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Inverting weak random operators

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Jonathan Gutierrez-Pavón, Carlos G. Pacheco

Probability · Submitted 13 Sep 2019

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

The authors' main claim In this paper, we find explicitly the inverse of such weak operators, by provinding the forms of the so-called Green kernel.

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arXiv ID1909.06257 (version 1)
Pith IDXDF6WSO4, Pith's permanent record of this paper and its review
CitationCarlos G. Pacheco, Jonathan Gutierrez-Pav\'on. "Inverting weak random operators." arXiv:1909.06257 (2019). https://pith.science/paper/XDF6WSO4
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,391 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.17 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.17 · mentions code or data availability 0.01.

The abstract, sentence by sentence

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

We analyze two weak random operators, initially motivated from processes in random environment. Intuitively speaking these operators are ill-defined, but using bilinear forms one can deal with them in a rigorous way. This point of view can be found for instance in the work Skorohod \cite{Skorohod}, and it remarkably helps to carry out specific calculations. In this paper, we find explicitly the inverse of such weak operators, by provinding the forms of the so-called Green kernel. We show how this approach helps to analyze the spectra of the operators. In addition, we provide the existence of strong operators associated to our bilinear forms. Important tools that we use are the Sturm-Liouville theory and the stochastic calculus.

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 analyze two weak random operators, initially motivated from processes in random environment. background 0.57 0.80 0.03 asserted 0.98
s1 Intuitively speaking these operators are ill-defined, but using bilinear forms one can deal with them in a rigorous way. method 0.54 0.23 0.04 asserted 0.87
s2 This point of view can be found for instance in the work Skorohod \cite{Skorohod}, and it remarkably helps to carry out specific calculations. background 0.98 0.05 0.03 asserted 0.95
s3 In this paper, we find explicitly the inverse of such weak operators, by provinding the forms of the so-called Green kernel. contribution 0.71 0.97 0.77 asserted 1.00
s4 We show how this approach helps to analyze the spectra of the operators. contribution 0.65 0.93 0.12 asserted 1.00
s5 In addition, we provide the existence of strong operators associated to our bilinear forms. result 0.77 0.95 0.10 asserted 1.00
s6 Important tools that we use are the Sturm-Liouville theory and the stochastic calculus. method 1.00 0.53 0.03 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 analyze two weak random operators, initially motivated from processes in random environment. Intuitively speaking these operators are ill-defined, but using bilinear forms one can deal with them in a rigorous way. This point of view can be found for instance in the work Skorohod \cite{Skorohod}, and it remarkably helps to carry out specific calculations. In this paper, we find explicitly the inverse of such weak operators, by provinding the forms of the so-called Green kernel. We show how this approach helps to analyze the spectra of the operators. In addition, we provide the existence of strong operators associated to our bilinear forms. Important tools that we use are the Sturm-Liouville theory and the stochastic calculus.

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