arxiv: 2605.10561 · v2 · submitted 2026-05-11 · ❄️ cond-mat.stat-mech

Recognition: 2 theorem links

· Lean Theorem

The diffusion equation for non-Markovian Gaussian stochastic processes

Alessandro Taloni , Gianni Pagnini , Aleksei Chechkin

Authors on Pith no claims yet

Pith reviewed 2026-05-15 05:39 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech

keywords non-Markovian diffusionGaussian velocity processesWick contractionsFokker-Planck generalizationprobability density evolutionstochastic processesmemory effectscharacteristic function

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The pith

Gaussian velocity processes produce an exact closed non-Markovian diffusion equation for position probability density.

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

The paper derives the exact evolution equation for the probability density of particle displacements driven by arbitrary Gaussian velocity processes, without assuming Markovianity or stationarity. It begins with the characteristic function of the position density and applies Wick's theorem to build a hierarchy of equations whose terms are sums of geometrically connected contractions. If the derivation holds, this yields a single closed diffusion equation that generalizes the Fokker-Planck equation while recovering exact Gaussianity for the position only when the hierarchy is taken to infinite order. A sympathetic reader would care because many physical systems involve velocity fluctuations that are Gaussian yet correlated in time, and the equation supplies a systematic route to their position statistics.

Core claim

Starting from the characteristic function of the density of the position, we construct a systematic hierarchy of equations based on Wick's theorem, in which the dynamics is governed by sums of geometrically connected Wick contractions. This approach yields a closed non-Markovian diffusion equation that generalizes the Fokker-Planck description and preserves Gaussianity only in the infinite-order limit.

What carries the argument

The hierarchy of equations whose terms are sums of geometrically connected Wick contractions, which closes to a single non-Markovian diffusion equation for the position density.

If this is right

The equation reduces to the ordinary Fokker-Planck equation when the velocity process is Markovian.
The position distribution is exactly Gaussian only when every order of the contraction hierarchy is retained.
The same closed equation applies to non-stationary velocity processes whose two-time correlations are time-dependent.
Any Gaussian velocity correlation function can be inserted to obtain the corresponding position diffusion equation.

Where Pith is reading between the lines

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

Truncating the hierarchy at low finite order may produce useful approximate diffusion equations for particular correlation shapes that coincide with known fractional or memory-kernel models.
The derivation supplies a direct test: solve the closed equation analytically for an Ornstein-Uhlenbeck velocity and verify that the result matches the exact position variance obtained from the underlying Langevin equation.
The same contraction technique could be applied to the joint statistics of multiple particles driven by correlated Gaussian velocities.

Load-bearing premise

The velocity process must be Gaussian so that Wick's theorem applies and the hierarchy of connected contractions can be constructed and summed to a closed equation.

What would settle it

Generate trajectories from a specific non-Markovian Gaussian velocity process, compute the position probability density numerically, and compare it to the solution of the derived closed equation; any systematic mismatch would show the closure fails.

Figures

Figures reproduced from arXiv: 2605.10561 by Aleksei Chechkin, Alessandro Taloni, Gianni Pagnini.

read the original abstract

We derive the exact evolution equation for the probability density function of particle displacements generated by arbitrary Gaussian velocity processes, when neither Markovianity and nor stationarity are assumed. Starting from the characteristic function of the density of the position, we construct a systematic hierarchy of equations based on Wick's theorem, in which the dynamics is governed by sums of geometrically connected Wick contractions. This approach yields a closed non-Markovian diffusion equation that generalizes the Fokker-Planck description and preserves Gaussianity only in the infinite-order limit.

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.

