Pith. sign in

REVIEW 4 major objections 5 minor 53 references

Mean-field optimal control with stochastic leaders

T0 review · 4 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The optimal control of a finite leader-follower system converges to the solution of a low-dimensional mean-field control problem as the number of followers grows, so the limit problem can be solved instead of the high-dimensional one.

desk verdict Legitimate extension of the partial mean-field program, but the main convergence claim for optimal controls has a real compactness gap that needs repairing before the result can be trusted as stated. read the letter →

arxiv 2512.19201 v2 pith:EXSA3WQH submitted 2025-12-22 math.OC math.PR

Sebastian Zimper , Ana Djurdjevac , Carsten Hartmann , Christof Schütte , Nataša Djurdjevac Conrad This is my paper · ORCID
classification math.OCmath.PR MSC 93E2060H1035Q84
keywords mean-fieldoptimalcontrolMcKean-VlasovSDEpropagationofchaosleader-followersystemsFokker-PlanckequationpolicygradientHegselmann-KrausemodelΓ-convergence
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper studies a large population of stochastic followers steered by a single stochastic leader, and asks whether controlling the leader in the infinite-follower limit is a good proxy for controlling the finite system. Its central claim: under Lipschitz regularity and compactness assumptions, the finite-N optimal control problem Γ-converges to a mean-field optimal control problem whose state is the conditional law of a representative follower given the leader's noise, and the optimal controls converge accordingly. If true, this means a low-dimensional control problem—one leader plus a density equation—can produce asymptotically optimal policies for the high-dimensional interacting system. The paper also establishes propagation of chaos for non-constant diffusion coefficients, uniqueness of the coupled nonlinear Fokker-Planck equation, and a gradient-descent algorithm with explicit derivative formulas, demonstrated by steering a noisy Hegselmann-Krause opinion model to consensus.

What carries the argument

The central object is the conditional McKean-Vlasov dynamics (the partial mean-field limit): an SDE for the representative follower whose coefficients depend on g_t = Law(X_t | F^Y_t), coupled to the controlled Itô SDE for the leader. The identity doing the work is the Γ-convergence of the finite-N cost J_N to the mean-field cost J (Theorem 4.4), built on a propagation-of-chaos estimate (Theorem 3.4) for non-constant diffusion, and on the compactness and parametric control class (Assumption 4.1) that allow subsequential convergence of minimizers via Arzelà–Ascoli. For numerics, the paper replaces the McKean-Vlasov SDE by its coupled nonlinear Fokker-Planck equation, whose uniqueness (Theorem

What would settle it

Take a system satisfying all assumptions (e.g., a smooth bounded interaction kernel on the torus) and compute both the finite-N optimal control and the mean-field optimal control; if their difference does not vanish as N grows, Corollary 4.5 fails. A sharper test of the compactness assumption: allow opinions on the real line with Gaussian noise and check whether the sequence of finite-N minimizers has a unique accumulation point—two distinct limits would show compactness is essential.

Watch

Extended reading notes

Core claim

The core discovery is that the finite-N optimal control problem and the mean-field optimal control problem are linked by Γ-convergence: the value and the minimizers of the finite problem pass to the N→∞ limit. The limiting state is g_t = Law(X_t | F^Y_t), the conditional law of a representative follower given the Brownian motion driving the leader, and the leader's controlled Itô SDE is coupled to this conditional law. The paper proves the propagation-of-chaos convergence of the empirical measure to g_t (Theorem 3.4), the Γ-convergence of the cost functionals (Theorem 4.4) under a parametric control class u = h(t)f(y,μ) on a compact state space, and the resulting convergence of optimal contr

Load-bearing premise

The whole convergence argument rests on the followers' state space being compact and on controls being restricted to the product form u(t,y,μ)=h(t)f(y,μ) with bounded Lipschitz f and h in a fixed compact set; if either condition fails, the claimed convergence of optimal controls is not proved.

