Optimal response for stochastic differential equations in $\mathbb{T}^d$ with perturbations on the drift term
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arXiv
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866913075768066048 |
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| author | Del Sarto, Gianmarco Flandoli, Franco Galatolo, Stefano Jain, Sakshi Ni, Angxiu |
| author_facet | Del Sarto, Gianmarco Flandoli, Franco Galatolo, Stefano Jain, Sakshi Ni, Angxiu |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_27404 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Optimal response for stochastic differential equations in $\mathbb{T}^d$ with perturbations on the drift term Del Sarto, Gianmarco Flandoli, Franco Galatolo, Stefano Jain, Sakshi Ni, Angxiu Dynamical Systems Optimization and Control Probability 37H30, 37N35, 60H10, 37C30 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. |
| title | Optimal response for stochastic differential equations in $\mathbb{T}^d$ with perturbations on the drift term |
| topic | Dynamical Systems Optimization and Control Probability 37H30, 37N35, 60H10, 37C30 |
| url | https://arxiv.org/abs/2604.27404 |