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Pontryagin Maximum Principle for McKean-Vlasov Stochastic Reaction-Diffusion Equations

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Pith's one-line read For the stochastic control of McKean–Vlasov reaction-diffusion systems, an optimal control must minimize the associated Hamiltonian pointwise, and the paper makes this condition explicit through an adjoint backward SPDE with well-posed…

desk verdict Substantial technical advance in control of McKean-Vlasov SPDEs, but Theorem 5.6 is false as stated when the L^q control budget is active; the switching argument breaks and a scalar LQ counterexample kills the pointwise PMP over all of U. read the letter →

arxiv 2507.16288 v2 pith:DQLAVAMC submitted 2025-07-22 math.PR math.OC

Johan Benedikt Spille , Wilhelm Stannat This is my paper · ORCID
classification math.PRmath.OC MSC 93E2049K4549N8060H1535K57
keywords PontryaginmaximumprincipleMcKean-Vlasovstochasticreaction-diffusionequationbackwardSPDELionsderivativeoptimalcontrolvariationalapproachtomean-fieldcompactnessmethod
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 proves a Pontryagin maximum principle — a first-order necessary condition for optimality — for stochastic control of McKean–Vlasov stochastic reaction-diffusion equations, where the drift and diffusion coefficients depend on the state, the control, and the probability law of the state. The main result (Theorem 5.6) states that at an optimal control the Hamiltonian is minimized pointwise, almost surely in time and randomness, by the optimal control value; the companion gradient formula expresses the derivative of the cost as the expectation of the Hamiltonian's control derivative. To reach it, the authors prove Gateaux differentiability of the control-to-state map, a new existence and uniqueness theorem for the linear backward McKean–Vlasov SPDE that serves as the adjoint equation, and — when controls are deterministic — existence of a minimizer via a compactness method whose folklore ingredients are proved in full in the appendix. If correct, the paper supplies the missing first-order calculus for a class of mean-field distributed-parameter control problems, with explicit examples including the linear-quadratic case and a reaction-diffusion equation with cubic nonlinearity.

What carries the argument

The load-bearing object is the infinite-dimensional Lions derivative. For a map $h:\mathcal{P}_2(H)\to V^*$, one lifts to $\hat{h}(X) = h(\mathcal{L}(X))$ and differentiates in the Hilbert-space variable; the derivative $D\hat{h}(X)$ is then factored through the law $\mu = \mathcal{L}(X)$ by a Radon–Nikodym theorem for vector measures, giving $\partial_\mu h(\mu)(X)$ with $D\hat{h}(X)Y = \mathbb{E}[\partial_\mu h(\mu)(X)Y]$. Under the resulting 'Λ-continuously L-differentiable' condition (Definition 2.1), the law-derivatives $\partial_\mu F$, $\partial_\mu B$, $\partial_\mu f$, $\partial_\mu g$ become well-defined Hilbert-space operators rather than scalars, which is what makes the adjoint equation (19) meaningful and solvable. The proof of the maximum principle then runs through Galerkin approximation of the backward SPDE, a fixed-point contraction handling the law-coupling, and — for the existence theorem — tightness on the path spaces $L^2([0,T],V)_{\text{weak}}$, $C([0,T],H_{\text{weak}})$, and $L^2([0,T],H)$ together with the generalized Skorokhod embedding.