Editorial extensions

If this is right

  • Solving the low-dimensional mean-field control problem gives policies that are asymptotically optimal for the finite high-dimensional system, so one can design controls without simulating all N followers.
  • The optimal control problem for the finite system is strictly convex under the stated conditions, so gradient and Newton schemes with explicit derivative formulas converge to the unique optimum in a chosen finite basis.
  • The propagation-of-chaos result extends to non-constant diffusion coefficients for the followers, making the partial mean-field limit available beyond the constant-diffusion setting.
  • The PDE/SDE representation (coupled Fokker-Planck) is a tractable surrogate for the conditional McKean-Vlasov system, because the cost depends only on (Y, g), enabling spatial discretization to an SDE-ODE system.
  • The algorithm produces discrete-time Markov controls that match practical settings in which leaders observe the population at discrete polling times.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The compactness assumption is the real limitation: on unbounded state spaces (e.g., opinions on the real line with fat-tailed noise) the subsequential compactness argument breaks; a natural extension would add coercive state costs or moment penalties to recover tightness of minimizers.
  • The parametric control class u(t,y,μ)=h(t)f(y,μ) excludes controls depending on higher-order statistics of the population; the optimal mean-field control may require such dependence, and a richer parameterization would be needed to guarantee convergence.
  • The numerical Hegselmann-Krause example uses a discontinuous interaction kernel and a torus domain, so it does not satisfy the paper's Lipschitz assumptions; its good behavior hints that the results may extend to discontinuous interactions, but that extension remains unproven.
  • The explicit Fréchet-derivative formulas for the finite-particle cost suggest a testable route: compute the same derivatives for the discretized PDE/SDE surrogate and verify that the gradient descent converges to the limiting optimum as the spatial grid refines—the paper leaves this convergence analysis open.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper studies a system of N identical follower particles and one leader, where the control acts only on the leader. The main theoretical results are a propagation-of-chaos limit for the follower empirical measure to a conditional McKean-Vlasov SDE coupled to the leader’s SDE (Theorem 3.4), well-posedness and uniqueness of the associated nonlinear Fokker-Planck/SDE system (Theorem 5.11), and a Γ-convergence statement (Theorem 4.4) from which the paper claims that finite-N optimal controls converge to the mean-field optimal control (Corollary 4.5). The paper also proposes a gradient-descent/policy-gradient algorithm for finite-particle and mean-field control and illustrates it on a noisy Hegselmann-Krause opinion-dynamics model. The abstract states the stronger claim that the unique optimal control of the finite-agent system converges to the optimal control of the limiting system.

Significance. If the convergence-of-optimal-controls result were rigorously established, the paper would make a useful contribution to partial mean-field control with a fixed number of leaders: it would justify controlling the low-dimensional limiting system as an asymptotically optimal strategy for the finite system. The propagation-of-chaos result for non-constant diffusion coefficients and the uniqueness result for the coupled Fokker-Planck/SDE system are plausible and potentially valuable. The numerical scheme is clearly presented and the Hegselmann-Krause example is relevant. However, the central compactness/convergence claim is not proved as written, so the main theoretical contribution is currently conditional.

