Wick multiplication and its relationship with integration and stochastic differentiation on spaces of nonregular test functions in the Lévy white noise analysis

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2024-05-11 DOI:10.15330/cmp.16.1.61-83

Kachanovsky N.A

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引用次数: 0

Abstract

We deal with spaces of nonregular test functions in the Lévy white noise analysis, which are constructed using Lytvynov's generalization of a chaotic representation property. Our goal is to study properties of a natural multiplication $-$ a Wick multiplication on these spaces, and to describe the relationship of this multiplication with integration and stochastic differentiation. More exactly, we establish that the Wick product of nonregular test functions is a nonregular test function; show that when employing the Wick multiplication, it is possible to take a time-independent multiplier out of the sign of a generalized stochastic integral; establish an analog of this result for a Pettis integral (a weak integral); obtain a representation of the generalized stochastic integral via formal Pettis integral from the Wick product of the original integrand by a Lévy white noise; and prove that the operator of stochastic differentiation of first order on the spaces of nonregular test functions satisfies the Leibnitz rule with respect to the Wick multiplication.

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莱维白噪声分析中的维克乘法及其与非规则测试函数空间上的积分和随机微分的关系

我们处理的是莱维白噪声分析中的非规则测试函数空间，这些空间是利用莱特维诺夫对混沌表示性质的概括而构建的。我们的目标是研究这些空间上的自然乘法 $$-$ Wick 乘法的性质，并描述这种乘法与积分和随机微分的关系。更确切地说，我们确定非规则检验函数的威克乘积是一个非规则检验函数；证明当使用威克乘法时，有可能从广义随机积分的符号中取出一个与时间无关的乘数；为佩蒂斯积分（弱积分）建立一个类似的结果；通过莱维白噪声对原始积分的威克乘积，从形式佩提斯积分中获得广义随机积分的表示；并证明非规则检验函数空间上的一阶随机微分算子满足关于威克乘法的莱布尼茨规则。

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.