What would settle it

Take the reaction-diffusion example of Section 7.2 ($F_1(x) = -x^3$, $d \le 2$), pick an admissible control $\alpha$, solve the state equation (1) and the adjoint equation (16)–(18), and compare the gradient predicted by Corollary 5.5, $dJ(\alpha)\cdot\beta = \mathbb{E}[\int_0^T H_\alpha(\beta_t)\,dt]$, with a finite-difference evaluation of $J$ along a small step $\beta$; any discrepancy beyond numerical tolerance would falsify the adjoint calculus on which Theorem 5.6 rests.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is a set of first-order optimality conditions for the control problem (1)–(2): for every optimal control $\alpha^*$, the Hamiltonian $H(t, X^{\alpha^*}_t, \mathcal{L}(X^{\alpha^*}_t), \alpha, P_t, Q_t)$ is minimized over the control space at $\alpha = \alpha^*_t$ for $P\otimes dt$-almost every $(t,\omega)$, where $(P,Q)$ is the unique solution of the adjoint backward McKean–Vlasov SPDE (16)–(18). The same adjoint calculus yields the explicit gradient identity $dJ(\alpha)\cdot\beta = \mathbb{E}\big[\int_0^T H_\alpha(t, X^\alpha_t, \mathcal{L}(X^\alpha_t), \alpha_t, P_t, Q_t)(\beta_t)\,dt\big]$. The supporting results are Gateaux differentiability of the control-to-state map with derivative characterized by the linearized SPDE (11), existence and uniqueness of the adjoint backward SPDE under monotonicity and growth conditions that are not Lipschitz uniform in $\omega$, and, for deterministic controls, existence of an optimal control obtained by tightness on three path spaces followed by a generalized Skorokhod representation.

Load-bearing premise

The argument rests on the new infinite-dimensional Lions derivative — a vector-measure Radon–Nikodym factorization of the lifted map's derivative — being a sound calculus, since the paper imports it as a black box and uses it to define every law-derivative in the adjoint equation; if that derivative machinery has a gap, the adjoint equation, the gradient formula, and the maximum principle lose their foundation.

Editorial extensions

If this is right

  • Theorem 5.6 gives a checkable necessary condition: any candidate control that fails the pointwise Hamiltonian inequality cannot be optimal, so the condition can prune search spaces before evaluating costs.
  • Corollary 5.5 turns the cost functional into a differentiable object with an explicit gradient $dJ(\alpha)\cdot\beta = \mathbb{E}[\int_0^T H_\alpha(\beta)\,dt]$, opening the door to descent algorithms for law-dependent distributed control.
  • Theorem 6.4 guarantees at least one deterministic minimizer under a compact embedding of $V$ into $H$ and the continuity assumptions 6.1–6.2, covering linear-quadratic costs and polynomial reaction-diffusion nonlinearities.
  • The well-posedness result for the linear backward McKean–Vlasov SPDE extends adjoint calculus to coefficients that are not Lipschitz uniform in $\omega$, a regime earlier forward-backward mean-field SPDE treatments did not cover.
  • The appendix supplies full proofs of the compactness lemmas and the nonmetric Skorokhod representation, converting previously folklore tools into checkable statements.

Reading between the lines

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

  • Where the Hamiltonian is convex in the control (Assumption 2.10), the necessary inequality of Theorem 5.6 becomes sufficient as well in the linear-quadratic case: any control satisfying it is globally optimal, so the theorem doubles as a verification result there.
  • Because the derivative calculus is stated for Banach-space-valued functions of the law, the same adjoint construction could carry over to costs and dynamics depending on higher moments or on the law of derived processes, provided the vector-measure factorization remains available.
  • The tightness route used for deterministic controls suggests a concrete extension: proving existence of adapted optimal controls would require tightness in the joint law of $(X, \alpha)$ plus passage to the limit inside the constraint set $A$, a step the current proof avoids by minimizing over deterministic controls only.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. The paper studies optimal control of a class of McKean-Vlasov semilinear SPDEs in the variational (monotone operator) framework. The main advertised results are: (i) well-posedness of the state equation under a progressive measurability and integrability constraint on controls (the set A defined by an L^q budget); (ii) Gateaux differentiability of the control-to-state map and of the cost functional; (iii) existence and uniqueness of a linear McKean-Vlasov backward SPDE serving as the adjoint equation; (iv) a representation formula for the gradient of the cost; (v) a Pontryagin maximum principle (Theorem 5.6) asserting that an optimal control minimizes the Hamiltonian pointwise over the whole control space U; and (vi) existence of optimal deterministic controls via a compactness method. The technical novelty is the use of an infinite-dimensional Lions derivative for Banach-valued functions developed in the companion preprint [1].