major comments (4)
  1. [§4, Corollary 4.5 and Appendix 8.2] The proof of Corollary 4.5 is invalid. It asserts: 'From the Assumption 4.1 and the Arzela-Ascoli theorem we have that any sequence (f_i), for f_i ∈ C(K×P_2(K);R^l) has a uniformly converging subsequence to f*'. Assumption 4.1 only requires each f to be Lipschitz and bounded; no common Lipschitz constant or common bound is imposed. For example, on K=[0,1], f_N(y,μ)=sin(N y_1) is bounded by 1 and N-Lipschitz, so every f_N satisfies Assumption 4.1, but the sequence has no uniformly convergent subsequence. Likewise, the claim that compactness of K gives a uniformly convergent subsequence of (h_N) is false without equicontinuity: h_N(t)=e^{2π i N t} maps [0,T] into a compact circle but has no uniformly convergent subsequence. Thus the existence of a convergent subsequence of finite-N minimizers is not established. The proof also assumes that minimizers u*_N of (3) exist; no existence theorem
  2. [Abstract and §2, Eq. (3)] The abstract claims that 'the unique optimal control of the finite agent system converges to the optimal control of the limiting system.' The paper does not prove existence or uniqueness of minimizers of J_N over the unrestricted class U in §2; it only states 'If there exists a unique minimizer...' and Theorem 5.2 concerns a finite-dimensional admissible subset and gives at most one critical point. Corollary 4.5, even if its compactness argument were valid, gives only a subsequential limit, not convergence of a unique optimizer. The claims should be weakened to subsequential convergence (and uniqueness proved separately, or removed).
  3. [§4, Theorem 4.4 and Lemma 4.2] The Γ-convergence statement is not formulated with respect to a specified topology on U. Lemma 4.2 assumes a particular sequence satisfying uniform convergence h_N→h, f_N→f and, crucially, that all f_N share a common Lipschitz constant and common bound. In Γ-convergence, the liminf inequality must hold for every sequence converging in the underlying topology. If the topology is uniform convergence on the relevant function space, there are sequences converging to an admissible f whose Lipschitz constants are unbounded (e.g., f_N(y)=(1/N)sin(N^2 y_1) converges uniformly to 0 but has Lipschitz constant N); such sequences are outside Lemma 4.2. Thus Theorem 4.4 currently proves only a restricted form of Γ-convergence, not the full statement. The authors should define the topology on U and either prove the full liminf inequality in that topology or explicitly restrict the theorem and Corollar
  4. [§6, numerical example] The numerical experiments use the Hegselmann-Krause model with a discontinuous interaction kernel a(r)=1_{r≤R} and with the torus as state space. This does not satisfy Assumption 2.1 (global Lipschitz drift) or the compact-subset-of-R^d setting of Assumption 4.1. The paper acknowledges at the start of Section 6 that not all theoretical assumptions are satisfied. As written, the numerics are illustrative rather than a validation of the convergence theorems; this should be stated more prominently, and the relation of the torus example to the compact-K framework should be clarified.
minor comments (5)
  1. [Abstract] The abstract contains the phrase 'convergences' where 'converges' is intended, and the uniqueness claim is unsupported; see major comment.
  2. [Assumption 4.1 and Corollary 4.5 proof] Assumption 4.1 defines f with values in R^d, while the Corollary 4.5 proof writes f_i∈C(K×P_2(K);R^l) with R^l. This notational inconsistency should be fixed.
  3. [§4, paragraph after Eq. (8)] Corollary 4.5 says the mean-field optimal control u* 'exists a.s.' for a deterministic control problem; the phrase 'a.s.' is unclear and should be removed or explained.
  4. [§5.2] In Corollary 5.3 and Remark 5.4, the statement 'when N=0 in (1)' is confusing since N denotes the number of followers; it should say 'when there are no followers' or otherwise explain the notation.
  5. [§6] The statement that more Monte-Carlo samples 'had no significant effect' is informal; reporting a convergence check or confidence interval would strengthen the numerical section.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central Γ-convergence and mean-field-limit theorems are proved via external results [11,12,25]; the paper's self-citations are non-load-bearing background/numerical references.

full rationale

The derivation chain is self-contained against external benchmarks. The central claims are (i) the propagation-of-chaos result Theorem 3.4, proved in Appendix 8.1 by a synchronous-coupling/Grönwall argument extending Carmona–Zhu [12]; (ii) the Γ-convergence of J_N to J (Theorem 4.4), which follows from Theorem 3.4, the continuity estimates in Lemmas 4.2–4.3, and the recovery sequence J_N(u)→J(u) via (33); and (iii) the uniqueness of the coupled PDE/SDE system (Theorem 5.11), proved by adapting Kurtz–Xiong [25]. None of these steps reduces to its inputs by construction: J_N(u)→J(u) for fixed u is the content of the propagation-of-chaos estimate, not a definitional identity, and the liminf inequality of Lemma 4.3 is a genuine analysis argument. The self-citations ([10], [13], [29], [52], [53]) appear only in the literature review, in describing the origin of the gradient-descent algorithm, and in off-hand remarks; the derivative formulas used numerically are quoted from the external preprint [14] (Lie, arXiv:2106.09149), and the control-convergence theorem relies on external [11,12,25]. No parameter is fitted and no external benchmark is 'predicted': the Hegselmann–Krause experiment is a demonstration with chosen parameters, so the fitted-input-called-prediction pattern is absent. Limitations flagged in the text—the compactness restriction ('this assumption is often made when studying the mean-field limit of interacting particle systems') and the admission that the numerical example does not satisfy all assumptions ('Although not all theoretical assumptions are satisfied, the numerical results show good qualitative behaviour')—are honest scope statements, not circularity. The most serious issue identified by the skeptic is a correctness gap in Corollary 4.5: the proof asserts that Assumption 4.1 plus Arzelà–Ascoli yields a uniformly convergent subsequence of (f_i), but Assumption 4.1 imposes no common bound or common Lipschitz constant across the sequence, so the assertion does not follow as written; nor does Corollary 4.5 verify that the actual minimizers u*_N satisfy the uniformity conditions required by Lemma 4.2/Theorem 4.4. This is an invalid-inference/omitted-proof problem, repairable by strengthening Assumption 4.1, not an equivalence-by-construction problem, so it does not raise the circularity score. Likewise, Corollary 5.3 ('This result can be adapted in a straightforward manner') is an unproved extension of the external Theorem 5.2; again a rigor

