arxiv: 2604.27404 · v1 · submitted 2026-04-30 · 🧮 math.DS · math.OC· math.PR

Recognition: unknown

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

Angxiu Ni, Franco Flandoli, Gianmarco Del Sarto, Sakshi Jain, Stefano Galatolo

Pith reviewed 2026-05-07 08:03 UTC · model grok-4.3

classification 🧮 math.DS math.OCmath.PR

keywords stochastic differential equationslinear responseoptimal responseinvariant measurestransfer operatorsFourier methodstorusperturbations

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The pith

A linear response formula for invariant densities on the torus yields explicit optimal drift perturbations that maximize the first-order change in any observable.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies stochastic differential equations on the d-dimensional flat torus with bounded drift, additive non-degenerate noise, and small perturbations to the drift. It shows that the stationary probability density and the long-run average of any observable change linearly with respect to these perturbations. This linear response relation is used to pose an optimization problem: among all admissible small perturbations, find the one that produces the largest possible first-order increase in the chosen observable's average. The authors prove that an optimal perturbation always exists; when the space of perturbations is a Hilbert space, the optimizer is unique and admits an explicit description that can be computed via Fourier series. The resulting numerical method works in both low- and high-dimensional examples.

Core claim

For SDEs on the torus with L^∞ drift and perturbation coefficients and additive non-degenerate noise, the stationary measure and observable expectations admit a linear response formula under drift perturbations. The associated optimal response problem of maximizing the first-order variation of a fixed observable over admissible perturbations admits solutions; in a Hilbert-space setting these solutions are unique and characterized explicitly, which produces a practical Fourier-based numerical scheme applicable in arbitrary dimensions.

What carries the argument

The linear response formula obtained by differentiating the transfer operator with respect to the drift perturbation parameter, together with the Hilbert-space inner-product optimization whose solution is recovered explicitly from the adjoint of the response operator.

If this is right

Expectations of observables vary differentiably with respect to drift perturbations, so first-order approximations are valid for small changes.
An optimal perturbation always exists that maximizes the first-order response of any prescribed observable.
In a Hilbert-space setting the optimizer is unique and given by an explicit formula involving the adjoint of the response operator.
The explicit characterization supplies a Fourier-series numerical method that remains practical in high dimensions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same linear-response-plus-optimization structure could be used to design small external controls that steer long-term averages toward target values in periodically bounded stochastic systems.
Because the optimizer is recovered from an adjoint operator, the method may extend efficiently to sensitivity analysis in applications such as fluid flow or population dynamics on periodic domains.
Verifying the differentiability assumption for the transfer operator would be the main step needed to apply the same optimal-response result to multiplicative noise or non-flat manifolds.

Load-bearing premise

The drift and perturbation coefficients must be bounded, the noise must be additive and non-degenerate, and the perturbations must lie in a function space that guarantees existence of stationary measures and differentiability of the transfer operators.

What would settle it

A concrete bounded drift perturbation on the torus for which the change in the stationary density cannot be approximated to first order by the linear response formula, or two distinct Hilbert-space perturbations that produce identical maximal first-order variation of the same observable.

Figures

Figures reproduced from arXiv: 2604.27404 by Angxiu Ni, Franco Flandoli, Gianmarco Del Sarto, Sakshi Jain, Stefano Galatolo.

**Figure 1.** Figure 1: Typical orbit of the two-dimensional Kuramoto system (31), shown as a scatter plot in the phase space T 2 view at source ↗

**Figure 2.** Figure 2: Squared H5 norms of the non-normalized Fourier basis functions B j ⃗n for the two-dimensional Kuramoto example in (31), shown in logarithmic scale. In the ergodic kernel-differentiation algorithm, we set the total time T = 105 and the decorrelation time W = 4. The code is at https://github.com/niangxiu/optrKD. On a 3GHz 8-core computer, the computation time for computing the linear responses of all 2 × 11 … view at source ↗

**Figure 3.** Figure 3: Computed response coefficients C j ⃗n = R(B˜j ⃗n) for the normalized Fourier basis functions in the two-dimensional Kuramoto example. Left: coefficients for the first component (j = 1). Right: coefficients for the second component (j = 2) view at source ↗

**Figure 4.** Figure 4: Vector field of the drift F and of the optimal perturbation ηopt for the twodimensional Kuramoto example. Blue arrows represent 1 5 F, and red arrows represent 1 2 ηopt. The figure shows the spatial structure of the perturbation that maximizes the linear response under the H5 constraint. basis function. In particular, it is larger than the linear response of B˜1 (1,0), which, by view at source ↗

