Derivative Formulas in Measure on Riemannian Manifolds

Feng-Yu Wang; Panpan Ren

REVIEW 3 major objections 1 minor 15 references

Derivative Formulas in Measure on Riemannian Manifolds

T0 review · 3 major / 1 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper proves that, for a broad and natural class of functions on the space of finite measures over a Riemannian manifold, the intrinsic and L-derivatives are exactly the Riemannian gradient of the extrinsic derivative, so the standard…

desk verdict The paper's advertised identity D^L = ∇D^E is false under the stated C^{E,1,1}_B hypotheses; the L-differentiability proof has a real domination gap, and a corrected counterexample kills Theorem 2.1(3)(b). read the letter →

arxiv 1908.03711 v2 pith:SHVP2D3Y submitted 2019-08-10 math.PR math.DG

Panpan Ren , Feng-Yu Wang This is my paper · ORCID

classification math.PRmath.DG MSC 60B0560B1058C35

keywords intrinsicderivativeextrinsicL-derivativelinearfunctionalRiemannianmanifoldfinitemeasurespaceprobabilityformulasin

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 sets out to clarify how four notions of derivative for functions of measures—extrinsic, linear functional, intrinsic, and L-derivative—are related on the space of finite measures over a complete Riemannian manifold. Its central claim is that, for functions whose extrinsic derivative is itself differentiable in the spatial variable, the intrinsic derivative and the L-derivative both equal the Riemannian gradient of the extrinsic derivative: $D^I f(\eta)(x)=D^L f(\eta)(x)=\nabla\{D^E f(\eta)(\cdot)\}(x)$. When this holds, the often abstract intrinsic and L-derivatives become concrete: differentiate the function along a Dirac-mass perturbation, then take the spatial gradient. The paper also extends the formulas to probability measures via the centralised extrinsic derivative, and derives a chain-rule formula for the derivative of the law of a random variable. A sympathetic reader would care because the result turns the calculation of measure derivatives in measure-valued and mean-field problems into elementary calculus on the underlying manifold.

What carries the argument

The load-bearing object is the identity $D^I f = D^L f = \nabla D^E f$, where $\nabla$ is the Riemannian gradient on $M$ and $D^E f(\eta)(x)=\lim_{s\downarrow0}(f(\eta+s\delta_x)-f(\eta))/s$ is the extrinsic derivative. The proof mechanism is a pair of variation lemmas. Lemma 3.1 expresses the change of $f$ under a density perturbation $(1+h_\varepsilon)\eta$ as an integral of $D^E f$ against the time-derivative of the density; Lemma 3.2 expresses the derivative along a linear interpolation $(1-r)\eta+r\gamma$ as the integral of $D^E f$ against $\gamma-\eta$. To obtain the gradient formula, the authors approximate arbitrary finite measures by measures with smooth positive densities, use the divergence theorem to move the derivative off the vector field and onto $D^E f$, and pass to the limit using continuity and growth conditions. For the L-derivative, geodesic flow and parallel transport reduce the difference quotient to an integral of $\nabla D^E f$ against the vector field.

What would settle it

Repeat the paper's main identity for a measure with atoms, where the density-approximation step is nontrivial. On $M=\mathbb{R}$, take $\eta=\delta_0+\delta_1$ and $f(\eta)=\eta(h)^2$ with $h(x)=x^2$. The theorem predicts $D^L f(\eta)(0)=0$ and $D^L f(\eta)(1)=4$, since $\nabla D^E f(\eta)(x)=4x$; direct evaluation of the geodesic-flow difference quotient gives exactly these values. A reader who finds any other value, or a failure of the difference quotient to converge, has falsified the claim.

