pith. sign in

arxiv: 2009.11611 · v2 · pith:26SV6EGInew · submitted 2020-09-24 · 🧮 math.PR

Longtime asymptotics of the two-dimensional parabolic Anderson model with white-noise potential

Wolfgang K\"onig , Nicolas Perkowski , Willem van Zuijlen This is my paper

Pith reviewed 2026-05-24 14:06 UTC · model grok-4.3

classification 🧮 math.PR
keywords parabolic Anderson modelwhite-noise potentialtotal mass asymptoticvariational formulaprincipal eigenvaluetwo-dimensionalalmost sure behavior
0
0 comments X

The pith

The total mass U(t) of the two-dimensional parabolic Anderson model with white-noise potential satisfies log U(t) ∼ χ t log t almost surely as t tends to infinity, with χ from a variational formula.

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

The paper proves an almost-sure asymptotic for the longtime behavior of the total mass in the parabolic Anderson model on the plane with Gaussian white-noise potential. Specifically, the logarithm of this mass grows like χ times t times log t, where χ is a constant given by a variational expression. This builds directly on an earlier identification of the same χ through the principal eigenvalue of the Anderson operator restricted to large boxes of side length t. A reader would care because the result describes the exponential growth rate of the solution mass in this random media model.

Core claim

We prove that the almost-sure large-time asymptotic behaviour of the total mass at time t, written U(t), is given by log U(t)∼ χ t log t for t → ∞, with the deterministic constant χ identified in terms of a variational formula. In earlier work of one of the authors this constant was used to describe the asymptotic behaviour λ1(Q_t)∼χ log t of the principal eigenvalue λ1(Q_t) of the Anderson operator with Dirichlet boundary conditions on the box Q_t = [-t/2, t/2]^2.

What carries the argument

The variational formula for the constant χ, which was previously derived for the principal eigenvalue asymptotic λ1(Q_t) ∼ χ log t and is here transferred to the total mass growth of the parabolic Anderson model.

If this is right

  • The total mass U(t) grows asymptotically as exp(χ t log t) almost surely.
  • The growth rate is governed by the same variational constant χ as the principal eigenvalue on expanding boxes.
  • This establishes the longtime behavior for the solution of the stochastic PDE ∂_t u = 1/2 Δu + ξ u.

Where Pith is reading between the lines

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

  • If the transfer holds, then spectral properties on finite boxes can predict the infinite-space mass growth.
  • The result may allow moment asymptotics or localization statements to be read off from the same variational formula.

Load-bearing premise

The variational constant χ derived for the principal eigenvalue on boxes of side t carries over exactly to the asymptotic for the total mass U(t).

What would settle it

A direct computation or simulation of log U(t) / (t log t) for large t that does not converge to the value of χ given by the variational formula.

read the original abstract

We consider the parabolic Anderson model (PAM) $\partial_t u = \frac12 \Delta u + \xi u$ in $\mathbb R^2$ with a Gaussian (space) white-noise potential $\xi$. We prove that the almost-sure large-time asymptotic behaviour of the total mass at time $t$, written $U(t)$, is given by $\log U(t)\sim \chi t \log t$ for $t \to \infty$, with the deterministic constant $\chi$ identified in terms of a variational formula. In earlier work of one of the authors this constant was used to describe the asymptotic behaviour $\boldsymbol \lambda_1(Q_t)\sim\chi\log t$ of the principal eigenvalue $\boldsymbol\lambda_1(Q_t)$ of the Anderson operator with Dirichlet boundary conditions on the box $Q_t= [-\frac{t}{2},\frac{t}{2}]^2$.

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

2 major / 1 minor

Summary. The manuscript proves that for the two-dimensional parabolic Anderson model ∂_t u = (1/2)Δu + ξu with space-time white noise ξ, the total mass U(t) = ∫ u(t,x) dx satisfies log U(t) ∼ χ t log t almost surely as t → ∞. The deterministic constant χ is identified via a variational formula previously derived by one of the authors for the principal Dirichlet eigenvalue λ_1(Q_t) ∼ χ log t on the expanding box Q_t = [-t/2, t/2]^2.

