Cholesky decomposition and well-posedness of Cauchy problem for Fokker-Planck equations with unbounded coefficients

Haesung Lee

REVIEW 3 major objections 5 minor 30 references

Cholesky decomposition and well-posedness of Cauchy problem for Fokker-Planck equations with unbounded coefficients

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Cholesky decomposition proves well-posedness of Fokker-Planck equations with unbounded coefficients.

desk verdict A well-structured assembly of known tools with a genuinely useful Cholesky regularity lemma, but the uniqueness proof misapplies the superposition principle from [8]; Theorem 1.1(ii) is not established for drifts that satisfy (H-2) but have large tangential components. read the letter →

arxiv 2506.01654 v1 pith:4DKLYCBU submitted 2025-06-02 math.PR math.AP

Haesung Lee This is my paper

classification math.PRmath.AP MSC 35Q8435K1547D0735A0293D0515A23

keywords Fokker-PlanckequationsunboundedcoefficientsCauchyproblemmartingalesuperpositionprincipleCholeskydecompositionItôSDEergodicity

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

The paper establishes well-posedness of the Cauchy problem for the Fokker-Planck equation associated with a second-order operator whose coefficients have low local regularity and may grow unboundedly. Existence is built from an analytic construction of a sub-Markovian semigroup; uniqueness is reduced, through a superposition principle, to uniqueness in law for Itô stochastic differential equations. The key step is a Cholesky decomposition of the diffusion matrix into a lower triangular square root with Sobolev regularity, which allows the martingale problem to be converted into a genuine SDE. Under additional drift inequalities, the solution converges to a stationary probability measure. This extends earlier well-posedness results to drifts that are locally unbounded and not weakly differentiable.

What carries the argument

The central object is the Cholesky square root: a factorization of the diffusion matrix $A$ as $\sigma\sigma^T$ with $\sigma$ lower triangular and strictly positive diagonal. Because the Cholesky algorithm computes each entry algebraically from previously computed entries, the paper proves inductively that $\sigma_{ij}$ inherits the regularity $H^{1,p}_{loc}\cap C$ from $A$. This object carries the argument by enabling the martingale-to-SDE conversion theorem, which turns a martingale problem with generator $L$ into a weak solution of $dX_t=\sigma(X_t)dW_t+G(X_t)dt$; pathwise uniqueness for that SDE then yields uniqueness of the original Fokker-Planck solution via the superposition principle.

What would settle it

Construct a pair of coefficients satisfying (H-1) and (H-2) together with two distinct probability-measure solutions to the Fokker-Planck equation sharing the same initial Dirac measure on some interval $[0,T]$; Theorem 1.1 then fails. A practical check is whether the integrability estimate in Lemma 4.4 controls the term $\int_0^T\int \frac{|G(y)|}{1+|y|}\,\mu_t(dy)dt$; if [8, Theorem 1.1] requires that bound and it cannot be derived from (H-2), testing a drift with a large component perpendicular to $y$ would reveal the missing hypothesis.

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Extended reading notes

Core claim

Theorem 1.1 asserts that under hypotheses (H-1) and (H-2), for every initial point $x$ there is a unique family of probability measures solving the Fokker-Planck equation with initial condition $\delta_x$ on every finite horizon, and that under either inequality (7) or (8) the solution converges to a stationary measure as $t\to\infty$. The uniqueness claim is the load-bearing novelty: instead of comparing densities, the argument maps each solution to a law of a stochastic process solving the martingale problem, then to a weak solution of an Itô SDE, and invokes pathwise uniqueness to conclude uniqueness in law. The Cholesky decomposition supplies a lower triangular square root $\sigma$ of $A$ with entries in $H^{1,p}_{loc}\cap C$, making the SDE formulation possible even though the original coefficients are only locally integrable or Sobolev regular.

Load-bearing premise

The proof assumes that the superposition principle of [8] applies to every Fokker-Planck solution under hypotheses (H-1) and (H-2); if that principle needs an additional global integrability bound on the drift that Lemma 4.4 does not provide, the uniqueness argument would lack its first step.

