Optimal response for stochastic differential equations in ${\mathbb{T}}^d$ with perturbations on the drift term

Abstract

We study stochastic differential equations on the $d$-dimensional flat torus ${\mathbb{T}}^d$ with drift and perturbation coefficients in $L^{\infty }({\mathbb{T}}^d;{\mathbb{R}}^d)$ and additive non-degenerate noise. For the associated transfer operators, we analyse the dependence of the stationary measure and of the expectation of a given observable on small perturbations of the drift. In this framework, we prove a linear response formula for the invariant density and for the expectation of a given observable. We then address an optimal response problem, namely the determination of admissible perturbations that maximise the first-order variation of a prescribed observable. We establish existence of optimal perturbations and, in a Hilbert space framework, prove uniqueness and provide an explicit characterisation of the optimiser. This yields a practical Fourier-based numerical method, which we implement in several numerical examples, including both low and high-dimensional settings.

11 figures (14 pictures)