Desk Editor's Note private letter to a colleague

The paper gives a systematic Wick-based hierarchy for non-Markovian Gaussian velocities that closes to a diffusion equation, but the claim that Gaussianity appears only at infinite order conflicts with the exact Gaussianity of the integrated position.

read the letter

The core contribution is a derivation that starts from the characteristic function of the position and applies Wick's theorem to generate a hierarchy of geometrically connected contractions for arbitrary Gaussian velocity processes. This produces a closed non-Markovian diffusion equation without assuming Markovianity or stationarity, which is a clean generalization of the usual Fokker-Planck setup for this class of processes. The construction itself is formally straightforward and could be useful for anyone who needs an exact equation when the velocity correlation is given but non-stationary or non-Markovian. Credit is due for keeping the derivation parameter-free and for making the connectedness explicit rather than hiding it in an ad-hoc closure. The soft spot is the abstract's assertion that Gaussianity is recovered only in the infinite-order limit. Because the velocity is Gaussian, the position X(t) = integral V(s) ds is exactly Gaussian at every finite t, so its PDF is exactly Gaussian and must obey the simple time-dependent diffusion equation with D(t) equal to the integrated velocity autocorrelation. If the summed hierarchy produces anything else, the closure step or the geometric summation must be introducing or failing to cancel higher cumulants. That tension is worth checking against the explicit low-order terms in the paper. The work is aimed at researchers who already work with generalized Langevin or non-Markovian diffusion models and want an exact route rather than phenomenological memory kernels. A reader who cares about reproducible derivations for Gaussian cases will find the hierarchy worth examining, though they will want to verify it reproduces the known exact result. It is coherent enough on its own terms to deserve a serious referee, mainly to clarify the Gaussianity point and confirm the closure is exact rather than approximate.

Referee Report

1 major / 0 minor

Summary. The manuscript derives an exact evolution equation for the probability density function of particle displacements generated by arbitrary non-Markovian, non-stationary Gaussian velocity processes. Starting from the characteristic function of the position and applying Wick's theorem, the authors construct a hierarchy of equations governed by sums of geometrically connected Wick contractions; this hierarchy is summed to produce a closed non-Markovian diffusion equation that generalizes the Fokker-Planck description, with the property that the position distribution remains Gaussian only in the infinite-order limit of the hierarchy.

Significance. If the derivation is internally consistent, the result would supply a systematic, diagram-based closure procedure for non-Markovian diffusion equations arising from Gaussian velocities, extending standard Fokker-Planck theory to cases without Markovianity or stationarity. The explicit use of connected Wick contractions offers a reproducible, algebraic route to the equation that could be checked in simple limits and applied to physical models such as persistent random walks or colored-noise-driven particles.

major comments (1)

[Abstract and hierarchy summation] Abstract and § on hierarchy summation: the claim that Gaussianity of the position PDF is recovered only in the infinite-order limit is in tension with the exact Gaussianity of X(t) = ∫_0^t V(s) ds when V is Gaussian. For any finite t the characteristic function is exactly exp(−k² σ²(t)/2), so the PDF must satisfy the exact local equation ∂_t p = D(t) ∂_{xx} p with D(t) = ∫_0^t ⟨V(t)V(s)⟩ ds. The manuscript must demonstrate that the summed connected-contraction equation reduces identically to this form at every finite order; otherwise the closure procedure introduces spurious higher-order terms whose cancellation is not guaranteed.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying a key consistency issue with the Gaussian character of the displacement process. We address the major comment below and will revise the manuscript to correct the imprecise statement regarding Gaussianity.

read point-by-point responses

Referee: [Abstract and hierarchy summation] Abstract and § on hierarchy summation: the claim that Gaussianity of the position PDF is recovered only in the infinite-order limit is in tension with the exact Gaussianity of X(t) = ∫_0^t V(s) ds when V is Gaussian. For any finite t the characteristic function is exactly exp(−k² σ²(t)/2), so the PDF must satisfy the exact local equation ∂_t p = D(t) ∂_{xx} p with D(t) = ∫_0^t ⟨V(t)V(s)⟩ ds. The manuscript must demonstrate that the summed connected-contraction equation reduces identically to this form at every finite order; otherwise the closure procedure introduces spurious higher-order terms whose cancellation is not guaranteed.