Assumptions & free parameters 7 free parameters · 7 assumptions · 0 invented entities

The theoretical results introduce no fitted parameters and no new entities. The numerical demonstration uses hand-chosen model parameters (k, k_L, σ, R, λ, N, m) and a standard Fokker-Planck discretization; these do not affect the proof of the convergence theorems but are needed to reproduce Figures 1-5.

free parameters (7)
  • λ (control cost weight) = 0.01
    Hand-chosen in the numerical HK experiment (§6); controls the trade-off between consensus cost and control effort. Not fitted to data.
  • σ (noise amplitude) = 0.05, 0.1, 0.2
    Three values used in §6; hand-chosen to test weak/strong noise.
  • k, k_L (HK interaction strengths) = k=10, k_L=5
    Parameters of the Hegselmann-Krause model in §6; hand-chosen.
  • R (HK confidence radius) = 0.15
    Interaction radius in the HK model (§6).
  • N (number of followers) = 99
    Finite population size in the numerical example (§6).
  • m (number of control intervals) = 5 or 6
    Piecewise-constant control basis size (§6); text inconsistently says 'five evenly spaced intervals (m=6)'.
  • Monte-Carlo samples = 10^4
    Number of MC paths used to estimate expectations in Algorithm 1 (§6).
assumptions (7)
  • domain assumption Standard Itô SDE well-posedness: coefficients b, σX, c, σY and u are measurable, uniformly continuous in t and globally Lipschitz in state and measure (Assumption 2.1); square-integrable initial data (Assumptions 2.2-2.3).
    Invoked for existence/uniqueness of the finite and limiting systems (Section 2-3).
  • domain assumption Running cost r satisfies the growth/Lipschitz condition in Assumption 2.4.
    Needed to make the cost functional well-defined.
  • domain assumption Compact state space K⊂R^d and admissible controls of the form u(t,y,μ)=h(t)f(y,μ) with h∈C([0,T];K), f Lipschitz and bounded (Assumption 4.1).
    Crucial for the subsequential compactness argument in Corollary 4.5; acknowledged as restrictive in Section 4.
  • domain assumption σY is invertible and uniformly elliptic (Assumption 5.1).
    Required for the explicit Girsanov derivative formulas (12)-(13) used by the numerical scheme.
  • domain assumption Coefficients b and σX are bounded (Assumption 5.8).
    Needed for the L^2 bound in Lemma 5.9 and uniqueness of the non-linear Fokker-Planck equation (Theorem 5.11).
  • standard math Girsanov/Cameron-Martin theorem and Banach fixed point / Arzela-Ascoli / Gronwall arguments.
    Standard tools used in the proofs.
  • domain assumption The finite-N optimal control problem has a (unique) minimizer (Section 2, 'If there exists a unique minimiser...').
    Existence of minimizers for J_N is not proved; Corollary 4.5 relies on a sequence of such minimizers.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Mean-field optimal control with stochastic leaders." pith.science (2026). https://pith.science/paper/EXSA3WQH

@misc{pith2026251219201,
  author       = {Pith},
  title        = {Pith review of: Mean-field optimal control with stochastic leaders},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EXSA3WQH}},
  note         = {Machine review of arXiv:2512.19201}
}
read the original abstract

