The diffusion equation for non-Markovian Gaussian stochastic processes

Alessandro Taloni , Gianni Pagnini , Aleksei Chechkin

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

classification ❄️ cond-mat.stat-mech

keywords equationdensitydiffusionfunctiongaussiannon-markovianprocesseswick

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The pith

Derives a closed non-Markovian diffusion equation for Gaussian velocity processes via a systematic Wick contraction hierarchy that generalizes Fokker-Planck and preserves Gaussianity only at infinite order.

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

Particles have velocities that follow a Gaussian random process, which can remember past values over time. The authors begin with the characteristic function that encodes the position distribution. They apply Wick's theorem to generate a hierarchy of equations whose terms are sums of connected contractions. Summing this hierarchy produces a diffusion equation that includes memory and is non-Markovian. The position distribution remains exactly Gaussian only when every term in the infinite hierarchy is kept.

Core claim

This approach yields a closed non-Markovian diffusion equation that generalizes the Fokker-Planck description and preserves Gaussianity only in the infinite-order limit.

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.

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.

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.

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The diffusion equation for non-Markovian Gaussian stochastic processes

Core claim

Load-bearing premise

discussion (0)