Optimal response for stochastic differential equations by local kernel perturbations

TL;DR

This paper investigates how to determine the optimal infinitesimal kernel perturbation in stochastic differential equations to maximally influence the expected value of an observable, providing theoretical conditions and numerical methods.

Contribution

It establishes conditions for the unique existence of optimal perturbations and introduces a numerical approach to approximate them in stochastic dynamical systems.

Findings

Optimal perturbation uniquely exists under certain conditions.

Numerical method effectively approximates the optimal perturbation.

Illustrative examples demonstrate practical applicability.

Abstract

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.