Significance. If the central claims were correct, the paper would be a significant contribution: it would provide the first Pontryagin maximum principle for mean-field SPDEs with general distribution dependence in both drift and diffusion coefficients, and it also proves a new well-posedness result for linear McKean-Vlasov backward SPDEs. The paper contains several useful and carefully proved ingredients: strong a priori estimates, Lipschitz continuity of the control-to-state map, a detailed Galerkin construction for the adjoint equation, and a self-contained proof of the compactness method used for the existence of deterministic optimal controls. However, the main advertised theorem, Theorem 5.6, is false as stated, and this is not a matter of presentation: the claimed pointwise Hamiltonian inequality fails under an active L^q budget constraint. Consequently the central contribution cannot stand, despite the value of some auxiliary results.

major comments (3)
  1. [Theorem 5.6 and Section 5.3] The proof of the Pontryagin maximum principle switches to the control β_t = α 1_C(t) + α*_t 1_{C^c}(t), where C is a progressively measurable set and α∈U, and asserts that β is admissible. This is not true in general. The admissible set A is defined in Section 2 by ∫_0^T ‖α_t‖_U^q dt ≤ K P-a.s.; for the switched control, ∫_0^T ‖β_t‖^q dt = ∫_{C^c} ‖α*_t‖^q dt + ∫_C ‖α‖^q dt. If α* already saturates the budget (∫‖α*‖^q = K) and ‖α‖^q exceeds ‖α*_t‖^q on a positive-measure subset of C, this integral is strictly larger than K, so β∉A and the variational inequality dJ(α*)·(β−α*)≥0 cannot be invoked. The failure is not merely a gap in the argument: the conclusion of Theorem 5.6 is false when the integral control constraint is active. A concrete counterexample within the paper's assumptions is the deterministic scalar LQ problem with H=U=V=R, L=-1, F(t,x,µ,α)=-x+α, B=0, f=x^2+α^2, g=0, q=4, X0=1, and K small enough that the constraint ∫|α|^4 dt ≤ K is active. Assumptions 2.4–2.10 are satisfied. For the constrained optimizer α*, the KKT condition is 2α*_t + P_t + 4λ|α*_t|^2α*_t = 0 with λ>0, whereas the Hamiltonian H(α)=P_t(-X_t+α)+X_t^2+α^2 is minimized pointwise at α=-P_t/2. Since λ>0 and P_t is not identically zero, α*_t≠-P_t/2 on a set of positive measure, so H(α*_t)>H(-P_t/2), contradicting the theorem. A correct PMP in this setting would need an additional Lagrange multiplier for the L^q budget constraint, and the theorem as stated cannot be repaired by minor edits.
  2. [Theorem 4.1 and Corollaries 4.3, 5.5] The Gateaux differentiability statements overstate the valid domain. Theorem 4.1 claims differentiability of the control-to-state map at every α∈A in every direction β∈A. Since A is a closed L^q-ball, for a boundary point α with ∫‖α‖^q dt = K and for a generic β∈A, the segment α+εβ leaves A for all sufficiently small ε>0; indeed the proof explicitly restricts to 'ε>0 such that α+εβ∈A' (page 11). Thus the derivative is not defined in the stated sense. The same issue propagates to Corollaries 4.3 and 5.5, which assert the gradient representation for all α,β∈A. A rigorous formulation should either restrict directions to the tangent cone of A at α, or extend the control-to-state map to the full space L^q(Ω×[0,T],U) (or at least to all progressively measurable L^q controls with finite q-th moment), where the Gateaux derivative can be defined with respect to the ambient norm. This is not merely cosmetic: the proof of Theorem 5.6 uses the derivative only for the feasible direction β−α*, but the statements as written are incorrect and need correction.
  3. [Section 2.1, Equation (4); Assumptions 2.7 and 2.9] The paper depends critically on the infinite-dimensional Lions derivative machinery of the unpublished companion preprint [1], in particular the vector-measure Radon–Nikodym factorization in Equation (4) and the resulting Λ-continuous L-differentiability. This machinery is used to define ∂_µF, ∂_µB, ∂_µf in the adjoint equation (16)–(18) and in Assumptions 2.7 and 2.9. The present manuscript uses this as a black box, and the referee cannot verify the correctness of [1]. Since the well-posedness of the adjoint equation (Theorem 5.3) and the gradient and PMP theorems all rest on this derivative calculus, the authors should either make the required results self-contained in the paper or provide a published, verifiable reference. This is a verification risk even setting aside the counterexample to Theorem 5.6.
minor comments (6)
  1. [Section 2.2, (H3b)] In the statement of Assumption (H3b), '∥α−β|^2_U' is a typo and should read '∥α−β∥^2_U'.
  2. [Theorem 4.1, proof] In the fixed-point part of the proof, the metric is defined as δ(Z^{(1)},Z^{(1)}); this should read δ(Z^{(1)},Z^{(2)}).
  3. [Remark 5.2 and Section 5.1] The notation θ̂ is redefined in Section 5.1 after being used with a different meaning in Section 4. The two conventions should be distinguished more clearly, for instance by using a different symbol for the independent-copy argument.
  4. [Section 5.3] In the proof of Theorem 5.6, the sentence 'Let C ... and α∈A' should read 'α∈U', since β_t uses the pointwise value α in the control space U.
  5. [Assumption 6.2(i)] The phrase 'for for any' is a typo and should read 'for any'.
  6. [Section 7.2] The example is introduced with d≤3 but the existence-of-optimal-control subsection restricts to d≤2; this restriction should be stated explicitly at the beginning of the example and not only later in the text.