We consider interacting agent systems with a large number of stochastic agents influenced by a fixed number of external stochastic lead agents. Such settings arise, for example in models of opinion dynamics, where a small number of leaders can steer the behaviour of a large population of followers. In this context, we study a partial mean-field limit where the number of followers tends to infinity, while the number of leaders stays constant. The partial mean-field limit dynamics is then given by a McKean-Vlasov stochastic differential equation (SDE) for the followers, coupled to a controlled It\^o-SDE governing the dynamics of the lead agents. For a given cost functional that the lead agents seek to minimise, we show that the unique optimal control of the finite agent system converges to the optimal control of the limiting system. This establishes that the low-dimensional control of the partial (mean-field) system provides an effective approximation for controlling the high-dimensional finite agent system. In addition, we propose a stochastic gradient descent algorithm that can efficiently approximate the mean-field control. Our theoretical results are illustrated on opinion dynamics model with lead agents, where the control objective is to drive the followers to reach consensus.

Figures

Figures reproduced from arXiv: 2512.19201 by the authors.

Figure 1
Figure 1. Left: Simulation of the noisy Hegselmann-Krause model for [PITH_FULL_IMAGE:figures/full_fig_p020_1.png] view at source ↗
Figure 2
Figure 2. Results obtained by implementing the gradient descent based on Algorithm [PITH_FULL_IMAGE:figures/full_fig_p021_2.png] view at source ↗
Figure 3
Figure 3. Left: System dynamics under the optimal discrete-time Markov control com [PITH_FULL_IMAGE:figures/full_fig_p022_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Similar to Fig [PITH_FULL_IMAGE:figures/full_fig_p023_4.png]
Figure 5
Figure 5. Figure 5: Left: System dynamics of the limiting non-linear Fokker-Planck equation [PITH_FULL_IMAGE:figures/full_fig_p024_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

53 extracted references · 1 canonical work pages

  1. [12]

    A probabilistic approach to mean field games with major and minor players

    Ren´ e Carmona and Xiuneng Zhu. A probabilistic approach to mean field games with major and minor players. The Annals of Applied Probability, 26(3):1535– 1580, 2016

    2016

  2. [25]

    Kurtz and Jie Xiong

    Thomas G. Kurtz and Jie Xiong. Particle representations for a class of nonlinear SPDEs. Stochastic Processes and their Applications, 83(1):103–126, 1999. 26

    1999

  3. [1]

    Ergodic theorems for weakly interacting infinite systems and the voter model

    Richard A Holley and Thomas M Liggett. Ergodic theorems for weakly interacting infinite systems and the voter model. The Annals of Probability, pages 643–663, 1975

    1975

  4. [2]

    Collective motion

    Tam´ as Vicsek and Anna Zafeiris. Collective motion. Physics Reports, 517(3):71– 140, 2012. Collective motion

    2012

  5. [3]

    Large deviations for a mean field model of systemic risk

    Josselin Garnier, George Papanicolaou, and Tzu-Wei Yang. Large deviations for a mean field model of systemic risk. SIAM Journal on Financial Mathematics, 4(1):151–184, 2013

    2013

  6. [4]

    Mathematical modeling of collective behavior in socio-economic and life sciences

    Giovanni Naldi, Lorenzo Pareschi, and Giuseppe Toscani, editors. Mathematical modeling of collective behavior in socio-economic and life sciences. Modeling and Simulation in Science, Engineering and Technology Ser. Springer Science+Business Media, New York, 2010

    2010

  7. [5]

    Feedback loops in opinion dynamics of agent-based models with multiplicative noise

    Natasa Djurdjevac Conrad, Jonas K¨ oppl, and Ana Djurdjevac. Feedback loops in opinion dynamics of agent-based models with multiplicative noise. Entropy, 24(10), 2022

    2022

  8. [6]

    Topics in propagation of chaos

    Alain-Sol Sznitman. Topics in propagation of chaos. In Ecole d’Et´ ede Probabilit´ es de Saint-Flour XIX — 1989, pages 165–251, Berlin, Heidelberg, 1991. Springer Berlin Heidelberg

    1989

Show all 53 references
  1. [7]

    The flow of human crowds

    Roger L Hughes. The flow of human crowds. Annual review of fluid mechanics, 35(1):169–182, 2003

    2003

  2. [8]

    A consensus-based model for global optimization and its mean-field limit

    Ren´ e Pinnau, Claudia Totzeck, Oliver Tse, and Stephan Martin. A consensus-based model for global optimization and its mean-field limit. Mathematical Models and Methods in Applied Sciences, 27(01):183–204, 2017

