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arxiv: 2606.23034 · v1 · pith:WRVVTPC3new · submitted 2026-06-22 · 🧮 math.AP · math.OC· math.PR

The Mortensen observer on the space of probability measures

Martin Morange (ANANKE) This is my paper

Pith reviewed 2026-06-26 07:47 UTC · model grok-4.3

classification 🧮 math.AP math.OCmath.PR
keywords Wasserstein spaceMortensen observerviscosity solutionsHamilton-Jacobi-Bellman equationsemi-Lagrangian schemeGamma-convergencedeterministic filteringprobability measures
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The pith

A minimum-energy observer on the Wasserstein space of measures yields a value function that is the unique viscosity solution to a Hamilton-Jacobi-Bellman equation.

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

The paper formulates a deterministic filtering problem on the space of probability measures with finite second moment by minimizing an action that penalizes both the kinetic cost of transporting the measure and the mismatch with partial observations. The resulting value function lives on the infinite-dimensional manifold equipped with the Wasserstein metric and satisfies a Hamilton-Jacobi-Bellman equation whose gradient is the Wasserstein gradient. Under regularity and growth conditions on the observation term, the value function is continuous, admits minimizing trajectories, and solves the equation in two complementary viscosity senses, one geometric and one obtained by lifting to a Hilbert space. A comparison principle then gives uniqueness, and a semi-Lagrangian discretization is shown to converge in the Gamma sense to the value function.

Core claim

We study a deterministic filtering problem formulated directly on the Wasserstein space of probability measures with finite second moment. Motivated by the Mortensen minimum-energy observer, we consider the reconstruction of an evolving probability density from partial observations by minimizing an action functional combining a kinetic transport cost and a time-dependent observation mismatch. The resulting value function is defined on the infinite-dimensional manifold (P_2(R^d), W_2) and satisfies a Hamilton-Jacobi-Bellman equation involving the Wasserstein gradient. Under suitable regularity and growth assumptions on the observation functional, we establish dynamic programming principles, c

What carries the argument

The value function on (P_2(R^d), W_2) obtained by minimizing the action functional that combines kinetic transport cost and observation mismatch.

If this is right

  • The value function satisfies the dynamic programming principle.
  • Continuity of the value function and existence of minimizing trajectories hold.
  • Viscosity solution properties are established in both geometric and Hilbertian formulations.
  • A comparison principle yields uniqueness of the viscosity solution.
  • The semi-Lagrangian scheme Gamma-converges to the value function.

Where Pith is reading between the lines

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

  • This observer could be used to reconstruct particle distributions from partial sensor data in applications like fluid dynamics or biology.
  • The two viscosity notions may allow comparison with other infinite-dimensional control problems on metric spaces.
  • Explicit solutions for quadratic observation functionals could test the scheme's accuracy.
  • The drift extension suggests applicability to controlled transport equations in mean-field settings.

Load-bearing premise

The observation functional satisfies suitable regularity and growth assumptions that enable the dynamic programming principle, continuity of the value function, existence of minimizers, and the viscosity solution properties including the comparison principle.

What would settle it

An observation functional that violates the growth conditions for which the comparison principle fails or the Gamma-convergence of the scheme does not hold.

read the original abstract

We study a deterministic filtering problem formulated directly on the Wasserstein space of probability measures with finite second moment. Motivated by the Mortensen minimum-energy observer, we consider the reconstruction of an evolving probability density from partial observations by minimizing an action functional combining a kinetic transport cost and a time-dependent observation mismatch. The resulting value function is defined on the infinite-dimensional manifold $(P_2(R^d), W_2)$ and satisfies a Hamilton-Jacobi-Bellman equation involving the Wasserstein gradient. Under suitable regularity and growth assumptions on the observation functional, we establish dynamic programming principles, continuity of the value function, existence of minimizing trajectories, and viscosity solution properties of the associated Hamilton-Jacobi equation. We provide two complementary notions of viscosity solutions: a geometric formulation based on subdifferentials in Wasserstein space, and a Hilbertian formulation inspired by Lions' lifting approach. This allows us to prove a comparison principle and uniqueness of solutions. Extensions to transport equations with drift are also discussed. Finally, we introduce a semi-Lagrangian scheme in order to approximate the value function, and show $\Gamma$-convergence of the scheme.

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.

Referee Report

0 major / 3 minor

Summary. The manuscript formulates a deterministic filtering problem on the Wasserstein space (P_2(R^d), W_2) motivated by the Mortensen minimum-energy observer. It defines a value function via minimization of an action functional combining kinetic transport cost and time-dependent observation mismatch, establishes the dynamic programming principle, continuity of the value function, and existence of minimizers under regularity/growth assumptions on the observation functional. The value function is shown to be a viscosity solution to the associated Hamilton-Jacobi-Bellman equation in both a geometric (subdifferential) sense and a Hilbertian (Lions lifting) sense; a comparison principle yields uniqueness. Extensions to transport equations with drift are discussed, and a semi-Lagrangian scheme is introduced with a proof of its Γ-convergence to the value function.