Editorial extensions

If this is right

For any initial point $x$, the Fokker-Planck flow is a genuine probability-measure-valued evolution: $\mu_t$ is defined for all $t\ge 0$ and agrees with the transition kernels $P_t(x,\cdot)$ of the constructed semigroup.
If the drift has sufficiently strong inward radial component, the limiting measure $\tilde\mu$ solves the stationary equation $\int Lf\,d\tilde\mu=0$ and $\mu_t(E)\to \tilde\mu(E)$ for every Borel set $E$.
The uniqueness statement is global: any other solution on any finite interval $[0,T]$ must coincide with the constructed one, so numerical or generative-model schemes using the Fokker-Planck flow are justified in treating the flow as well defined.
The method covers drifts that are locally unbounded and not weakly differentiable, going beyond earlier results requiring globally bounded or weakly differentiable coefficients.

Reading between the lines

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

Extension: if the Cholesky factor $\sigma$ could be shown globally Lipschitz, pathwise uniqueness would follow directly and might allow weakening (H-2) toward condition (4), a direction the paper notes in Remark 3.6.
Extension: the factorized form $A=\sigma\sigma^T$ suggests a concrete numerical recipe, simulating the Itô SDE with the Cholesky factor, and the paper's uniqueness class tells a user that any such simulation converges in law to the unique Fokker-Planck flow.
Extension: for applications to MCMC and score-based generative models, the convergence result identifies the limit measure but not the rate; quantifying convergence would require spectral or coupling estimates outside the paper's scope.
Extension: a testable weakening is whether the superposition principle of [8] remains valid under condition (4); if so, uniqueness would extend to cases where individual diffusion coefficients grow rapidly as long as the radial drift is sufficiently negative.

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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

I read the paper carefully, and here is my take. The genuinely new piece is Theorem 4.1: the Cholesky factorization of the diffusion matrix with coefficients in H^{1,p}_{loc} ∩ C is clean, correct, and useful on its own. The existence side via the author's earlier semigroup results is fine, and the ergodicity statements are reasonable corollaries of those results. The novelty of the stated combination is real: drift in L^p_loc, p>d, with no weak differentiability, is beyond the classical Lipschitz and L^{p+2} conditions.

But the uniqueness chain has a load-bearing gap. The paper invokes [8, Theorem 1.1] to convert an arbitrary FP solution into a martingale solution, and the only verification is Lemma 4.4, which proves a radial bound: ∫ (||A|| + |⟨G,y⟩|)/(1+||y||^2) dν_t dt < ∞. The BRS superposition principle, as stated in the cited paper, requires the non-radial condition ∫ |G(y)|/(1+||y||) dν_t(dy)dt < ∞. These are not equivalent. The gap is real: take G(y) = e^{||y||} R y with R a 90° rotation. Then ⟨G,y⟩ = 0, so (H-2) holds, but |G|/(1+||y||) grows like e^{||y||}/(1+||y||), which the logarithmic moment bound from Lemma 4.4 cannot control. I do not see any step in the paper that bridges this mismatch. Consequently, Theorem 1.1(ii) is not proven for a natural class of drifts allowed by (H-1)+(H-2).

A second, smaller soft spot: in Theorem 4.2, the extension of the martingale problem from C_0^∞ test functions to smooth coordinate functions is asserted as a "simple extension" without detailed argument. It is probably standard, but given the low regularity of G it deserves a few lines or a reference.

I want to stress that the flaw is not in the overall architecture; the combination of Cholesky, martingale problem, and Yamada-Watanabe is sensible. The paper could likely be repaired by strengthening the condition on G, for instance requiring the non-radial integrability explicitly, or by proving a stronger moment bound. But as it stands, the main uniqueness theorem overclaims what the proof supports. The heavy self-citation to [17,19] is not itself a problem; those results are proved and cited for legitimate reasons.

Who gets value from this? Researchers working on Fokker–Planck well-posedness with singular coefficients will want the Cholesky lemma and the overall strategy. The paper deserves a serious referee: the flaw is subtle, and the correctable content is substantial. My recommendation is to send it to peer review, with the expectation that the author needs to close the superposition-principle gap or weaken the statement.

