Optimal response for stochastic differential equations by local kernel perturbations

We consider a random dynamical system on Rd, 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 ϕ:Rd→R, we study which infinitesimal perturbation in P produces the greatest change in expectation of ϕ. 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.