Circularity Check

1 steps flagged · score 4.0 of 10

Load-bearing self-citation for the infinite-dimensional Lions derivative, but the PMP derivation itself is not circular by construction.

  1. self citation load bearing [Section 1 (Introduction), p. 2; Section 2.1, pp. 3-4]
    "The key technical novelty is the use of a recent extension of the Lions derivative to Banach space valued functions [1], which allows us to rigorously define the adjoint equation in our infinite-dimensional setting."

    Reference [1] is a preprint by Vogler and Stannat, and Stannat is a co-author of the present paper. The paper explicitly states that this extension is what 'makes this paper possible.' The Λ-continuous L-differentiability in Definition 2.1, the vector-measure Radon-Nikodym factorization in Equation (4), and the resulting derivative objects ∂µF, ∂µB, ∂µf used in Assumptions 2.7/2.9 and in the adjoint equation (16)-(18) are all taken from [1] as a black box. Theorems 5.3 and 5.6 therefore rest on a same-author preprint rather than on an independently verified theorem. This is load-bearing self-citation. It is not, however, a tautological reduction: the PMP does not reduce by construction to the definition of the Hamiltonian or to a fitted parameter.

full rationale

The derivation chain from the state equation to the Gâteaux derivative of the control-to-state map, to the adjoint BSDE, and to the Pontryagin maximum principle is a substantive adjoint-calculus argument. Corollary 5.5 follows from the Itô product rule and the terminal condition of the adjoint equation, and Theorem 5.6 uses convexity together with the variational inequality dJ(α*)·(β−α*)≥0. No fitted parameter is relabeled as a prediction, no known result is renamed, and the Hamiltonian minimization is not equivalent to an assumption by construction. The main circularity-adjacent element is the paper's dependence on [1] for the infinite-dimensional Lions derivative; the paper itself calls this extension the enabling novelty. Since this is a load-bearing same-group preprint and the central differentiability object is defined through it, the score is 4 rather than 0. The skeptical objection about the admissibility of the switched control β in §5.3 is a correctness concern about the variational inequality step, not a circular reduction, so it does not increase the circularity score.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