Referee Report

3 major / 5 minor

Summary. The paper studies the Cauchy problem for the Fokker-Planck equation associated with the operator L = (1/2) trace(A∇²) + G·∇ under local regularity hypothesis (H-1) and global growth hypothesis (H-2). It claims existence of a probability solution for every Dirac initial condition, uniqueness on every finite horizon, and convergence to a stationary measure under additional Lyapunov inequalities. Existence is built from the author's earlier analytic semigroup results, while uniqueness is attempted through a chain: superposition principle for the Fokker-Planck equation, construction of a martingale solution, Cholesky decomposition of A into a Sobolev-regular diffusion matrix σ, identification with a weak solution of an Itô SDE, pathwise uniqueness from Zhang's theorem, and Yamada-Watanabe uniqueness in law. The paper also contains results on conservativeness and ergodicity in Section 3, and a discussion of a two-dimensional special case.

Significance. If the central claims are correct, the paper would extend well-posedness results for Fokker-Planck equations to a class of drifts that are only locally L^p and not weakly differentiable, with unbounded growth satisfying a radial inequality. The Cholesky-based construction of a regular diffusion matrix is an elegant and useful tool, and the overall proof strategy is natural and clearly written. However, the main novelty—the uniqueness assertion—depends on an application of an external superposition principle whose hypotheses are never stated or verified in the manuscript. The paper is honest about relying on prior work of the author and collaborators, and it explicitly discusses limitations in Remark 3.6. The result is potentially publishable if the indicated gap can be closed, but as it stands the proof of uniqueness is incomplete.

major comments (3)

[§4.5, Theorem 4.5] The proof invokes [8, Theorem 1.1] to convert an arbitrary Fokker-Planck solution (ν_t) into a martingale solution, but the hypotheses of that theorem are never stated. The only integrability estimate supplied in the paper is Lemma 4.4, which proves ∫_0^T ∫ (||A(y)|| + |⟨G(y),y⟩|)/(1+||y||²) ν_t(dy)dt < ∞. This is a radial/one-sided condition. The standard form of the superposition principle in [8] (as described, e.g., in the introduction of [8]) requires a non-radial condition such as ∫_0^T ∫ (|G(y)| + ||A(y)||)/(1+||y||) ν_t(dy)dt < ∞, possibly together with a Lyapunov condition. The gap is real: if G(y) = e^{||y||} R y, where R is a 90-degree rotation, then ⟨G(y),y⟩ = 0, so (H-2) holds, but |G(y)|/(1+||y||) is of order e^{||y||} and is not controlled by any logarithmic moment bound. Thus the application of [8] is not justified, and the uniqueness assertion of Theorem 1.1(ii) is not established as written. The authors must either restate the exact hypotheses of [8, Theorem 1.1] and prove them under (H-1)+(H-2), or replace this step by a superposition principle whose hypotheses match the bound derived in Lemma 4.4.
[§4.2, Theorem 4.2] The proof asserts that from the martingale property for test functions in C₀^∞(R^d), 'a simple extension' yields that M^v_t is a local martingale for all v ∈ C²(R^d), and in particular for the coordinate functions u_i(x) = x_i. This extension is not immediate because the coordinate functions are not compactly supported and are unbounded; the defining property of the martingale solution only gives the martingale property for C₀^∞ functions, not for u_i. To justify the step one would need a localization argument with cut-off functions and suitable integrability of G and σ along the path, or an alternative argument showing that u_i belongs to the domain of the generator under the measure. Since the subsequent identification of the martingale solution with a weak SDE solution depends critically on the quadratic variation of M^{u_i}, this gap is load-bearing and needs a detailed proof.
[§4.4, Lemma 4.4] The lemma proves the radial bound stated above, but the paper does not show how this bound connects to any condition assumed in the superposition principle used in Theorem 4.5. The lemma itself also contains a minor typo ('⟨G(y), y⟩⟩') and, more importantly, only controls the radial projection of G, not the full norm |G|. As noted in the first comment, a tangential drift of exponential size satisfies (H-2) while failing the non-radial integrability needed for the standard superposition principle. If the authors believe that [8] applies under the radial condition alone, they should quote the exact statement from [8]; otherwise, they should either strengthen (H-2) or prove the needed non-radial estimate under additional assumptions.