Significance. If the derivations hold, the work provides a rigorous extension of minimum-energy observers and optimal control to the infinite-dimensional setting of probability measures equipped with the Wasserstein metric. The dual viscosity formulations, comparison principle, and Γ-convergence result together supply both analytic foundations and a concrete numerical approximation tool for filtering problems on measures. This bridges optimal transport, viscosity theory on metric spaces, and deterministic filtering in a self-contained manner.

minor comments (3)
  1. [Introduction and §2] The precise statement of the regularity and growth assumptions on the observation functional (used for dynamic programming, continuity, and viscosity properties) should be collected in a single numbered assumption block early in the paper rather than distributed across sections.
  2. [Viscosity solution sections] In the definition of the two viscosity notions, the relationship between the geometric subdifferential test functions and the Hilbertian lifting should be made explicit, including whether one implies the other under the stated assumptions.
  3. [Numerical scheme section] The Γ-convergence proof for the semi-Lagrangian scheme would benefit from an explicit statement of the discretization parameters and the precise topology in which convergence holds.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, the detailed summary of its contributions, and the recommendation for minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation proceeds from the definition of the value function via minimization of an action functional on (P_2(R^d), W_2), through the dynamic programming principle, continuity, existence of minimizers, and viscosity solution properties (both geometric and Hilbertian) to the Wasserstein HJB equation, followed by a comparison principle for uniqueness and Gamma-convergence of the semi-Lagrangian scheme. All steps rely on standard optimal control and optimal transport arguments under explicit regularity/growth assumptions on the observation functional; no equation or property reduces by construction to a fitted parameter, self-definition, or self-citation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on background results from optimal transport (Wasserstein geometry) and viscosity solution theory in metric spaces, plus domain-specific assumptions on the observation functional; no free parameters or invented entities are introduced.

axioms (2)
  • standard math The space (P2(R^d), W2) is a complete separable metric space with the standard properties of Wasserstein geometry.
    Invoked as the state space throughout the formulation and analysis.
  • domain assumption The observation functional satisfies regularity and growth conditions sufficient for dynamic programming, continuity, and viscosity properties to hold.
    Explicitly stated as the hypothesis under which all main results are proved.

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Reference graph

Works this paper leans on

13 extracted references · 10 canonical work pages

  1. [1]

    Viscosity solutions of centralized control prob- lems in measure spaces

    334 pp.isbn: 978-3-7643-8721-1. [AJZ24] A. Aussedat, O. Jerhaoui, and H. Zidani. “Viscosity solutions of centralized control prob- lems in measure spaces”. en. In:ESAIM: Control, Optimisation and Calculus of Variations 30 (2024), p

    2024

  2. [2]

    Numerische Mathematik , author =

    issn: 1292-8119, 1262-3377. doi: 10 . 1051 / cocv / 2024081. url: https : / / www . esaim - cocv . org / articles / cocv / abs / 2024 / 01 / cocv240040 / cocv240040 . html (visited on 05/26/2026). [BB00] J.-D. Benamou and Y. Brenier. “A computational fluid mechanics solution to the Monge- Kantorovich mass transfer problem”. In:Numerische Mathematik 84 (Ja...

    work page doi:10.1007/s002110050002 2024

  3. [3]

    Partial Differential Equations48(2023), no

    [Ber23] C. Bertucci. “Monotone solutions for mean field games master equations: continuous state space and common noise”. In:Communications in Partial Differential Equations 48.10-12 (2023), pp. 1245–1285.doi: 10.1080/03605302.2023.2276564 . eprint: https://doi.org/10.1080/ 03605302.2023.2276564. url: https://doi.org/10.1080/03605302.2023.2276564. [Ber24]...

    work page doi:10.1080/03605302.2023.2276564 2023

  4. [4]

    Asymptotic approaches in inverse problems for depolymeriza- tion estimation

    url: https: //doi.org/10.1007/s00526-024-02718-4 (visited on 05/26/2026). [DM26] M. Doumic and P. Moireau. “Asymptotic approaches in inverse problems for depolymeriza- tion estimation”. In:Inverse Problems and Imaging 20.0 (2026), pp. 105–155. issn: 1930-

    work page doi:10.1007/s00526-024-02718-4 2026

  5. [5]