No empirical parameters; the 'free parameters' are mathematical exponents/thresholds chosen to make the a priori estimates close. The central claim rests on the self-cited Lions derivative framework [1] and a bundle of monotonicity/growth hypotheses.

free parameters (3)
  • q (integrability exponent for controls)
    Chosen by hand to satisfy q > p+2 and q > p′+2; used in a priori estimates and uniform integrability. Not fitted to data.
  • K (upper bound on control L^q-norm)
    Defines the admissible set A; used for estimates and weak compactness. Hand-chosen.
  • p, p′ (polynomial growth exponents)
    Appear in growth conditions (H4), (H4′); assumed q > p+2 and q > p′+2. These are hypotheses, not fitted.
assumptions (6)
  • domain assumption Gelfand triple V⊂H≅H*⊂V* and operator L with discrete spectrum satisfying coercivity (Assumption 2.4)
    Standard variational SPDE framework; used throughout for well-posedness.
  • domain assumption Lions derivative extension to Banach-valued functions from [1], including vector measure Radon-Nikodym factorization
    Load-bearing for defining ∂_µF, ∂_µB, ∂_µf in the adjoint equation; cited as [1], same group preprint.
  • domain assumption Monotonicity, coercivity, growth, and differentiability assumptions on F,B,f,g (Assumptions 2.5, 2.7, 2.9)
    These hypotheses ensure existence/uniqueness of state, adjoint, and derivative formulas.
  • domain assumption Convexity of F,B,f in control (Assumption 2.10)
    Needed to pass from gradient equality to pointwise Hamiltonian inequality in Theorem 5.6.
  • domain assumption Compact embedding V↪H and continuity conditions (Assumptions 6.1, 6.2)
    Used only for existence of deterministic optimal controls (Theorem 6.4).
  • standard math Atomless filtered probability space with cylindrical Wiener process on H
    Ensures existence of lifts and copies; standard in stochastic analysis.

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Cite this review

Pith. "Pith review of Pontryagin Maximum Principle for McKean-Vlasov Stochastic Reaction-Diffusion Equations." pith.science (2026). https://pith.science/paper/DQLAVAMC

@misc{pith2026250716288,
  author       = {Pith},
  title        = {Pith review of: Pontryagin Maximum Principle for McKean-Vlasov Stochastic Reaction-Diffusion Equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DQLAVAMC}},
  note         = {Machine review of arXiv:2507.16288}
}
read the original abstract

We consider the stochastic control of a semi-linear stochastic partial differential equations (SPDE) of McKean-Vlasov type. Based on a recent novel approach to the Lions derivative for Banach space valued functions, we prove the Gateaux differentiability of the control to state map and, using adjoint calculus, we derive explicit representations of the gradient of the cost functional and a Pontryagin maximum principle. On the way, we also prove a novel existence and uniqueness result for linear McKean-Vlasov backward SPDE. Furthermore, for deterministic controls, we prove the existence of optimal controls using a martingale approach and a novel compactness method. This result is complemented in the appendix with a rigorous proof of folklore results on the compactness method in the variational approach to SPDE. Our setting uses the variational approach to SPDE with monotone coefficients, allowing for a polynomial perturbation and allowing the drift and diffusion coefficients to depend on the state, the distribution of the state and the control.