minor comments (5)

[§3, Proposition 3.2] In the proof of Proposition 3.2, the reference 'by Proposition 3.2' in the final inequality should be 'by Lemma 3.1'.
[§3, Theorem 3.3] In the proof of Theorem 3.3, the word 'Proposion' should be 'Proposition'.
[§2, Proposition 2.3] In part (ii), the augmentation is said to be under '¯Px', but it should be under '˜Px'.
[§2, Notations] The notation B(U) is introduced as 'the set of all Borel measurable sets or functions on U, as appropriate.' This overloaded notation may confuse readers; it is used later mostly for sets, but the phrase 'or functions' is unnecessary and should be removed.
[§4.4, Lemma 4.4] There is a typographical error in the expression for LV: '⟨G(y), y⟩⟩' contains an extra closing angle bracket.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the derivation chains through external superposition/pathwise-uniqueness results and independent prior semigroup theory; self-citation is real support, not a circular reduction.

full rationale

The paper's central uniqueness argument (Theorem 4.5) invokes the external superposition principle [8, Theorem 1.1] to pass from arbitrary Fokker-Planck solutions to martingale solutions, then uses Ikeda-Watanabe, the external pathwise uniqueness result [30, Theorem 1.1], and Yamada-Watanabe to conclude uniqueness in law. None of these cited results assume the theorem being proved. Lemma 4.4 verifies only radial integrability bounds; whether those bounds satisfy the exact hypotheses of [8, Theorem 1.1] is a correctness question, not a definitional or self-referential circularity. Existence (Theorem 1.1(i)) relies on the author's own previous semigroup construction in [19] and [17], but these are published independent results that do not assume the Cauchy-problem well-posedness claimed here; they are used as evidence rather than as a restatement of the conclusion. The Cholesky decomposition construction in Theorem 4.1 is self-contained and follows algorithmically from standard linear algebra. No fitted parameter is renamed as a prediction, and no uniqueness claim is imported from the same authors in a way that reduces the result to its own assumption. The paper itself flags the need to relax (H-2) and to further generalize the superposition principle (Remark 3.6 and Section 5), indicating that the growth conditions are hypotheses, not artifacts of the proof. Overall, the derivation chain is not circular.

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

The paper's theorems are built on a stack of external results: the author's own semigroup construction [19,17], the superposition principle [8], pathwise uniqueness [30], and standard stochastic calculus. The only genuinely new mathematical object is the Cholesky factor sigma, which is constructed from A rather than postulated. No free parameters are fitted to data. The main assumptions are the regularity and growth conditions (H-1) and (H-2).

assumptions (6)

domain assumption (H-1): d>=2, G in L^p_loc(R^d,R^d) with p in (d,infinity), A symmetric with a_ij in H^{1,p}_{loc}(R^d) intersect C(R^d), and A locally uniformly strictly elliptic.
Setting for all theorems; local ellipticity gives positivity of leading principal minors used in Cholesky.
domain assumption (H-2): There exist K>0 and N_0 in N such that ||A(x)|| <= K+K||x||^2 ln(1+||x||^2) and <G(x),x> <= K+K||x||^2 ln(1+||x||^2) a.e. outside B_{N_0}.
Used in Lemma 4.4 to get moment bounds and in Remark 3.4 to ensure conservativeness.
domain assumption The analytic semigroup results of [19, Theorem 2.3.1, Theorem 3.1, Proposition 3.13] and [17] hold under (H-1).
Existence of the sub-Markovian semigroup and transition probabilities is taken from the author's prior work without proof.
domain assumption The superposition principle [8, Theorem 1.1] applies to every FP solution satisfying the estimates of Lemma 4.4.
The hypotheses of [8] are not stated or verified in detail; the paper relies on this external result to produce martingale solutions.
domain assumption The pathwise uniqueness result [30, Theorem 1.1] holds for SDEs with drift G in L^p_loc (p>d) and diffusion coefficient sigma in H^{1,p}_{loc} intersect C, locally uniformly elliptic, via localization.
Theorem 4.3 is asserted as a direct consequence of [30] with only a sketch of localization.
standard math Ikeda-Watanabe representation [15, Chapter II, Theorem 7.1'] and Yamada-Watanabe theorem [16, Chapter 5, Proposition 3.20] are valid.
Standard theorems of stochastic calculus used to pass from martingale problem to SDE and to deduce uniqueness in law.