    2017

  3. [9]

    Mean-field sparse op- timal control

    Massimo Fornasier, Benedetto Piccoli, and Francesco Rossi. Mean-field sparse op- timal control. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 372(2028):20130400, 2014

  4. [10]

    Modelling opinion dynamics under the impact of influencer and media strategies

    Luzie Helfmann, Natasa Djurdjevac Conrad, Philipp Lorenz-Spreen, and Christof Sch¨ utte. Modelling opinion dynamics under the impact of influencer and media strategies. Scientific Reports, 13, 2023

    2023

  5. [11]

    Mean-field sparse optimal control of systems with additive white noise

    Giacomo Ascione, Daniele Castorina, and Francesco Solombrino. Mean-field sparse optimal control of systems with additive white noise. SIAM Journal on Mathematical Analysis, 55(6), 2023. 25

    2023

  6. [13]

    Efficient rare event simulation by optimal nonequilibrium forcing

    Carsten Hartmann and Christof Sch¨ utte. Efficient rare event simulation by optimal nonequilibrium forcing. Journal of Statistical Mechanics: Theory and Experiment, 2012(11), 2012

    2012

  7. [14]

    Fr´ echet derivatives of expected functionals of solutions to stochastic differential equations

    Han Cheng Lie. Fr´ echet derivatives of expected functionals of solutions to stochastic differential equations. Unpublished preprint, 2021. arXiv:2106.09149

    2021 arXiv

  8. [15]

    Probability Theory and Stochastic Modelling Ser

    Ren´ e Carmona and Fran¸ cois Delarue.Probabilistic Theory of Mean Field Games with Applications I. Probability Theory and Stochastic Modelling Ser. Springer, 2018

    2018

  9. [16]

    Control of McKean- Vlasov dynamics versus mean field games

    Ren´ e Carmona, Fran¸ cois Delarue, and Aim´ e Lachapelle. Control of McKean- Vlasov dynamics versus mean field games. Mathematics and Financial Economics, 7(2):131–166, 2012

    2012

  10. [17]

    Propagation of chaos: A review of models, methods and applications

    Louis-Pierre Chaintron and Antoine Diez. Propagation of chaos: A review of models, methods and applications. I. models and methods. Kinetic and Related Models, 15(6):895, 2022

    2022

  11. [18]

    Propagation of chaos: A review of models, methods and applications

    Louis-Pierre Chaintron and Antoine Diez. Propagation of chaos: A review of models, methods and applications. II. applications. Kinetic and Related Models, 15(6):1017, 2022

    2022

  12. [19]

    Mean Field Games and Mean Field Type Control Theory

    Alain Bensoussan, Jens Frehse, and Phillip Yam. Mean Field Games and Mean Field Type Control Theory. Springer New York, 2013

    2013

  13. [20]

    Dynamic programming for mean-field type control

    Mathieu Lauri` ere and Olivier Pironneau. Dynamic programming for mean-field type control. Comptes Rendus. Math´ ematique, 352(9):707–713, 2014

    2014

  14. [21]

    A maximum principle for SDEs of mean- field type

    Daniel Andersson and Boualem Djehiche. A maximum principle for SDEs of mean- field type. Applied Mathematics & Optimization, 63(3):341–356, 2010

    2010

  15. [22]

    Forward–backward stochastic differential equations and controlled McKean-Vlasov dynamics

    Ren´ e Carmona and Fran¸ cois Delarue. Forward–backward stochastic differential equations and controlled McKean-Vlasov dynamics. The Annals of Probability, 43(5), 2015

    2015

  16. [23]

    Limit theory for controlled McKean-Vlasov dynamics

    Daniel Lacker. Limit theory for controlled McKean-Vlasov dynamics. SIAM Journal on Control and Optimization, 55(3):1641–1672, 2017

    2017

  17. [24]

    McKean-Vlasov optimal control: Limit theory and equivalence between different formulations

    Mao Fabrice Djete, Dylan Possama ¨ ı, and Xiaolu Tan. McKean-Vlasov optimal control: Limit theory and equivalence between different formulations. Mathematics of Operations Research, 47(4):2891–2930, 2022

    2022

  18. [26]

