A Small-Noise Analysis of Controlled Functional Differential Equations with Gaussian Noise

What the math gives to ML

The paper turns exponential small-noise costs for controlled, path-dependent systems into deterministic zero-sum games: an adverse Cameron–Martin perturbation competes with a quadratic energy penalty. The transferable asset is not the particular functional differential equation, but the variational replacement of Gaussian expectation by an explicit worst-case perturbation in the noise reproducing-kernel Hilbert space. This suggests a principled correlated adversarial-training objective for sequence models, where perturbations are constrained by the covariance geometry of realistic temporal noise rather than by an isotropic norm. The approach is most promising for recurrent, state-space, diffusion, and long-context models whose hidden-state dynamics are sensitive to structured time-correlated disturbances.