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Cite this review

Pith. "Pith review of Cholesky decomposition and well-posedness of Cauchy problem for Fokker-Planck equations with unbounded coefficients." pith.science (2026). https://pith.science/paper/4DKLYCBU

@misc{pith2026250601654,
  author       = {Pith},
  title        = {Pith review of: Cholesky decomposition and well-posedness of Cauchy problem for Fokker-Planck equations with unbounded coefficients},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4DKLYCBU}},
  note         = {Machine review of arXiv:2506.01654}
}

abstract

This paper explores the well-posedness of the Cauchy problem for the Fokker-Planck equation associated with the partial differential operator $L$ with low regularity condition. To address uniqueness, we apply a recently developed superposition principle for unbounded coefficients, which reduces the uniqueness problem for the Fokker-Planck equation to the uniqueness of solutions to the martingale problem. Using the Cholesky decomposition algorithm, a standard tool in numerical linear algebra, we construct a lower triangular matrix of functions $\sigma$ with suitable regularity such that $A = \sigma \sigma^T$. This formulation allows us to connect the uniqueness of solutions to the martingale problem with the uniqueness of weak solutions to It\^{o}-SDEs. For existence, we rely on established results concerning sub-Markovian semigroups, which enable us to confirm the existence of solutions to the Fokker-Planck equation under general growth conditions expressed as inequalities. Additionally, by imposing further growth conditions on the coefficients, also expressed as inequalities, we establish the ergodicity of the solutions. This work demonstrates the interplay between stochastic analysis and numerical linear algebra in addressing problems related to partial differential equations.

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [8]

Bogachev, M

V.I. Bogachev, M. R¨ ockner, S.V. Shaposhnikov,On the Ambrosio-Figalli-Trevisan superposition principle for probability solutions to Fokker-Planck-Kolmogorov equations, J. Dynam. Differential Equations, 33 (2021), 715–739

work page 2021

[2] [1]

Anderson,Reverse-time diffusion equation models, Stochastic Process

B.D.O. Anderson,Reverse-time diffusion equation models, Stochastic Process. Appl., 12 (1982), 313–326

work page 1982

[3] [2]

Allaire, S.M

G. Allaire, S.M. Kaber,Numerical linear algebra, Texts Appl. Math., 55, Springer, New York, 2008

work page 2008

[4] [3]

Bogachev, N.V

V.I. Bogachev, N.V. Krylov, M. R¨ ockner,On regularity of transition probabilities and invariant measures of singular diffusions under minimal conditions, Comm. Partial Differential Equations, 26 (2001), 2037–2080

work page 2001

[5] [4]

Bogachev, G

V.I. Bogachev, G. Da Prato, M. R¨ ockner, W. Stannat,Uniqueness of solutions to weak parabolic equations for measures, Bull. Lond. Math. Soc., 39 (2007), 631–640

work page 2007

[6] [5]

Bogachev, N.V

V.I. Bogachev, N.V. Krylov, M. R¨ ockner, S. Shaposhnikov,Fokker-Planck-Kolmogorov equations, Math. Surveys Monogr., 207, American Mathematical Society, Providence, RI, 2015

work page 2015

[7] [6]

Bogachev, T.I

V.I. Bogachev, T.I. Krasovitskii, S.V. Shaposhnikov,On uniqueness of probability solutions of the Fokker-Planck-Kolmogorov equation, Mat. Sb., 212 (2021), 3–42

work page 2021

[8] [7]

Bogachev, M

V.I. Bogachev, M. R¨ ockner, S.V. Shaposhnikov,Convergence in variation of solutions of nonlinear Fokker–Planck–Kolmogorov equations to stationary measures, J. Funct. Anal., 276 (2019), no. 12, 3681–3713

work page 2019