    Mean field games and applications: Numerical aspects

    Yves Achdou, Pierre Cardaliaguet, Fran¸ cois Delarue, Alessio Porretta, Filippo San- tambrogio, Yves Achdou, and Mathieu Lauri` ere. Mean field games and applications: Numerical aspects. Mean Field Games: Cetraro, Italy 2019, pages 249–307, 2020

    2019

  19. [27]

    Kushner and Paul G

    Harold J. Kushner and Paul G. Dupuis. Numerical methods for stochastic control problems in continuous time. Springer-Verlag, Berlin, Heidelberg, 1992

    1992

  20. [28]

    Numeri- cal methods for backward stochastic differential equations: A survey

    Jared Chessari, Reiichiro Kawai, Yuji Shinozaki, and Toshihiro Yamada. Numeri- cal methods for backward stochastic differential equations: A survey. Probability Surveys, 20:486–567, 2023

    2023

  21. [29]

    Variational approach to rare event simulation using least-squares regression.Chaos, 29(6), 2019

    Carsten Hartmann, Omar Kebiri, Lara Neureither, and Lorenz Richter. Variational approach to rare event simulation using least-squares regression.Chaos, 29(6), 2019

    2019

  22. [30]

    Adaptive importance sampling with forward-backward stochastic differential equations

    Omar Kebiri, Lara Neureither, and Carsten Hartmann. Adaptive importance sampling with forward-backward stochastic differential equations. In Stochastic dynamics out of equilibrium. Lecture notes from the IHP trimester, Institut Henri Poincar´ e(IHP), Paris, France, April – Jul...

    2017

  23. [31]

    Stochastic Approximation and Recursive Algorithms and Applications

    Harold J Kushner and George Yin. Stochastic Approximation and Recursive Algorithms and Applications. Springer, New York, 2003

    2003

  24. [32]

    Regression meth- ods for stochastic control problems and their convergence analysis

    Denis Belomestny, Anastasia Kolodko, and John Schoenmakers. Regression meth- ods for stochastic control problems and their convergence analysis. SIAM Journal on Control and Optimization, 48(5):3562–3588, 2010

    2010

  25. [33]

    Applications of the cross-entropy method to importance sampling and optimal con- trol of diffusions

    Wei Zhang, Han Wang, Carsten Hartmann, Marcus Weber, and Christof Sch¨ utte. Applications of the cross-entropy method to importance sampling and optimal con- trol of diffusions. SIAM Journal on Scientific Computing, 36(6):A2654–A2672, 2014

    2014

  26. [34]

    Solving high-dimensional Hamilton–Jacobi– Bellman PDEs using neural networks: perspectives from the theory of controlled dif- fusions and measures on path space

    Nikolas N¨ usken and Lorenz Richter. Solving high-dimensional Hamilton–Jacobi– Bellman PDEs using neural networks: perspectives from the theory of controlled dif- fusions and measures on path space. Partial differential equations and applications, 2(4):48, 2021

    2021

  27. [35]

    Policy gradient methods

    Jan Peters and J Andrew Bagnell. Policy gradient methods. In Encyclopedia of Machine Learning, pages 774–776. Springer, 2011

    2011

  28. [36]

    Noisy Hegselmann- Krause systems: Phase transition and the 2R-conjecture

    Chu Wang, Qianxiao Li, Weinan E, and Bernard Chazelle. Noisy Hegselmann- Krause systems: Phase transition and the 2R-conjecture. Journal of Statistical Physics, 166(5):1209–1225, January 2017

    2017

  29. [37]

    Opinion dynamics and bounded confi- dence: Models, analysis and simulation

    Rainer Hegselmann and Ulrich Krause. Opinion dynamics and bounded confi- dence: Models, analysis and simulation. Journal of Artificial Societies and Social Simulation, 5(3), 2002. 27

    2002

  30. [38]

    On the control through lead- ership of the Hegselmann-Krause opinion formation model

    Suttida Wongkaew, Marco Caponigro, and Alfio Borz ` ı. On the control through lead- ership of the Hegselmann-Krause opinion formation model. Mathematical Models and Methods in Applied Sciences, 25(03):565–585, 2014

    2014

  31. [39]

    Kinetic description of opti- mal control problems and applications to opinion consensus

    Giacomo Albi, Michael Herty, and Lorenzo Pareschi. Kinetic description of opti- mal control problems and applications to opinion consensus. Communications in Mathematical Sciences, 13(6):1407–1429, 2015