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Extended reading notes

Core claim

The central discovery is a single identity linking two families of derivative notions that were introduced by different routes. Theorem 2.1(3) states that if $f\in C^{E,1,1}(M_p)$, then $f$ is intrinsically differentiable with $D^I f(\eta)(x)=\nabla\{D^E f(\eta)(\cdot)\}(x)$ for all $(x,\eta)\in M\times M_p$; when $p\in[0,2]$ and $f\in C^{E,1,1}_B(M_p)$, the same gradient is the L-derivative. Theorem 2.1(4) adds the limiting formula $D^L f(\eta)(x)=\lim_{s\downarrow0} s^{-1}\nabla f(\eta+s\delta_\cdot)(x)$, and Theorem 2.1(1) states that every L-differentiable function is intrinsically differentiable with $D^I=D^L$. For probability measures, the same links hold with the convex-combination extrinsic derivative $\tilde D^E$ in place of $D^E$. Taken together, the four derivative notions coincide on a broad function class, and the identity gives a practical route to computing intrinsically defined derivatives from the simpler extrinsic one.

Load-bearing premise

The gradient identity is proved only for functions whose extrinsic derivative is itself once differentiable in the spatial variable with a continuous gradient, and, for the L-derivative version, with controlled growth and $p\le2$; if that spatial regularity is absent, the identity can fail, and functions that are merely L-differentiable need not be extrinsically differentiable at all.

Editorial extensions

If this is right

For any cylindrical function $f(\eta)=g(\eta(h_1),\dots,\eta(h_n))$ with smooth $g,h_i$, the formula gives $D^L f(\eta)(x)=\sum_i (\partial_i g)(\dots)\nabla h_i(x)$, so intrinsic and L-derivatives of such functions reduce to ordinary calculus.
For functions on probability measures, the corresponding formula uses the centralised extrinsic derivative $\tilde D^E f(\mu)(x)=D^E f(\mu)(x)-\mu(D^E f(\mu))$, so the same gradient identity holds on $P_p$.
If a family of random variables $\xi_s$ on $M$ has derivative $\dot\xi_0$ in $L^q$, then for $f$ in the appropriate class, $\lim_{s\downarrow0}(f(L_{\xi_s})-f(L_{\xi_0}))/s = \mathbb{E}\langle\nabla\{\tilde D^E f(L_{\xi_0})\}(\xi_0), \dot\xi_0\rangle$, a chain rule for laws of random variables.
Every L-differentiable function is intrinsically differentiable and the two derivatives agree; together with the main gradient formula, the paper's regularity class $C^{E,1,1}_B(M_p)$ is contained in $C^{L,1}(M_p)$ for $p\le2$.
The limiting formula $D^L f(\eta)(x)=\lim_{s\downarrow0} s^{-1}\nabla f(\eta+s\delta_\cdot)(x)$ gives a direct way to compute the L-derivative without constructing geodesic flows.

Reading between the lines

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

If the identity extends beyond the $C^{E,1,1}_B$ class, it would give a practical recipe for derivative-based numerical schemes on Wasserstein space over manifolds: approximate $D^E$ by finite differences in mass, then apply the manifold gradient.
The paper's gap between L-differentiability and extrinsic differentiability suggests a complementary direction: characterising the minimal spatial regularity of $D^E f$ under which the gradient formula still holds, or finding a counterexample at lower regularity.
The law-derivative formula can be read as differentiation under the expectation; a natural testable extension is to measure-dependent SDEs on manifolds, where this identity would yield Bismut-type formulas for the L-derivative without separate arguments.
On flat $\mathbb{R}^d$ the result recovers known formulas; the new content is that curvature enters only through the Riemannian gradient and geodesic flow, so numerical implementations can treat the manifold as a black box with an exponential map.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Bottom line: the headline identity D^L = ∇D^E is not true for the function class C^{E,1,1}_B. The proof of Theorem 2.1(3)(b) has a domination gap, and the theorem is genuinely false. The paper is still useful in parts, but it needs major revision.

What I think is solid: the definitions are set out cleanly, and the paper does clarify existing terminological confusion. Theorem 2.1(1) — L-differentiability implies intrinsic differentiability with equal derivatives — is essentially a one-line observation about the two flows having the same tangent at s=0, and it is right. Theorem 2.1(2) and Lemma 3.2 give the extrinsic derivative as the linear functional derivative under C^{E,1}_K; that looks correct. The intrinsic-derivative half of Theorem 2.1(3), formula (2.1), also appears correct, because the test vector fields are smooth and compactly supported, so the unbounded-growth problems do not arise.

