk0,l, computed fiberwise in y. This produces the identical expression and uniqueness. The extra L∞ bound follows from the paper’s Fredholm/compactness argument on L∞ as well. Thus, both are correct; the proofs differ (projection vs. Lagrange multiplier).