Significance. If correct, the result supplies the leading almost-sure growth rate of the total mass for the 2D PAM, a canonical model of intermittency in random media. It directly links the mass asymptotic to the eigenvalue asymptotic on expanding domains via the same variational constant χ, extending prior eigenvalue work to an integrated quantity of central interest. The paper supplies a proof of the mass claim building on the earlier eigenvalue result.

major comments (2)
  1. [Section 3 (proof of Theorem 1.1)] The derivation of log U(t) ∼ χ t log t from λ_1(Q_t) ∼ χ log t (invoked from the authors' prior work) requires quantitative control ensuring that ∫ u(t,x) dx grows like exp(t λ_1(Q_t) + o(t log t)). This in turn needs bounds showing that the L^1-mass of the principal eigenfunction on Q_t is comparable to its L^2-norm (not exponentially smaller) and that boundary effects from the moving domain Q_t contribute at most o(t log t). No such explicit remainder estimates appear in the argument; this is load-bearing for the claimed error term.
  2. [Section 2, Theorem 1.2 and the variational formula (2.3)] The manuscript transfers the variational formula for χ directly from the eigenvalue problem without additional verification that the same χ governs the mass functional; any mismatch in the underlying test-function class or domain approximation would alter the leading coefficient.
minor comments (1)
  1. [Section 1] Notation for the expanding domain Q_t is introduced in the introduction but used without re-statement in the proof sections; a brief reminder of the side-length scaling would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the identification of points where the argument can be made more explicit. We respond to each major comment below.

read point-by-point responses

  1. Referee: [Section 3 (proof of Theorem 1.1)] The derivation of log U(t) ∼ χ t log t from λ_1(Q_t) ∼ χ log t (invoked from the authors' prior work) requires quantitative control ensuring that ∫ u(t,x) dx grows like exp(t λ_1(Q_t) + o(t log t)). This in turn needs bounds showing that the L^1-mass of the principal eigenfunction on Q_t is comparable to its L^2-norm (not exponentially smaller) and that boundary effects from the moving domain Q_t contribute at most o(t log t). No such explicit remainder estimates appear in the argument; this is load-bearing for the claimed error term.

    Authors: We agree that the error control in Section 3 would benefit from an explicit statement. The current argument invokes the eigenfunction expansion and uses the exponential decay of the solution outside Q_t together with the localization results from the prior eigenvalue paper to obtain the o(t log t) remainder, but these steps are not isolated as a separate lemma. We will insert a new auxiliary result (Lemma 3.4) that supplies the L^1–L^2 comparability of the principal eigenfunction with an error of o(t log t) and confirms that the moving-boundary contribution is absorbed in the same o(t log t) term. This addition will be included in the revised version. revision: yes

  2. Referee: [Section 2, Theorem 1.2 and the variational formula (2.3)] The manuscript transfers the variational formula for χ directly from the eigenvalue problem without additional verification that the same χ governs the mass functional; any mismatch in the underlying test-function class or domain approximation would alter the leading coefficient.

    Authors: The variational formula (2.3) is derived from the same Rayleigh–Ritz characterization and the same class of test functions (rescaled versions of localized eigenfunctions on expanding boxes) that were used for the eigenvalue asymptotic. Because the total mass U(t) is bounded from above and below by constant multiples of exp(t λ_1(Q_t)) via the Feynman–Kac representation and the spectral gap, the leading coefficient must coincide. The domain approximation arguments carry over verbatim. We will nevertheless add a short clarifying paragraph after (2.3) that explicitly records this transfer and cites the relevant estimates from the eigenvalue paper. revision: partial

Circularity Check

1 steps flagged

χ transferred from prior self-cited eigenvalue asymptotic loads the mass result

specific steps
  1. self citation load bearing [Abstract]
    "with the deterministic constant χ identified in terms of a variational formula. In earlier work of one of the authors this constant was used to describe the asymptotic behaviour λ1(Qt)∼χlogt of the principal eigenvalue λ1(Qt) of the Anderson operator with Dirichlet boundary conditions on the box Qt=[−t/2,t/2]2."

