Optimal response for stochastic differential equations by local kernel perturbations

We consider a random dynamical system on $\mathbb{R}^d$, whose dynamics is defined by a stochastic differential equation. The annealed transfer operator associated with such systems is a kernel operator. Given a set of feasible infinitesimal perturbations $P$ to this kernel, with support in a certain compact set, and a specified observable function $\phi: \mathbb{R}^d \to \mathbb{R}$, we study which infinitesimal perturbation in $P$ produces the greatest change in expectation of $\phi$. We establish conditions under which the optimal perturbation uniquely exists and present a numerical method to approximate the optimal infinitesimal kernel perturbation. Finally, we numerically illustrate our findings with concrete examples.