Authors: We agree that when the velocity V(t) is a Gaussian process, the integrated displacement X(t) is exactly Gaussian for every finite t, with characteristic function exp(−k² σ²(t)/2) where σ²(t) = 2∫_0^t ds ∫_0^s du ⟨V(s)V(u)⟩. Consequently p(x,t) must obey the exact diffusion equation ∂_t p = D(t) ∂_xx p with D(t) = ∫_0^t ⟨V(t)V(s)⟩ ds. Our derivation begins from the exact characteristic function and applies Wick’s theorem to generate the hierarchy of connected contractions; the summation is intended to produce an exact closed equation. The statement that Gaussianity is preserved “only in the infinite-order limit” was an imprecise formulation meant to indicate that finite truncations of the hierarchy yield approximate non-Gaussian corrections, while the complete sum recovers the exact Gaussian evolution. We acknowledge that this wording creates an apparent tension and will revise both the abstract and the hierarchy-summation section. In the revised manuscript we will (i) remove the misleading phrase, (ii) explicitly demonstrate that the fully summed connected-contraction equation reduces identically to ∂_t p = D(t) ∂_xx p by showing that all coefficients of higher-order derivatives (∂_x^{2n} p for n>1) vanish when the velocity correlations are Gaussian, and (iii) verify the reduction in the stationary limit where D becomes constant. These changes will be incorporated in the next version. revision: yes

Circularity Check

0 steps flagged

Derivation proceeds from characteristic function via standard Wick theorem without self-referential reduction or fitted inputs renamed as predictions

full rationale

The paper begins from the characteristic function of the position PDF for Gaussian velocity processes and applies the established Wick theorem to construct a hierarchy of connected contractions. The resulting closed non-Markovian equation is obtained by summing the hierarchy; no step equates a derived quantity to a fitted parameter or prior self-citation that is itself unverified. The position remains exactly Gaussian by linearity for any finite t, and the derivation does not introduce or rely on any self-definitional loop, ansatz smuggled via citation, or renaming of known results. This is the normal case of a self-contained first-principles construction from standard tools.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the assumption that the velocity is a Gaussian process (allowing Wick's theorem) and that the characteristic function of position can be used to generate a closed hierarchy; these are standard domain assumptions rather than new postulates.

axioms (2)

domain assumption Velocity process is Gaussian
Invoked to apply Wick's theorem for constructing the hierarchy of connected contractions.
standard math Characteristic function encodes the position density and admits a systematic expansion
Standard starting point in stochastic process theory used to derive the evolution equation.

pith-pipeline@v0.9.0 · 5379 in / 1256 out tokens · 55153 ms · 2026-05-15T05:39:33.614358+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

This approach yields a closed non-Markovian diffusion equation that generalizes the Fokker-Planck description and preserves Gaussianity only in the infinite-order limit.
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the dynamics is governed by sums of geometrically connected Wick contractions

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

43 extracted references · 43 canonical work pages · 1 internal anchor

[1]

Fokker,Over Browns’ che Bewegingen in Het Stral- ingsveld, Ph.D

A. Fokker,Over Browns’ che Bewegingen in Het Stral- ingsveld, Ph.D. thesis, Physics and Mathematics Faculty, University of Leiden Leiden, The Netherlands (1913)

work page 1913
[2]

A. D. Fokker, Die mittlere energie rotierender elektrischer dipole im strahlungsfeld, Annalen der Physik348, 810 (1914)

work page 1914
[3]

Planck, ¨Uber einen satz der statistischen dynamik und seine erweiterung in der quantentheorie

M. Planck, ¨Uber einen satz der statistischen dynamik und seine erweiterung in der quantentheorie. sitzungsberichte der k¨ oniglich-preussischen akademie der wissenschaften zu berlin (1917)

work page 1917
[4]

Smoluchowski,Einige Beispiele Brown’scher Moleku- larbewegung unter Einfluss ¨ ausserer Kr¨ afte(Acad

M. Smoluchowski,Einige Beispiele Brown’scher Moleku- larbewegung unter Einfluss ¨ ausserer Kr¨ afte(Acad. Lit- terarum Cracoviensis, 1913)

work page 1913
[5]

Hoare, The linear gas, Advances in chemical physics , 135 (1971)

M. Hoare, The linear gas, Advances in chemical physics , 135 (1971)

work page 1971
[6]