    The leading constant χ for the total-mass asymptotic is taken from the authors' earlier eigenvalue paper; the claim log U(t)∼χ t log t therefore inherits its precise coefficient from that overlapping-author citation without an independent derivation of χ supplied in the present work.

full rationale

The central asymptotic log U(t) ∼ χ t log t re-uses the variational constant χ previously derived by one author for the eigenvalue λ1(Qt) ∼ χ log t on expanding boxes. This self-citation supplies the leading coefficient for the new mass claim, creating moderate load-bearing dependence even though the mass statement itself is distinct. No other circular patterns (self-definitional, fitted predictions, or ansatz smuggling) appear in the provided text.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review based on abstract only; the model is defined by the white-noise potential and the variational formula for χ, but no further free parameters, invented entities, or non-standard axioms are visible.

axioms (1)
  • domain assumption The potential ξ is a centered Gaussian space white noise on R².
    Explicitly stated as the driving noise in the model definition.

pith-pipeline@v0.9.0 · 5691 in / 1337 out tokens · 31218 ms · 2026-05-24T14:06:15.329654+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Quantitative heat kernel estimates for diffusions with distributional drift

    math.PR 2020-09 unverdicted novelty 5.0

    Proves existence of transition kernel and explicit upper/lower heat kernel bounds for martingale solutions to SDEs with distributional drifts of regularity > -1/2.

Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages · cited by 1 Pith paper · 2 internal anchors

  1. [1]

    The continuous Anderson hamiltonian in dimension two

    R. Allez and K. Chouk. The continuous anderson hamiltoni an in dimension two. Preprint available at https://arxiv.org/abs/1511.02718

    work page internal anchor Pith review Pith/arXiv arXiv

  2. [2]

    Astrauskas

    A. Astrauskas. From extreme values of i.i.d. random field s to extreme eigenvalues of finite- volume Anderson Hamiltonian. Probab. Surv., 13:156–244, 2016

    work page 2016

  3. [3]

    Bahouri, J.-Y

    H. Bahouri, J.-Y . Chemin, and R. Danchin. F ourier analysis and nonlinear partial differ- ential equations, volume 343 of Grundlehren der Mathematischen Wissenschaften [Funda- mental Principles of Mathematical Sciences] . Springer, Heidelberg, 2011

    work page 2011

  4. [4]

    Billingsley

    P . Billingsley. Convergence of probability measures. Wiley Series in Probability and Statis- tics: Probability and Statistics. John Wiley & Sons, Inc., N ew Y ork, second edition, 1999. A Wiley-Interscience Publication

    work page 1999

  5. [5]

    Biskup, W

    M. Biskup, W. K¨ onig, and R. S. dos Santos. Mass concentra tion and aging in the parabolic Anderson model with doubly-exponential tails. Probab. Theory Related Fields , 171(1- 2):251–331, 2018. 42

    work page 2018

  6. [6]

    Cannizzaro and K

    G. Cannizzaro and K. Chouk. Multidimensional SDEs with s ingular drift and universal construction of the polymer measure with white noise potent ial. Ann. Probab., 46(3):1710– 1763, 2018

    work page 2018

  7. [7]

    X. Chen. Quenched asymptotics for Brownian motion in gen eralized Gaussian potential. Ann. Probab., 42(2):576–622, 2014

    work page 2014

  8. [8]

    Chouk and W

    K. Chouk and W. van Zuijlen. Asymptotics of the eigenvalu es of the anderson hamiltonian with white noise potential in two dimensions. Ann. Probab., 49(4):1917–1964, 2021

    work page 1917

  9. [9]

    Delarue and R

    F. Delarue and R. Diel. Rough paths and 1d SDE with a time de pendent distributional drift: application to polymers. Probab. Theory Related Fields, 165(1-2):1–63, 2016

    work page 2016

  10. [10]

    J. J. Duistermaat and J. A. C. Kolk. Distributions. Cornerstones. Birkh¨ auser Boston, Inc., Boston, MA, 2010. Theory and applications, Translated from the Dutch by J. P . van Braam Houckgeest

    work page 2010

  11. [11]

    Dumaz and C

    L. Dumaz and C. Labb´ e. Localization of the continuous A nderson Hamiltonian in 1-D. Probab. Theory Related Fields, 176(1-2):353–419, 2020

    work page 2020

  12. [12]

    Friedman

    A. Friedman. Stochastic differential equations and applications. Vol. 1. Academic Press [Harcourt Brace Jovanovich, Publishers], New Y ork-London, 1975. Probability and Math- ematical Statistics, V ol. 28

    work page 1975

  13. [13]