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Reference graph

Works this paper leans on

38 extracted references · 29 canonical work pages · cited by 1 Pith paper

  1. [1]

    The lions derivative in infinite dimensions – ap- plication to higher order expansion of mean-field spdes, 2025

    Alexander Vogler and Wilhelm Stannat. The lions derivative in infinite dimensions – ap- plication to higher order expansion of mean-field spdes, 2025

    work page 2025

  2. [2]

    An introductory approach to duality in optimal stochastic control

    Jean-Michel Bismut. An introductory approach to duality in optimal stochastic control. SIAM Review, 20(1):62–78, 1978. ISSN 00361445, 10957200

    work page 1978

  3. [3]

    A general stochastic maximum principle for optimal control problems.SIAM Journal on Control and Optimization, 28(4):966–979, 1990

    Shige Peng. A general stochastic maximum principle for optimal control problems.SIAM Journal on Control and Optimization, 28(4):966–979, 1990. doi: 10.1137/0328054

    doi:10.1137/0328054 1990

  4. [4]

    Mean field games.Japanese Journal of Mathe- matics, 2(1):229–260, Mar 2007

    Jean-Michel Lasry and Pierre-Louis Lions. Mean field games.Japanese Journal of Mathe- matics, 2(1):229–260, Mar 2007. ISSN 1861-3624. doi: 10.1007/s11537-007-0657-8

    doi:10.1007/s11537-007-0657-8 2007

  5. [5]

    Springer International Publishing, Cham, 2018

    René Carmona and François Delarue.Probabilistic Theory of Mean Field Games with Ap- plications I: Mean Field FBSDEs, Control, and Games. Springer International Publishing, Cham, 2018. ISBN 978-3-319-58920-6. doi: 10.1007/978-3-319-58920-6

    doi:10.1007/978-3-319-58920-6 2018

  6. [6]

    Mean-field stochastic differential equations and associated pdes

    Rainer Buckdahn, Juan Li, Peng Shi-Ge, and Catherine Rainer. Mean-field stochastic differential equations and associated pdes. The Annals of Probability, 45, 07 2014. doi: 10.1214/15-AOP1076

    doi:10.1214/15-aop1076 2014

  7. [7]

    Dynamic programming for optimal control of stochastic mckean–vlasov dynamics

    Huyên Pham and Xiaoli Wei. Dynamic programming for optimal control of stochastic mckean–vlasov dynamics. SIAM Journal on Control and Optimization, 55(2):1069–1101,

    work page

  8. [8]

    Ency- clopedia of Mathematics and its Applications

    Giuseppe Da Prato and Jerzy Zabczyk.Stochastic Equations in Infinite Dimensions. Ency- clopedia of Mathematics and its Applications. Cambridge University Press, 2 edition, 2014. doi: 10.1017/CBO9781107295513

    doi:10.1017/cbo9781107295513 2014

Show all 38 references
  1. [9]

    Stochastic Partial Differential Equations: An Introduc- tion

    Wei Liu and Michael Röckner. Stochastic Partial Differential Equations: An Introduc- tion. Springer International Publishing, 2015. ISBN 978-3-319-22354-4. doi: 10.1007/ 978-3-319-22354-4

    2015

  2. [10]

    Springer International Publishing, Cham, 2021

    Étienne Pardoux.SPDEs as Infinite-Dimensional SDEs, pages 9–39. Springer International Publishing, Cham, 2021. ISBN 978-3-030-89003-2. doi: 10.1007/978-3-030-89003-2_2

    2021 doi

  3. [11]

    Stochasticmaximumprinciplefordistributedparametersystems

    A.Bensoussan. Stochasticmaximumprinciplefordistributedparametersystems. Journal of the Franklin Institute, 315(5):387–406, 1983. ISSN 0016-0032. doi: 10.1016/0016-0032(83) 90059-5. 28

    1983 doi

  4. [12]

    Stochastic maximum principle for optimal control of spdes

    Marco Fuhrman, Ying Hu, and Gianmario Tessitore. Stochastic maximum principle for optimal control of spdes. Comptes Rendus Mathematique, 350(13):683–688, 2012. ISSN 1631-073X. doi: 10.1016/j.crma.2012.07.009

    2012 doi

  5. [13]