**Figure 5.** Figure 5: Computed linear response in the two-dimensional Kuramoto example. The short line segments show the linear responses at γ = 0, while the dots show the values of µ γ (ϕ) for the perturbed systems F + γηopt (red circles), F + γB˜1 (1,0) (blue squares), and F + γB˜2 (10,10) (black triangles). All linear responses and µ γ (ϕ) are computed on orbits of same length. only on the first coordinate and to act only on… view at source ↗

**Figure 6.** Figure 6: Computed linear responses R(B˜ n) for the normalized basis functions of the reduced perturbation space H in the twenty-dimensional example. The rapid decay as n increases indicates that the truncation error from the finite Fourier basis is small view at source ↗

**Figure 7.** Figure 7: Optimal perturbation for the twenty-dimensional example with reduced perturbation space H. Left: vector field of the drift 1 5 F (blue arrows) and of the optimal perturbation 1 2 ηopt (red arrows), projected onto the first two coordinates; all remaining components vanish. Right: graph of the first component η 1 opt(x 1 ). at γ = 0 computed for different perturbations in view at source ↗

**Figure 8.** Figure 8: Linear response in the twenty-dimensional example. As in view at source ↗

**Figure 9.** Figure 9: Two-dimensional slices of the drift and of the optimal perturbation for the three-dimensional Lorenz example. Blue arrows represent 0.01[F 1 , F2 ], and red arrows represent 10[η 1 opt, η2 opt]. Left: slice at x 3 = 20. Right: slice at x 3 = 40 view at source ↗

**Figure 10.** Figure 10: Three-dimensional visualization of the optimal perturbation for the Lorenz example. Red arrows represent 5ηopt, plotted along a typical orbit of the system shown in black. We plot two-dimensional slices of ηopt in view at source ↗

**Figure 11.** Figure 11: Computed linear response for the three-dimensional Lorenz example. The figure compares the linear responses at γ = 0 with the values of the averaged observable µ γ (ϕ) for several perturbations, showing that the optimal perturbation produces the largest response and that the linear approximation is accurate near γ = 0. Then we plot µ γ (ϕ) versus γ and the linear responses at γ = 0 computed for different … view at source ↗

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

They give an explicit Hilbert-space characterization of the optimal L^infty drift perturbation that maximizes the first-order response of an observable for SDEs on the torus, plus a Fourier numerical scheme that they actually run.

read the letter

The main takeaway is that the authors turn a linear response formula into a concrete optimization problem and solve it with uniqueness plus an explicit Fourier characterization. They start with SDEs on T^d with additive nondegenerate noise and L^infty drift and perturbation, establish the first-order change in the invariant density and in expectations of observables, then maximize that change over admissible perturbations. Existence is shown in a general setting; in a Hilbert space they get uniqueness and a characterization that directly yields a computable Fourier method, which they test in both low- and high-dimensional examples. That combination of explicit optimizer and working numerics is the part that stands out from standard linear-response results in the literature they cite. The linear-response step itself follows from standard ergodic and transfer-operator arguments under their assumptions, so it is not surprising but is cleanly set up for the optimization that follows. The numerical examples are a genuine plus because they show the method scales at least to moderate dimensions without extra smoothing tricks. The soft spot is the regularity question raised in the stress-test note. With only L^infty coefficients the stationary density is positive and Hölder, but the source term in the linearized Fokker-Planck equation is div(h rho_0) with h merely bounded; that source sits in a low-regularity space. If the Hilbert space chosen for uniqueness is L^2 or H^1, it is not immediate that the response operator maps onto the right dual or that the optimizer remains admissible without further approximation. The abstract states the results hold under the given assumptions, but a referee will want to see the precise function spaces, any density arguments, and whether the Fourier characterization stays valid when the source is only distributional. No circularity or invented entities appear; the derivations rest on standard elliptic regularity and ergodicity for nondegenerate SDEs. This paper is for people working on sensitivity or design problems in stochastic dynamics on compact domains who need an explicit optimizer rather than abstract existence. A reader already comfortable with Fokker-Planck linearization will get the most out of the optimization and numerics sections. It deserves peer review: the claims are specific, the numerical component is reproducible in principle, and the potential regularity gap is checkable rather than fatal.

Referee Report

1 major / 2 minor

Summary. The paper proves a linear response formula for the invariant density and expectations of observables for SDEs on the d-dimensional torus with L^∞ drift and perturbation coefficients and additive non-degenerate noise. It then formulates an optimal response problem of finding admissible perturbations that maximize the first-order variation of a prescribed observable, establishes existence of optimizers, and in a Hilbert-space setting proves uniqueness together with an explicit characterization of the optimizer. This characterization yields a practical Fourier-based numerical method, which is implemented and tested in low- and high-dimensional examples.