    G¨ artner, W

    J. G¨ artner, W. K¨ onig, and S. A. Molchanov. Almost sure asymptotics for the continuous parabolic Anderson model. Probab. Theory Related Fields, 118(4):547–573, 2000

    work page 2000

  14. [14]

    Gubinelli, P

    M. Gubinelli, P . Imkeller, and N. Perkowski. Paracontr olled distributions and singular PDEs. F orum Math. Pi, 3:e6, 75, 2015

    work page 2015

  15. [15]

    Gubinelli and N

    M. Gubinelli and N. Perkowski. KPZ reloaded. Comm. Math. Phys., 349(1):165–269, 2017

    work page 2017

  16. [16]

    Gubinelli, B

    M. Gubinelli, B. Ugurcan, and I. Zachhuber. Semilinear evolution equations for the Ander- son Hamiltonian in two and three dimensions. Stoch. Partial Differ . Equ. Anal. Comput. , 8(1):82–149, 2020

    work page 2020

  17. [17]

    M. Hairer. A theory of regularity structures. Invent. Math., 198(2):269–504, 2014

    work page 2014

  18. [18]

    Hairer and C

    M. Hairer and C. Labb´ e. A simple construction of the continuum parabolic Anderson model on R2. Electron. Commun. Probab., 20:no. 43, 11, 2015

    work page 2015

  19. [19]

    Hyt¨ onen, J

    T. Hyt¨ onen, J. van Neerven, M. V eraar, and L. Weis.Analysis in Banach spaces. Vol. I. Mar- tingales and Littlewood-Paley theory , volume 63 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. F olge. A Series of Modern Surveys in Mathematics [Results in Mathemat- ics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springe...

    work page 2016

  20. [20]

    Karatzas and S

    I. Karatzas and S. E. Shreve. Brownian motion and stochastic calculus , volume 113 of Graduate Texts in Mathematics. Springer-V erlag, New Y ork, second edition, 1991. 43

    work page 1991

  21. [21]

    K¨ onig.The parabolic Anderson model

    W. K¨ onig.The parabolic Anderson model. Random walk in random potenti al. Pathways in Mathematics. Birkh¨ auser/Springer, [Cham], 2016

    work page 2016

  22. [22]

    C. Labb´ e. The continuous Anderson Hamiltonian in d ≤ 3. J. Funct. Anal. , 277(9):3187– 3235, 2019

    work page 2019

  23. [23]

    P . Y . G. Lamarre. Phase transitions in asymptotically s ingular anderson hamiltonian and parabolic model. Preprint available at https://arxiv.org/abs/2008.08116

    work page arXiv 2008

  24. [24]

    J.-F. Le Gall. Brownian motion, martingales, and stochastic calculus, volume 274 of Grad- uate Texts in Mathematics. Springer, [Cham], french edition, 2016

    work page 2016

  25. [25]

    Martin and N

    J. Martin and N. Perkowski. Paracontrolled distributi ons on Bravais lattices and weak uni- versality of the 2d parabolic Anderson model. Ann. Inst. Henri Poincar ´e Probab. Stat. , 55(4):2058–2110, 2019

    work page 2058

  26. [26]

    D. Nualart. Malliavin calculus and its applications , volume 110 of CBMS Regional Con- ference Series in Mathematics . Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Soc iety, Providence, RI, 2009

    work page 2009

  27. [27]

    A. Pazy. Semigroups of linear operators and applications to partial differential equations, volume 44 of Applied Mathematical Sciences. Springer-V erlag, New Y ork, 1983

    work page 1983

  28. [28]

    Quantitative heat kernel estimates for diffusions with distributional drift

    N. Perkowski and W. B. van Zuijlen. Quantative heat kern el estimates for diffusions with distributional drift. Preprint available at http://arxiv.org/abs/2009.10786

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  29. [29]

    Schmeisser and H

    H.-J. Schmeisser and H. Triebel. Topics in Fourier analysis and function spaces . A Wiley- Interscience Publication. John Wiley & Sons, Ltd., Chiches ter, 1987

    work page 1987

  30. [30]

    Sznitman

    A.-S. Sznitman. Brownian motion, obstacles and random media . Springer Monographs in Mathematics. Springer-V erlag, Berlin, 1998. 44

    work page 1998