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

TL;DR

This paper investigates the sensitivity of invariant measures and observables in stochastic differential equations on the torus to small drift perturbations, deriving a linear response formula and identifying optimal perturbations.

Contribution

It introduces a linear response formula for invariant densities, proves existence and uniqueness of optimal perturbations, and develops a Fourier-based numerical method for high-dimensional cases.

Findings

Derived a linear response formula for invariant measures.

Proved existence and uniqueness of optimal perturbations.

Implemented a Fourier-based numerical method in various examples.

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.