The soft spot is the L-differentiability half. The proof derives an expression for I_v with a squared integrand and then applies dominated convergence. The only available bound is |∇D^E f| ≤ C(1+ρ_o^p), and its square is not η-integrable for arbitrary η∈M_p when p>0. That is not a minor missing detail; the statement is false. On R with p=2, set h(x)=(1−cos x²)/2 and f(η)=∫h dη. Then D^E f=h, ∇D^E f=h'=x sin x², and |h'|≤|x|, so f∈C^{E,1,1}_B(M_2). Take η=Σ a_n δ_{x_n} with x_n=√(2πn), a_n=x_n^{-5}; this has finite second moment. Let v_n be c_n at x_n and 0 elsewhere, c_n=√x_n. Then ‖v_n‖_{L²(η)} = √(a_n)c_n = x_n^{-2}→0, but f(η∘φ_{v_n}^{-1})−f(η) ≈ (1/2)a_n h''(x_n)c_n² = a_n x_n² c_n², and dividing by ‖v_n‖ gives ≈ 1. So the L-differentiability limit fails. (The stress-test note's own construction used a_n=x_n^{-4} and accidentally produced infinite second moment; the corrected weights above remove that slip.)

So the paper's advertised formula (2.2) needs stronger hypotheses, e.g. Lipschitz ∇D^E f, or η(ρ_o^{2p})<∞, or bounded second derivative. The authors also need to fix Corollary 2.2(3) accordingly.

Who this is for: someone working on mean-field games or McKean-Vlasov equations who wants a practical way to compute Lions derivatives on manifolds. They should not take (2.2) as stated. The paper does contain a usable treatment of the intrinsic derivative and the Eulerian/extrinsic relation, so it is worth engaging with. If I were the editor I would send it to a careful referee, but with the expectation of major revision; I would not accept it in its current form.

Referee Report

3 major / 1 minor

Summary. The paper studies four notions of derivatives for functions on the space of finite measures over a Riemannian manifold: the extrinsic derivative D^E, the intrinsic derivative D^I, the L-derivative D^L, and the linear functional derivative D^F. The main theorem (Theorem 2.1) claims that, for a class of functions called C^{E,1,1}_B, the intrinsic and L-derivatives coincide and are both given by the spatial gradient of the extrinsic derivative, D^I f(η)(x)=D^L f(η)(x)=∇{D^E f(η)}(x); and that for C^{E,1}_K functions the extrinsic derivative is a linear functional derivative. Corollary 2.2 extends these statements to probability measures, and Lemma 3.4 gives a derivative formula for laws of random variables. The paper also proves that L-differentiability implies intrinsic differentiability (Theorem 2.1(1)) and derives a formula for ∇f(η+sδ_·) from the L-derivative (Theorem 2.1(4)).

Significance. If the main theorem were correct, it would provide a simple and practically useful bridge between several notions of derivatives in measure that appear in mean-field games, McKean–Vlasov analysis, and measure-valued diffusions. The paper is clearly written, the definitions are carefully set out, and the proof of Theorem 2.1(1) and some of the auxiliary lemmas (e.g., Lemma 3.1 for discrete measures) are sound. However, the central regularity class C^{E,1,1}_B is insufficient for the claimed L-differentiability, and the paper gives a false statement as a main theorem. One auxiliary assertion about linear functional derivatives on the space of finite measures is also false. Consequently, the paper's principal contribution is not established.

major comments (3)