Significance. If the linear-response and optimization results hold with the stated regularity, the work supplies a rigorous bridge between linear-response theory for rough-coefficient SDEs and a well-posed optimization problem whose solution is explicitly characterizable in Hilbert space. The resulting Fourier numerical scheme and its demonstration in high dimensions constitute a concrete, reproducible contribution that could be useful for applications in stochastic control and sensitivity analysis.

major comments (1)

[Section on linear response and the subsequent Hilbert-space optimization (around the statement of the linearized Fokker–] The differentiability of the stationary density ρ_ε at ε=0 with respect to L^∞ perturbations must be established in a topology compatible with the Hilbert space in which uniqueness of the optimizer is claimed. The linearized Fokker-Planck equation has source term div(h ρ_0) with h ∈ L^∞; it is not immediate that this source lies in the range of the linearized operator when the Hilbert space is, for example, L² or H¹. This regularity gap directly affects the validity of the uniqueness and explicit-characterization statements for the optimal perturbation.

minor comments (2)

[Introduction and statement of the optimal-response problem] The precise definition of the admissible perturbation class (the function space in which h lives and the norm in which the first-order variation is taken) should be stated explicitly before the optimization problem is formulated, rather than left implicit from the linear-response section.
[Numerical examples] A short remark on how the Fourier basis is chosen and truncated in the numerical examples would improve reproducibility, especially for the high-dimensional tests.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading, the positive assessment of the significance of the results, and the constructive comment on topological compatibility. We address the major comment below and will revise the manuscript accordingly to strengthen the presentation.

read point-by-point responses

Referee: [Section on linear response and the subsequent Hilbert-space optimization (around the statement of the linearized Fokker–] The differentiability of the stationary density ρ_ε at ε=0 with respect to L^∞ perturbations must be established in a topology compatible with the Hilbert space in which uniqueness of the optimizer is claimed. The linearized Fokker-Planck equation has source term div(h ρ_0) with h ∈ L^∞; it is not immediate that this source lies in the range of the linearized operator when the Hilbert space is, for example, L² or H¹. This regularity gap directly affects the validity of the uniqueness and explicit-characterization statements for the optimal perturbation.

Authors: We agree that explicit compatibility of topologies must be stated to connect the linear-response result (valid for L^∞ perturbations) with the Hilbert-space optimization. Because the additive noise is non-degenerate, the unperturbed density ρ_0 is smooth (in fact C^∞ on the compact torus). Consequently, for any h ∈ L^∞ the product h ρ_0 belongs to L^∞ and its distributional divergence defines a continuous linear functional on H^1, i.e., the source lies in H^{-1}. The linearized Fokker–Planck operator is an isomorphism from H^1 onto H^{-1} (by standard elliptic theory for the generator), so the first-order variation of the density exists in H^1. When the optimization is performed in the Hilbert space L^2 (or a closed subspace thereof), the map h ↦ first-order variation of the observable is a continuous linear functional on L^2, because the H^1-solution of the linearized equation is bounded in L^2 by Sobolev embedding on the torus. Riesz representation therefore yields existence, uniqueness, and the explicit characterization of the optimizer without further restrictions. We will insert a short paragraph (or remark) after the statement of the linearized equation that records these duality and embedding arguments, thereby removing any ambiguity about the topologies. This clarification does not alter the statements or proofs but makes the logical chain fully rigorous. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on independent functional-analytic results

full rationale

The paper derives a linear response formula for the stationary density and observable expectations under L^∞ drift perturbations of non-degenerate SDEs on the torus, then proves existence/uniqueness of optimal perturbations in a Hilbert-space setting. These steps invoke standard elliptic regularity for the Fokker-Planck operator, properties of transfer operators, and differentiability results for invariant measures under bounded measurable coefficients—none of which reduce by construction to the paper's own fitted quantities, self-definitions, or unverified self-citations. The abstract and described claims contain no renaming of known empirical patterns, ansatz smuggling, or load-bearing uniqueness theorems imported solely from the authors' prior work. The central claims remain independent of the target optimization result and are externally falsifiable via standard PDE/ergodic theory benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard background results from stochastic analysis rather than new postulates; no free parameters are fitted and no new entities are introduced.

axioms (2)

domain assumption Existence of a unique stationary measure for the unperturbed SDE with non-degenerate additive noise and L^infty drift on the compact torus T^d
Invoked to guarantee well-defined invariant densities and transfer operators before perturbation analysis.
domain assumption Differentiability of the stationary measure and observable expectations with respect to small L^infty perturbations of the drift
Required for the linear response formula to hold and for the subsequent optimization problem to be well-posed.