Driessler, On the spectrum of the rayleigh piston, Journal of Statistical Physics24, 595 (1981)

W. Driessler, On the spectrum of the rayleigh piston, Journal of Statistical Physics24, 595 (1981)

work page 1981
[7]

G. G. Stokeset al., On the effect of the internal friction of fluids on the motion of pendulums, (1851)

work page
[8]

Langevin, Comptes, Rendues146, 530 (1908)

P. Langevin, Comptes, Rendues146, 530 (1908)

work page 1908
[9]

N. G. Van Kampen, Stochastic differential equations, Physics reports24, 171 (1976)

work page 1976
[10]

Wick, The evaluation of the collision matrix, Phys- ical review80, 268 (1950)

G.-C. Wick, The evaluation of the collision matrix, Phys- ical review80, 268 (1950)

work page 1950
[11]

G. E. Uhlenbeck and L. S. Ornstein, On the theory of the brownian motion, Physical review36, 823 (1930)

work page 1930
[12]

J. L. Doob, The brownian movement and stochastic equa- tions, Annals of mathematics43, 351 (1942)

work page 1942
[13]

J. L. Doob, The elementary gaussian processes, The An- nals of Mathematical Statistics15, 229 (1944)

work page 1944
[14]

We recall that Markovianity, as well as stationarity or ergodicity, is a property of a chosen observable, not an intrinsic property of the underlying stochastic process

In general, time integration of a Markov process gen- erates non-Markovian dynamics, except in the singular case of white noise. We recall that Markovianity, as well as stationarity or ergodicity, is a property of a chosen observable, not an intrinsic property of the underlying stochastic process

work page
[15]

Risken, Fokker-planck equation, inThe Fokker-Planck equation: methods of solution and applications(Springer,

H. Risken, Fokker-planck equation, inThe Fokker-Planck equation: methods of solution and applications(Springer,

work page
[16]

Mori, Transport, collective motion, and brownian mo- tion, Progress of theoretical physics33, 423 (1965)

H. Mori, Transport, collective motion, and brownian mo- tion, Progress of theoretical physics33, 423 (1965)

work page 1965
[17]

H¨ aunggi and P

P. H¨ aunggi and P. Jung, Colored noise in dynamical sys- tems, Advances in chemical physics89, 239 (1994)

work page 1994
[18]

Zwanzig,Nonequilibrium statistical mechanics(Ox- ford university press, 2001)

R. Zwanzig,Nonequilibrium statistical mechanics(Ox- ford university press, 2001)

work page 2001
[19]

Balescu,Statistical dynamics: matter out of equilib- rium(World Scientific, 1997)

R. Balescu,Statistical dynamics: matter out of equilib- rium(World Scientific, 1997)

work page 1997
[20]

Balescu, V-langevin equations, continuous time ran- dom walks and fractional diffusion, Chaos, Solitons & Fractals34, 62 (2007)

R. Balescu, V-langevin equations, continuous time ran- dom walks and fractional diffusion, Chaos, Solitons & Fractals34, 62 (2007)

work page 2007
[21]

Dieterich, R

P. Dieterich, R. Klages, and A. V. Chechkin, Fluctu- ation relations for anomalous dynamics generated by time-fractional fokker–planck equations, New Journal of Physics17, 075004 (2015)

work page 2015
[22]

Zwanzig, Memory effects in irreversible thermodynam- ics, Physical Review124, 983 (1961)

R. Zwanzig, Memory effects in irreversible thermodynam- ics, Physical Review124, 983 (1961)

work page 1961
[23]

Nordholm and R

S. Nordholm and R. Zwanzig, A systematic derivation of exact generalized brownian motion theory, Journal of Statistical Physics13, 347 (1975)

work page 1975
[24]

H¨ anggi, Correlation functions and masterequa- tions of generalized (non-markovian) langevin equations, Zeitschrift f¨ ur Physik B Condensed Matter31, 407 (1978)

P. H¨ anggi, Correlation functions and masterequa- tions of generalized (non-markovian) langevin equations, Zeitschrift f¨ ur Physik B Condensed Matter31, 407 (1978)

work page 1978
[25]