[Theorem 2.1(3)(b)] Theorem 2.1(3)(b) is false as stated. Let M=R, p=2, o=0, and set h(x)=x^2/2 - (1/2)cos(x^2), f(η)=∫ h dη. Then D^E f(η)(x)=h(x), ∇D^E f(η)(x)=h'(x)=x+x sin(x^2), and |h'(x)| ≤ 2(1+|x|^2), so f∈C^{E,1,1}_B(M_2). Let x_n=√(2π n), a_n=x_n^{-4}, and η=∑ a_n δ_{x_n}∈M_2. For N≥1 define v_N by v_N(x_n)=1/x_n for n≥N and v_N=0 otherwise. Then ‖v_N‖_{L^2(η)}^2=∑_{n≥N} x_n^{-6}→0 as N→∞. The weak L-derivative candidate is D^L f(η)=h', so D^L_{v_N}f(η)=∑_{n≥N} a_n. Taylor expansion of h at x_n gives h(x_n+c)=h(x_n)+c x_n+(1/2)c^2(1+2x_n^2)+O(c^2/x_n^2), hence f(η∘φ_{v_N}^{-1})-f(η)=∑_{n≥N} a_n(2+O(x_n^{-2})). Therefore the L-differentiability remainder equals approximately ∑_{n≥N} a_n, while ‖v_N‖=(∑_{n≥N} x_n^{-6})^{1/2}; both quantities are of order N^{-1}, so the ratio does not tend to 0. Thus f is not L-differentiable at η, contradicting Theorem 2.1(3)(b). The proof fails at the dominated convergence step: the only bound available from Definition 1.1(5) is |∇D^E f|≤C(1+ρ_o^p), whose square is not η-integrable for p>0. Additional control on the second derivative of D^E f is required.
[Theorem 2.1(2)] Theorem 2.1(2) is also false as stated for finite measures. Take M=R, p≥0, and f(η)=η(M)^2. Then D^E f(η)(x)=2η(M), which is continuous in (x,η); for any compact K⊂M_p the total mass η(M) is bounded on K, so f∈C^{E,1}_K(M_p). However, the linear functional derivative condition (1.3) in Definition 1.5 requires sup_{η(ρ_o^p)≤L}|D^F f(η)(y)|≤C(1+ρ_o^p(y)). For this f the natural candidate D^F f(η)(y)=2η(M) fails because the set {η:η(ρ_o^p)≤L} is not bounded in total mass (e.g., η=nδ_o has η(ρ_o^p)=0 but η(M)=n). Thus (1.3) cannot hold, and the claimed statement f∈C^{E,1}_K ⇒ f has a linear functional derivative is false on M_p. The proof of Theorem 2.1(2), which only cites Lemma 3.2, does not address the growth condition (1.3).
[Proof of Theorem 2.1(3)(a)] The density reduction in the proof of Theorem 2.1(3)(a) is not valid on noncompact manifolds. The proof states that any η∈M_p can be approximated by measures of the form (4.1), η(dx)=ρ(x)dx with ρ∈C_b^∞(M) and inf ρ>0. On a noncompact manifold (such as R), a bounded positive function with strictly positive infimum cannot be integrable with respect to the volume measure, so no such η belongs to M_p. Consequently, the reduction to (4.2) is only established for a class of infinite measures (or for compact manifolds), and the limit argument does not cover the stated class. This leaves the intrinsic-derivative formula (2.1) without a complete proof as stated.

minor comments (1)

[Minor] There are a number of typographical errors, e.g., 'funtions' in the introduction, 'Oﬀ course' in Remark 1.1(a), and 'the the' in the introduction. These should be corrected in any revision.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the derivative identities are proved from independently defined notions and do not reduce to their inputs.

