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Random separation property for stochastic Allen-Cahn-type equations

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Federico Bertacco, Carlo Orrieri, Luca Scarpa

Probability · Submitted 13 Oct 2021

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Then, we show that a random separation property holds, i.e. almost every trajectory is strictly separated in space and time from the potential barriers.

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arXiv ID2110.06544 (version 1)
Pith ID6GR4GOWF, Pith's permanent record of this paper and its review
CitationCarlo Orrieri, Federico Bertacco, Luca Scarpa. "Random separation property for stochastic Allen-Cahn-type equations." arXiv:2110.06544 (2021). https://pith.science/paper/6GR4GOWF
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,219 input tokens.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.19 · human participants 0.01 · controlled comparison 0.12 · randomized assignment 0.02 · states quantitative result 0.76 · compares with prior results 0.19 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

We study a large class of stochastic $p$-Laplace Allen-Cahn equations with singular potential. Under suitable assumptions on the (multiplicative-type) noise we first prove existence, uniqueness, and regularity of variational solutions. Then, we show that a random separation property holds, i.e. almost every trajectory is strictly separated in space and time from the potential barriers. The threshold of separation is random, and we further provide exponential estimates on the probability of separation from the barriers. Eventually, we exhibit a convergence-in-probability result for the random separation threshold towards the deterministic one, as the noise vanishes, and we obtain an estimate of the convergence rate.

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 We study a large class of stochastic $p$-Laplace Allen-Cahn equations with singular potential. contribution 0.74 0.84 0.03 asserted 0.90
s1 Under suitable assumptions on the (multiplicative-type) noise we first prove existence, uniqueness, and regularity of variational solutions. result 0.66 0.94 0.04 asserted 1.00
s2 Then, we show that a random separation property holds, i.e. almost every trajectory is strictly separated in space and time from the potential barriers. result 0.95 0.95 0.60 asserted 1.00
s3 The threshold of separation is random, and we further provide exponential estimates on the probability of separation from the barriers. result 0.98 0.95 0.24 asserted 1.00
s4 Eventually, we exhibit a convergence-in-probability result for the random separation threshold towards the deterministic one, as the noise vanishes, and we obtain an estimate of the convergence rate. result 0.95 0.96 0.22 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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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

The paper
Original abstract

We study a large class of stochastic $p$-Laplace Allen-Cahn equations with singular potential. Under suitable assumptions on the (multiplicative-type) noise we first prove existence, uniqueness, and regularity of variational solutions. Then, we show that a random separation property holds, i.e. almost every trajectory is strictly separated in space and time from the potential barriers. The threshold of separation is random, and we further provide exponential estimates on the probability of separation from the barriers. Eventually, we exhibit a convergence-in-probability result for the random separation threshold towards the deterministic one, as the noise vanishes, and we obtain an estimate of the convergence rate.

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Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

What it cites · 2 sentences without a confident label

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  • By Itô's formula (see [ref] ) we have for all $t∈[0,T]$, $$-a.s. that [equation] On the left-hand side it holds by assumption B that [equation] whereas on the right-hand side the assumption C yields [equation] It follows for every $t∈[0,T]$ that [equation] Setting then$

    Doubly nonlinear stochastic evolution equations II · Pathwise estimates · 0.97

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