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An exactly solvable nonlinear model: Constructive effects of correlations between Gaussian noises

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P. F. Gora

Statistical Mechanics · Submitted 19 Feb 2006 · 13 references

Pith's reading: A nonlinear system with two correlated Gaussian white noises is claimed to have an exact stationary distribution showing stochastic localization, heavy tails, a variance minimum, and stochastic resonance under an external periodic signal.

T0 · AI review

How this review was made: An AI referee (deepseek-v4-flash) read the paper on August 28, 2026 and answered a fixed list of questions about it. T0 is the starting level every paper gets. The levels describe how deeply Pith has checked the paper, never the authors.

  1. T0 AI review An AI read the paper and answered a fixed list of questions.
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  3. T2 Compute The paper's code or calculations re-run with the same results.
  4. T3 Formal statement The claims written out in a formal math language a computer can read.
  5. T4 Proof Those claims proved, and the proof checked by computer.

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arXiv IDcond-mat/0602449 (version 1)
Pith IDKV2SIZXV, Pith's permanent record of this paper and its review
CitationP. F. Gora. "An exactly solvable nonlinear model: Constructive effects of correlations between Gaussian noises." arXiv:cond-mat/0602449 (2006). https://pith.science/paper/KV2SIZXV
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Claim paper
The paper

The paper claims

A nonlinear system with two correlated Gaussian white noises is claimed to have an exact stationary distribution showing stochastic localization, heavy tails, a variance minimum, and stochastic resonance under an external periodic signal.

The reading

The paper studies a particle whose position x obeys a stochastic equation with two random forces. One force multiplies x itself, the other multiplies |x| to a fractional power, and the two forces are correlated. By changing variables, the author turns the equation into a simpler form and solves for the long-time probability distribution of the particle's position.
Extended reading notes

Core claim

The paper's central assertion is that the stationary distribution of the system in Eq. (4) is exactly given by Eq. (20) (for |c|<1) and by Eq. (27) (for |c|=1 under condition (25)), and that this distribution exhibits stochastic localization, heavy tails, a nonmonotonic variance as q varies, and stochastic resonance when a periodic signal is added. If correct, this would be a rare exactly solvable nonlinear stochastic model with correlated noises.

Load-bearing premise

The load-bearing mathematical premise is the change of variables y=x/|x|^alpha used in Section 2. Converting the stationary density from y back to x requires the Jacobian |dy/dx|=(1-alpha)|x|^{-alpha}, so P_x(x)=P_y(y(x))(1-alpha)|x|^{-alpha}. The displayed formulas Eq. (17) and Eq. (20) omit this factor (they use |x|^{-alpha} where direct integration of Eq. (12) gives |x|^{-2alpha} and an extra 1/(1-alpha) in the arctangent argument). If this Jacobian check is correct, the exact distribution as printed does not solve the paper's own Fokker-Planck equation (10).

Original abstract

A system with two correlated Gaussian white noises is analysed. This system can describe both stochastic localization and long tails in the stationary distribution. Correlations between the noises can lead to a nonmonotonic behaviour of the variance as function of the intensity of one of the noises and to a stochastic resonance. A method for improving the transmission of external periodic signal by tuning parameters of the system discussed in this paper is proposed.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,482 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.06 · human participants 0.02 · controlled comparison 0.04 · randomized assignment 0.01 · states quantitative result 0.06 · compares with prior results 0.07 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

A system with two correlated Gaussian white noises is analysed. This system can describe both stochastic localization and long tails in the stationary distribution. Correlations between the noises can lead to a nonmonotonic behaviour of the variance as function of the intensity of one of the noises and to a stochastic resonance. A method for improving the transmission of external periodic signal by tuning parameters of the system discussed in this paper is proposed.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 A system with two correlated Gaussian white noises is analysed. contribution 0.74 0.78 0.04 asserted 0.95
s1 This system can describe both stochastic localization and long tails in the stationary distribution. result 0.46 0.59 0.20 hedged possibility 0.53
s2 Correlations between the noises can lead to a nonmonotonic behaviour of the variance as function of the intensity of one of the noises and to a stochastic resonance. result 0.99 0.80 0.53 hedged possibility 0.83
s3 A method for improving the transmission of external periodic signal by tuning parameters of the system discussed in this paper is proposed. contribution 0.95 0.82 0.26 asserted 0.89

What kind of paper the classifier reports this is: theoretical result 0.97 · evidence from proof or derivation 0.99.

What was examined, and what was not

sources read
  • arXiv listing abstract v1
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. Nonmonotonic behaviour of the variance of the distri
Figure 2. Panel (a): a fragment of a typical realization of the p
Figure 3. Signal-to-noise ratio for the system (28). Paramete
Figure 4. Same as Fig. 3 but with
Figure 5. Signal-to-noise ratio for the system (28) as a functi

Figures reproduced from arXiv:cond-mat/0602449 by the authors.

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