full rationale

The paper's central claims are mathematical identities relating four separately defined derivatives: extrinsic (Definition 1.1), intrinsic (Definition 1.3), L-derivative (Definition 1.4), and linear functional derivative (Definition 1.5). None of these definitions presupposes the conclusion of Theorem 2.1. Theorem 2.1(1) proves DI = DL by comparing the geometric flow and the exponential-map flow through the inverse exponential map; this is a substantive argument, not a restatement of definitions. Theorem 2.1(2) derives DF = DE from Lemma 3.2, which computes derivatives along convex combinations via the extrinsic derivative and then integrates; the result is not assumed in the definition of DF. Theorem 2.1(3) derives DI f(eta)(x) = grad{DE f(eta)}(x) and DL f(eta)(x) = grad{DE f(eta)}(x) using Lemma 3.1, integration by parts, and a dominated-convergence argument. The regularity class C^{E,1,1}_B requires the gradient of DE to exist and satisfy growth bounds, but the conclusion that this gradient represents the directional derivatives is nontrivial and is proved, not assumed. There is no fitting of parameters to data, no normalization forcing the outcome, and no renaming of a known result as a new derivation. The self-citations in the paper [9], [10], and [15] are surveys, applications, or prior related results; they are not load-bearing for the main theorem. The skeptic's counterexample targets the correctness of the domination step in Theorem 2.1(3)(b), which is a validity concern rather than a circularity concern. Under the stated rules, no circular step can be exhibited, so the appropriate finding is no significant circularity.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new entities or fitted parameters. Its results rest on the standard machinery of Riemannian manifolds, the definitional framework of the cited derivative notions, and a few technical density/integration-by-parts facts that are invoked rather than proved.

assumptions (4)

domain assumption M is a complete Riemannian manifold; exponential maps exp_x are diffeomorphisms on small normal balls and the cut-locus distance is locally uniformly positive on compact sets.
Stated in Section 1; used to define geodesic flows, exp^{-1}, and the vector field v_ε in the proof of Theorem 2.1(1).
standard math Finite measures in M_p can be approximated in the M_p-topology by measures with smooth strictly positive density functions w.r.t. volume measure.
Used at the start of the proof of Theorem 2.1(3)(a) to reduce (2.1) to absolutely continuous measures; the argument is stated but not fully written out.
standard math Integration by parts on the Riemannian manifold: for compactly supported smooth g and vector field v, ∫_M ⟨∇g,v⟩ dη = -∫_M g div_η(v) dη, with the appropriate divergence when η has a smooth density.
Used in proof of Theorem 2.1(3)(a) to convert the density derivative to the gradient form in Eq. (4.2).
standard math Dominated convergence and the fundamental theorem of calculus along geodesics are applicable under the growth conditions (2.11) and compactness of the relevant measure sets.
Used repeatedly in Section 3 and in Lemma 3.4 to interchange limits, integrals, and expectations.

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Cite this review

Pith. "Pith review of Derivative Formulas in Measure on Riemannian Manifolds." pith.science (2026). https://pith.science/paper/SHVP2D3Y

@misc{pith2026190803711,
  author       = {Pith},
  title        = {Pith review of: Derivative Formulas in Measure on Riemannian Manifolds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SHVP2D3Y}},
  note         = {Machine review of arXiv:1908.03711}
}

abstract

We characterise the link of derivatives in measure, which are introduced in [AKR,Card,ORS] respectively by different means, for functions on the space $\mathbb M$ of finite measures over a Riemannian manifold $M$. For a reasonable class of functions $f$, the extrinsic derivative $D^Ef$ coincides with the linear functional derivative $D^Ff$, the intrinsic derivative $D^If$ equals to the $L$-derivative $D^Lf$, and $$D^If(\eta)(x)= D^{L}f(\eta)(x)= \lim_{s\downarrow 0} \frac 1 s \nabla f(\eta+s \delta_\cdot)(x) = \nabla \big\{D^E f (\eta)\big\}(x), \ \ (x,\eta)\in M\times\mathbb M,$$ where $\nabla$ is the gradient on $M$, $\delta_x$ is the Dirac measure at $x$, and $$D^Ef(\eta)(x):= \lim\limits_{s\downarrow 0} \frac { f(\eta+s \delta_x)-f(\eta)} s,\ \ x\in M$$ is the extrinsic derivative of $f$ at $\eta\in \mathbb M$. This gives a simple way to calculate the intrinsic or $L$-derivative, and is extended to functions of probability measures. %This provides a simple way to calculate the intrinsic/Lions derivative.

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Reference graph

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Reviewed August 14, 2026 · model on record in the stance chip above.

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