Optimal response for stochastic differential equations by local kernel perturbations

What is it about?

Linear response theory provides a powerful framework for understanding the impact of perturbations on a given dynamical system. It investigates whether the resulting changes in the system’s statistical properties are differentiable with respect to the perturbations. This article studies an inverse problem related to the linear response, called Optimal Response, in which one searches for the optimal perturbation in order to change the statistical properties of the system in a wanted direction. We study this problem in a class of random dynamical system defined by a stochastic differential equation on a non-compact space, using a functional analytic tool known as the transfer operator. In particular we consider the problem of finding the optimal response in order to change as much as possible the average of a given observable, showing existence and uniqueness results for the solutions of this problem. We also show algorithms to approximate the optimal solution.

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Photo by Johannes Plenio on Unsplash

Photo by Johannes Plenio on Unsplash

Why is it important?

This theory is widely applied in physics, engineering, climate science, and statistical mechanics to analyse how systems evolve under external influences. Optimal response finds many applications, including climate sciences, for example in finding out which initial climatic actions can produce the greatest change in average temperature.

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