    Springer International Publishing, Cham, 2021

    Qi Lü and Xu Zhang.Pontryagin-Type Stochastic Maximum Principle and Beyond, page 387–475. Springer International Publishing, Cham, 2021. ISBN 978-3-030-82331-3. doi: 10.1007/978-3-030-82331-3_12

    2021 doi

  6. [14]

    N.U. Ahmed. A general class of mckean-vlasov stochastic evolution equations driven by brownian motion and lèvy process and controlled by lèvy measure.Discussiones Mathe- maticae, Differential Inclusions, Control and Optimization, 36(2):181–206, 2016

    2016

  7. [15]

    Stochastic control for mean-field stochastic partial differential equations with jumps.Journal of Optimization Theory and Applications, 176(3):559–584, Mar 2018

    Roxana Dumitrescu, Bernt Øksendal, and Agnès Sulem. Stochastic control for mean-field stochastic partial differential equations with jumps.Journal of Optimization Theory and Applications, 176(3):559–584, Mar 2018. ISSN 1573-2878. doi: 10.1007/s10957-018-1243-3

    2018 doi

  8. [16]

    Forward and backward mean-field stochastic partial differential equation and optimal control.Chinese Annals of Mathematics, Series B, 40(4):515–540, Jul 2019

    Maoning Tang, Qingxin Meng, and Meijiao Wang. Forward and backward mean-field stochastic partial differential equation and optimal control.Chinese Annals of Mathematics, Series B, 40(4):515–540, Jul 2019. ISSN 1860-6261. doi: 10.1007/s11401-019-0149-1

    2019 doi

  9. [17]

    Op- timal control of path-dependent mckean-vlasov sdes in infinite dimension.The Annals of Applied Probability, 33:2863–2918, 08 2023

    Andrea Cosso, Fausto Gozzi, Idris Kharroubi, Huyên Pham, and Mauro Rosestolato. Op- timal control of path-dependent mckean-vlasov sdes in infinite dimension.The Annals of Applied Probability, 33:2863–2918, 08 2023. doi: 10.1214/22-AAP1880

    2023 doi

  10. [18]

    Distribution-dependent stochastic porous media equations

    Jingyue Gao, Wei Hong, and Wei Liu. Distribution-dependent stochastic porous media equations. Stochastics and Dynamics, 22, 11 2022. doi: 10.1142/S0219493722400263

    2022 doi

  11. [19]

    Mckean-vlasov sde and spde with locally monotone coefficients

    Wei Hong, Shanshan Hu, and Wei Liu. Mckean-vlasov sde and spde with locally monotone coefficients. The Annals of Applied Probability, 34:2136–2189, 04 2024. doi: 10.1214/ 23-AAP2016

    2024

  12. [20]

    Deterministic control of stochastic reaction-diffusion equations

    Wilhelm Stannat and Lukas Wessels. Deterministic control of stochastic reaction-diffusion equations. Evolution Equations and Control Theory, 10(4):701–722, 2021. doi: 10.3934/ eect.2020087

    2021

  13. [21]

    Optimalcontrolofmeanfieldequationswithmono- tone coefficients and applications in neuroscience.Applied Mathematics & Optimization, 84 (2):1925–1968, Dec 2021

    AntoineHocquetandAlexanderVogler. Optimalcontrolofmeanfieldequationswithmono- tone coefficients and applications in neuroscience.Applied Mathematics & Optimization, 84 (2):1925–1968, Dec 2021. ISSN 1432-0606. doi: 10.1007/s00245-021-09816-1

    1925 doi

  14. [22]

    The Wasserstein distances, pages93–111

    CédricVillani. The Wasserstein distances, pages93–111. SpringerBerlinHeidelberg, Berlin, Heidelberg, 2009. ISBN 978-3-540-71050-9. doi: 10.1007/978-3-540-71050-9_6

    2009 doi

  15. [23]