    A Piecewise Rotation of the Circle, IPR Maps and Their Connection with Translation Surfaces

    doi: 10 . 3934 / ipi . 2025020. url: https : / / www . aimsciences . org / article / id / 6827171e4940ac5bf2c88f2e. [Eva08] L. C. Evans. “Weak KAM Theory and Partial Differential Equations”. en. In:Calculus of Varia- tions and Nonlinear Partial Differential Equations: With a historical overview by Elvira Mascolo. Ed. by L. Ambrosio, L. Caffarelli, M. G. C...

    work page doi:10.1007/978- 2008

  6. [6]

    Weak KAM from a PDE point of view: viscosity solutions of the Hamilton–Jacobi equation and Aubry set

    doi: 10.1137/1.9781611973051 . eprint: https://epubs.siam.org/doi/pdf/10.1137/1. 9781611973051. url: https://epubs.siam.org/doi/abs/10.1137/1.9781611973051. [Fat12] A. Fathi. “Weak KAM from a PDE point of view: viscosity solutions of the Hamilton–Jacobi equation and Aubry set”. en. In:Proceedings of the Royal Society of Edinburgh: Section A Mathematics 14...

    work page doi:10.1137/1.9781611973051 2012

  7. [7]

    Nonlinear filtering and large deviations:a pde-control theoretic approach

    url: https : / / www . sciencedirect . com / science / article/pii/S0022123615002104. 36 [JB88] M. R. James and J. S. Baras. “Nonlinear filtering and large deviations:a pde-control theoretic approach”. In:Stochastics 23.3 (1988), pp. 391–412.doi: 10.1080/17442508808833500. eprint: https : / / doi . org / 10 . 1080 / 17442508808833500. url: https : / / doi...

    work page doi:10.1080/17442508808833500 1988

  8. [8]

    Optimal density evolution with congestion: L infinity bounds via flow interchange techniques and applications to variational Mean Field Games

    [LS19] H. Lavenant and F. Santambrogio. “Optimal density evolution with congestion: L infinity bounds via flow interchange techniques and applications to variational Mean Field Games”. In:Com- munications in Partial Differential Equations43.12 (Mar. 2019), pp. 1761–1802.url: https: //hal.science/hal-01522084. [LM25] G. Le Ruz and P. Moireau. “Optimal filt...

    2019

  9. [9]

    Mayer control problem with probabilistic uncertainty on initial positions

    url: https://www.college-de-france.fr. [MQ18] A. Marigonda and M. Quincampoix. “Mayer control problem with probabilistic uncertainty on initial positions”. In:Journal of Differential Equations264.5 (Mar. 2018), pp. 3212–3252.issn: 0022-0396. doi: 10 . 1016 / j . jde . 2017 . 11

    2018

  10. [10]

    A convexity principle for interacting gases

    url: https : / / www . sciencedirect . com / science/article/pii/S0022039617306046 (visited on 05/26/2026). [McC97] R. J. McCann. “A convexity principle for interacting gases”. In:Advances in Mathematics128.1 (1997), pp. 153–179. [Moi18] P. Moireau. “A discrete-time optimal filtering approach for non-linear systems as a stable dis- cretization of the Mort...

    work page doi:10.1051/cocv/2017077 2026

  11. [11]

    Maximum-likelihood recursive nonlinear filtering

    [Mor68] R. E. Mortensen. “Maximum-likelihood recursive nonlinear filtering”. In:Journal of Optimization Theory and Applications 2.6 (1968), pp. 386–394.doi: 10 . 1007 / BF00925744. url: https : //doi.org/10.1007/BF00925744. [Ott01] F. Otto. “The geometry of dissipative evolution equations: the porous medium equation”. In: Communications in Partial Differe...

    work page doi:10.1007/bf00925744 1968

  12. [12]

    Springer, Berlin, Heidelberg, 2009

    Jan. 2008, pp. xxii+973. doi: 10.1007/978-3-540-71050-9. [Wil04] J. Willems. “Deterministic least squares filtering”. In: Journal of Econometrics 118.1 (2004). Contributions to econometrics, time series analysis, and systems identification: a Festschrift in honor of Manfred Deistler, pp. 341–373.issn: 0304-4076. doi: https : / / doi . org / 10 . 1016 / S0...

    work page doi:10.1007/978-3-540-71050-9 2008

  13. [13]

    Math Program

    url: https://www.sciencedirect.com/science/article/pii/ S0304407603001465. [YKW22] M.-C. Yue, D. Kuhn, and W. Wiesemann. “On linear optimization over Wasserstein balls”. In: Math. Program.195.1–2 (Sept. 2022), pp. 1107–1122.issn: 0025-5610. doi: 10.1007/s10107- 021-01673-8. url: https://doi.org/10.1007/s10107-021-01673-8. 38

    work page doi:10.1007/s10107- 2022