    2015

  32. [40]

    Pavliotis

    Sara Bicego, Dante Kalise, and Grigorios A. Pavliotis. Computation and control of unstable steady states for mean field multiagent systems. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 481(2311), 2025

    2025

  33. [41]

    Probability Theory and Stochastic Modelling Ser

    Ren´ e Carmona and Fran¸ cois Delarue.Probabilistic Theory of Mean Field Games with Applications II. Probability Theory and Stochastic Modelling Ser. Springer, 2018

    2018

  34. [42]

    William R. P. Hammersley, David ˇSiˇ ska, and Lukasz Szpruch. Weak existence and uniqueness for McKean-Vlasov SDEs with common noise. The Annals of Probability, 49(2):527 – 555, 2021

    2021

  35. [43]

    Quantitative relative entropy estimates for interacting particle sys- tems with common noise

    Paul Nikolaev. Quantitative relative entropy estimates for interacting particle sys- tems with common noise. SIAM Journal on Mathematical Analysis, 57(3), 2025

    2025

  36. [44]

    Robert A. Crowell. McKean-Vlasov SDEs: New results on existence of weak solutions and on propagation of chaos. Phd thesis, ETH Z¨ urich, 2025. Available at https://doi.org/10.3929/ethz-b-000723907

    2025 doi

  37. [45]

    William R. P. Hammersley. On the McKean-Vlasov dynamics with or without common noise. Phd thesis, The University of Edinburgh, 2021. Available athttps: //core.ac.uk/download/pdf/478960525.pdf

    2021

  38. [46]

    Jakobsen, Athena Picarelli, and Christoph Reisinger

    Espen R. Jakobsen, Athena Picarelli, and Christoph Reisinger. Improved order 1/4 convergence for piecewise constant policy approximation of stochastic control problems. Electronic Communications in Probability, 24, 2019

    2019

  39. [47]

    Improving control based importance sampling strategies for metastable diffusions via adapted metadynamics

    Enric Ribera Borrell, Jannes Quer, Lorenz Richter, and Christof Sch¨ utte. Improving control based importance sampling strategies for metastable diffusions via adapted metadynamics. SIAM Journal on Scientific Computing, 46(2):S298–S323, 2023

    2023

  40. [48]

    Stannat, A

    W. Stannat, A. Vogler, and L. Wessels. Neural network approximation of optimal controls for stochastic reaction–diffusion equations. Chaos: An Interdisciplinary Journal of Nonlinear Science, 33(9), 2023

    2023

  41. [49]

    Convergence analysis of machine learning algorithms for the numerical solution of mean field control and games: II—the finite horizon case

    Ren´ e Carmona and Mathieu Lauri` ere. Convergence analysis of machine learning algorithms for the numerical solution of mean field control and games: II—the finite horizon case. The Annals of Applied Probability, 32(6), 2022. 28

    2022

  42. [50]

    Consensus convergence with stochastic effects

    Josselin Garnier, George Papanicolaou, and Tzu-Wei Yang. Consensus convergence with stochastic effects. Vietnam Journal of Mathematics, 45(1–2), 2016

    2016

  43. [51]

    Structure preserving schemes for nonlin- ear Fokker–Planck equations and applications

    Lorenzo Pareschi and Mattia Zanella. Structure preserving schemes for nonlin- ear Fokker–Planck equations and applications. Journal of Scientific Computing, 74:1575–1600, 03 2018

    2018

  44. [52]

    From interacting agents to density-based modeling with stochastic PDEs

    Luzie Helfmann, Nataˇ sa Djurdjevac Conrad, Ana Djurdjevac, Stefanie Winkelmann, and Christof Sch¨ utte. From interacting agents to density-based modeling with stochastic PDEs. Communications in Applied Mathematics and Computational Science, 16(1), 2021

    2021

  45. [53]

    sup t≤T W 2 2 (gN,t, gt) # ≤E

    Ana Djurdjevac, Ann Almgren, and John Bell. A hybrid algorithm for systems of noninteracting particles with an external potential. Communications in Applied Mathematics and Computational Science, 20(1), 2025. 8 Appendix 8.1 Proofs for Section 3 To begin let us recall the follo...

    2025

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.