pith-pipeline@v0.9.0 · 5473 in / 1514 out tokens · 68638 ms · 2026-05-07T08:03:59.552469+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

41 extracted references

[1]

Amann.Linear and quasilinear parabolic problems

H. Amann.Linear and quasilinear parabolic problems. Vol. I, volume 89 ofMonographs in Mathematics. Birkh¨ auser Boston, Inc., Boston, MA, 1995

1995
[2]

Antown, D

F. Antown, D. Dragiˇ cevi´ c, and G. Froyland. Optimal linear responses for markov chains and stochastically perturbed dynamical systems.Journal of Statistical Physics, 170(6):1051–1087, 2018

2018
[3]

Antown, G

F. Antown, G. Froyland, and S. Galatolo. Optimal linear response for Markov Hilbert–Schmidt integral operators and stochastic dynamical systems.Journal of Nonlinear Science, 32(79):60, 2022. 42 GIANMARCO DEL SARTO, FRANCO FLANDOLI, STEFANO GALATOLO, SAKSHI JAIN, AND ANGXIU NI

2022
[4]

V. Baladi. Linear response, or else.ICM Seoul 2014 talk, 2014

2014
[5]

Baldi.Stochastic calculus

P. Baldi.Stochastic calculus. Universitext. Springer, Cham, 2017

2017
[6]

Bittracher, P

A. Bittracher, P. Koltai, S. Klus, R. Banisch, M. Dellnitz, and C. Sch¨ utte. Transition manifolds of complex metastable systems: Theory and data-driven computation of effective dynamics.Journal of Nonlinear Science, 28(2):471–512, 2017

2017
[7]

Bogachev, N

V. Bogachev, N. V Krylov, M. R¨ ockner, and S. Shaposhnikov.Fokker–Planck–Kolmogorov Equations, volume 207. American Mathematical Society, 2022

2022
[8]

Brezis.Functional Analysis, Sobolev Spaces and Partial Differential Equations

H. Brezis.Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer New York, 2010

2010
[9]

Carigi, T

G. Carigi, T. Kuna, and J. Br¨ ocker. Linear and fractional response for nonlinear dissipative SPDEs.Nonlinearity, 37(10):Paper No. 105002, 41, 2024

2024
[10]

Cazenave and A

T. Cazenave and A. Haraux.An introduction to semilinear evolution equations, volume 13 ofOxford Lecture Series in Mathematics and its Applications. The Clarendon Press, Oxford University Press, New York, 1998

1998
[11]

Conway.A course in functional analysis, volume 96 ofGraduate Texts in Mathematics

J. Conway.A course in functional analysis, volume 96 ofGraduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1990

1990
[12]

Da Prato.An introduction to infinite-dimensional analysis

G. Da Prato.An introduction to infinite-dimensional analysis. Universitext. Springer-Verlag, Berlin, 2006. Revised and extended from the 2001 original by Da Prato

2006
[13]

Del Sarto, S

G. Del Sarto, S. Galatolo, and S. Jain. Optimal response for stochastic differential equations by local kernel perturba- tions.Chaos: An Interdisciplinary Journal of Nonlinear Science, 35(7):073121, 2025

2025
[14]

Froyland and S

G. Froyland and S. Galatolo. Optimal linear response for expanding circle maps.Nonlinearity, 38, 2025

2025
[15]

Froyland and M

G. Froyland and M. Phalempin. Optimal linear response for anosov diffeomorphisms. arXiv preprint, April 2025. Submitted 23 Apr 2025; last revised 28 Nov 2025 (v2)

2025
[16]

Galatolo

S. Galatolo. Self-consistent transfer operators: Invariant measures, convergence to equilibrium, linear response and control of the statistical properties.Communications in Mathematical Physics, 395:715–772, 2022

2022
[17]

A linear response for dynamical systems with additive noise.Nonlinearity, 32(6):2269, may 2019

S Galatolo and P Giulietti. A linear response for dynamical systems with additive noise.Nonlinearity, 32(6):2269, may 2019

2019
[18]

Galatolo and A

S. Galatolo and A. Ni. Optimal response for hyperbolic systems by the fast adjoint response method.Nonlinearity, 38(11):115002, nov 2025

2025
[19]

Galatolo and M

S. Galatolo and M. Pollicott. Controlling the statistical properties of expanding maps.Nonlinearity, 30(7):2737, may 2017

2017
[20]

Ghil and V

M. Ghil and V. Lucarini. The physics of climate variability and climate change.Rev. Mod. Phys., 92:035002, Jul 2020

2020
[21]