    N.U. Ahmed. A note on radon-nikodym theorem for operator valued measures and its applications. Communications of the Korean Mathematical Society, 28, 04 2013. doi: 10. 4134/CKMS.2013.28.2.285

    2013

  16. [24]

    Dinculeanu

    N. Dinculeanu. Vector Measures. Hochschulbücher für Mathematik. Elsevier Science & Technology Books, 1967. ISBN 9780080121925

    1967

  17. [25]

    Diestel and J.J

    J. Diestel and J.J. Uhl.Vector Measures. Mathematical surveys and monographs. American Mathematical Society, 1977. ISBN 9780821815151

    1977

  18. [26]

    Springer International Publishing, Cham, 2023

    Zigang Pan.Measure-Theoretic Calculus in Abstract Spaces: On the Playground of Infinite- Dimensional Spaces. Springer International Publishing, Cham, 2023. ISBN 978-3-031- 21912-2. doi: 10.1007/978-3-031-21912-2

    2023 doi

  19. [27]

    David Blackwell and Lester E. Dubins. An extension of Skorohod’s almost sure represen- tation theorem. Proc. Amer. Math. Soc., 89(4):691–692, 1983. ISSN 0002-9939,1088-6826. doi: 10.2307/2044607. 29

    1983 doi

  20. [28]

    Backward Stochastic Differential Equations, pages 353–515

    Etienne Pardoux and Aurel Răşcanu. Backward Stochastic Differential Equations, pages 353–515. Springer International Publishing, Cham, 2014. ISBN 978-3-319-05714-9. doi: 10.1007/978-3-319-05714-9_5

    2014 doi

  21. [29]

    One-dimensional bsdes with finite and infinite time horizons

    ShengJun Fan, Long Jiang, and DeJian Tian. One-dimensional bsdes with finite and infinite time horizons. Stochastic Processes and their Applications, 121(3):427–440, 2011. ISSN 0304-4149. doi: 10.1016/j.spa.2010.11.008

    2011 doi

  22. [30]

    Jakubowski

    A. Jakubowski. Short communication:the almost sure skorokhod representation for subse- quences in nonmetric spaces.Theory of Probability & Its Applications, 42(1):167–174, 1998. doi: 10.1137/S0040585X97976052

    1998 doi

  23. [31]

    Martingale solutions and markov selections for stochastic partial differential equations.Stochastic Processes and their Appli- cations, 119(5):1725–1764, 2009

    Benjamin Goldys, Michael Röckner, and Xicheng Zhang. Martingale solutions and markov selections for stochastic partial differential equations.Stochastic Processes and their Appli- cations, 119(5):1725–1764, 2009. ISSN 0304-4149. doi: 10.1016/j.spa.2008.08.009

    2009 doi

  24. [32]

    Adams and J.J.F

    R.A. Adams and J.J.F. Fournier.Sobolev Spaces. Pure and Applied Mathematics. Elsevier,

  25. [33]

    S. Willard. General Topology. Addison-Wesley series in mathematics. Dover Publications,

  26. [34]

    Handbook of Analysis and Its Foundations

    Eric Schechter. Handbook of Analysis and Its Foundations. Academic Press, San Diego,

  27. [35]

    John L. Kelley. General Topology. Graduate Texts in Mathematics. Springer New York, NY, 1975. ISBN 978-0-387-90125-1

    1975

  28. [36]

    Topology, pages 21–67

    Aliprantis. Topology, pages 21–67. Springer Berlin Heidelberg, Berlin, Heidelberg, 2006. ISBN 978-3-540-29587-7. doi: 10.1007/3-540-29587-9_2. 30

    2006 doi

  29. [1997]

    doi: 10.1016/B978-0-12-622760-4.X5000-6

    ISBN 978-0-12-622760-4. doi: 10.1016/B978-0-12-622760-4.X5000-6

    doi

  30. [2017]

    doi: 10.1137/16M1071390

    doi

Pith tools

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