H¨ anggi, The functional derivative and its use in the description of noisy dynamical systems (1985)

P. H¨ anggi, The functional derivative and its use in the description of noisy dynamical systems (1985)

work page 1985
[26]

H¨ anggi, Colored noise in dynamical systems: A func- tional calculus approach, Noise in Nonlinear Dynamical Systems1, 307 (1989)

P. H¨ anggi, Colored noise in dynamical systems: A func- tional calculus approach, Noise in Nonlinear Dynamical Systems1, 307 (1989)

work page 1989
[27]

S. A. Adelman, Fokker–planck equations for simple non- markovian systems, The Journal of Chemical Physics64, 124 (1976)

work page 1976
[28]

R. F. Fox, Analysis of nonstationary, gaussian and non- gaussian, generalized langevin equations using methods of multiplicative stochastic processes, Journal of Statis- tical Physics16, 259 (1977)

work page 1977
[29]

G. K. Batchelor, Diffusion in a field of homogeneous tur- bulence: Ii. the relative motion of particles, inMathemat- ical Proceedings of the Cambridge Philosophical Society, Vol. 48 (Cambridge University Press, 1952) pp. 345–362

work page 1952
[30]

R. F. Fox, The generalized langevin equation with gaus- sian fluctuations, Journal of Mathematical Physics18, 2331 (1977)

work page 1977
[31]

Goldstone, Derivation of the brueckner many-body theory, Proceedings of the Royal Society of London

J. Goldstone, Derivation of the brueckner many-body theory, Proceedings of the Royal Society of London. Se- ries A. Mathematical and Physical Sciences239, 267 (1957)

work page 1957
[32]

M. E. Peskin,An Introduction to quantum field theory (CRC press, 2018)

work page 2018
[33]

Kleinert,Path integrals in quantum mechanics, statis- tics, polymer physics, and financial markets(World Sci- entific, 2009)

H. Kleinert,Path integrals in quantum mechanics, statis- tics, polymer physics, and financial markets(World Sci- entific, 2009)

work page 2009
[34]

N. N. Bogoliubov, D. V. Shirkov, and S. Chomet,Intro- duction to the theory of quantized fields, Vol. 3 (Inter- science New York, 1959)

work page 1959
[35]

J. E. Mayer and M. G. Mayer,Statistical mechanics, Vol. 28 (John Wiley & Sons New York, 1940)

work page 1940
[36]

Zinn-Justin,Quantum Field Theory and Critical Phe- nomena(Oxford University Press, 2002) Chap

J. Zinn-Justin,Quantum Field Theory and Critical Phe- nomena(Oxford University Press, 2002) Chap. 2

work page 2002
[37]

Itzykson and J.-B

C. Itzykson and J.-B. Zuber,Quantum Field Theory (McGraw–Hill, 1980) Chap. 6

work page 1980
[38]

OEIS Foundation Inc., The on-line encyclopedia of in- teger sequences, A000698,https://oeis.org/A000698, accessed: YYYY-MM-DD

work page
[39]

A combinatorial survey of identities for the double factorial

D. Callan, A combinatorial survey of identities for the double factorial, arXiv preprint arXiv:0906.1317 (2009)

work page internal anchor Pith review Pith/arXiv arXiv 2009
[40]

Klein,Zur statistischen Theorie der Suspensionen und L¨ osungen, Vol

O. Klein,Zur statistischen Theorie der Suspensionen und L¨ osungen, Vol. 16 (Hochschule Stockholm., 1921)

work page 1921
[41]

B. B. Mandelbrot and J. W. Van Ness, Fractional brow- nian motions, fractional noises and applications, SIAM review10, 422 (1968)

work page 1968
[42]

Wang and C

K. Wang and C. Lung, Long-time correlation effects and fractal brownian motion, Physics Letters A151, 119 (1990)

work page 1990
[43]

I. Podlubny,Fractional differential equations: an intro- duction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Vol. 198 (elsevier, 1998)

work page 1998