Hairer and A

M. Hairer and A. Majda. A simple framework to justify linear response theory.Nonlinearity, 23:909–922, 4 2010

2010
[22]

Kloeckner

B.. Kloeckner. The linear request problem.Proceedings of the American Mathematical Society, 146(7):2953–2962, 2018. Article electronically published on March 20, 2018

2018
[23]

Koltai, H

P. Koltai, H. Lie, and M. Plonka. Fr´ echet differentiable drift dependence of perron–frobenius and koopman operators for non-deterministic dynamics.Nonlinearity, 32:4232, 2019

2019
[24]

Lieberman.Second order parabolic differential equations

G. Lieberman.Second order parabolic differential equations. World Scientific Publishing Co., Inc., River Edge, NJ, 1996

1996
[25]

J. L. Lions and E. Magenes.Non-homogeneous boundary value problems and applications. Vol. I. Springer-Verlag, New York-Heidelberg, 1972

1972
[26]

Lucarini

V. Lucarini. Response theory for equilibrium and non-equilibrium statistical mechanics: Causality and generalized kramers-kronig relations.Journal of Statistical Physics, 131(3):543–558, 2008

2008
[27]

Lucarini, F

V. Lucarini, F. Ragone, and F. Lunkeit. Predicting climate change using response theory: Global averages and spatial patterns.Journal of Statistical Physics, 166(3–4):1036–1064, 2016

2016
[28]

Lunardi.Analytic semigroups and optimal regularity in parabolic problems

A. Lunardi.Analytic semigroups and optimal regularity in parabolic problems. Modern Birkh¨ auser Classics. Birkh¨ auser/Springer Basel AG, Basel, 1995

1995
[29]

A. Ni. Differentiating unstable diffusion, 2025

2025
[30]

A. Ni. Ergodic and foliated kernel-differentiation method for linear responses of random systems.J. Nonlinear Sci., 35(90), July 2025

2025
[31]

Pavliotis.Stochastic Processes and Applications: Diffusion Processes, the Fokker-Planck and Langevin Equations

G. Pavliotis.Stochastic Processes and Applications: Diffusion Processes, the Fokker-Planck and Langevin Equations. Springer New York, 2014

2014
[32]

Pazy.Semigroups of linear operators and applications to partial differential equations, volume 44 ofApplied Math- ematical Sciences

A. Pazy.Semigroups of linear operators and applications to partial differential equations, volume 44 ofApplied Math- ematical Sciences. Springer-Verlag, New York, 1983

1983
[33]

Rothe.Global solutions of reaction-diffusion systems

F. Rothe.Global solutions of reaction-diffusion systems. Lecture Notes in Mathematics. Springer, 1984 edition, July 1984

1984
[34]

D. Ruelle. Differentiation of srb states.Communications in Mathematical Physics, 187:227–241, 7 1997

1997
[35]

D. Ruelle. A review of linear response theory for general differentiable dynamical systems.Nonlinearity, 22(4):855, 2009

2009
[36]

Santos Gutierrez, N

M. Santos Gutierrez, N. Zagli, and G. Carigi. Markov matrix perturbations to optimize dynamical and entropy func- tionals, 2025

2025
[37]

Sch¨ utte, W

C. Sch¨ utte, W. Huisinga, and P. Deuflhard.Transfer Operator Approach to Conformational Dynamics in Biomolecular Systems, page 191–223. Springer Berlin Heidelberg, 2001. OPTIMAL RESPONSE FOR SDES INT d WITH PERTURBATIONS ON THE DRIFT TERM 43

2001
[38]

Sch¨ utte, S

C. Sch¨ utte, S. Klus, and C. Hartmann. Overcoming the timescale barrier in molecular dynamics: Transfer operators, variational principles and machine learning.Acta Numerica, 32:517–673, 2023

2023
[39]

Stroock and S

D. Stroock and S. Varadhan.Multidimensional diffusion processes. Classics in Mathematics. Springer-Verlag, Berlin, 2006

2006
[40]

Triebel.Interpolation Theory, Function Spaces, Differential Operators

H. Triebel.Interpolation Theory, Function Spaces, Differential Operators. North Holland, 1978

1978
[41]

A. Zvonkin. A transformation of the phase space of a diffusion process that removes the drift.Mathematics of the USSR-Sbornik, 22(1):129, 1974. Technische Universit ¨at Darmstadt, F achbereich Mathematik, Schlossgartenstr. 7, 64289 Darmstadt, Ger- many Email address:delsarto@mathematik.tu-darmstadt.de Scuola Normale Superiore, Classe di Scienze, P.za dei ...

1974