Linear response formulas are established for invariant densities and observables of perturbed SDEs on the torus, followed by existence, uniqueness, and explicit characterization of optimal drift perturbations that maximize first-order observable variation, with Fourier numerics demonstrated in low-
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Optimal response for stochastic differential equations in $\mathbb{T}^d$ with perturbations on the drift term
Linear response formulas are established for invariant densities and observables of perturbed SDEs on the torus, followed by existence, uniqueness, and explicit characterization of optimal drift perturbations that maximize first-order observable variation, with Fourier numerics demonstrated in low-