Pub Date : 2022-08-31DOI: 10.1142/s0219025722500187
A. Riahi
{"title":"Noncommutative quantum decomposition of Gegenbauer white noise process","authors":"A. Riahi","doi":"10.1142/s0219025722500187","DOIUrl":"https://doi.org/10.1142/s0219025722500187","url":null,"abstract":"","PeriodicalId":50366,"journal":{"name":"Infinite Dimensional Analysis Quantum Probability and Related Topics","volume":"46 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80444190","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-08-30DOI: 10.1142/s0219025722500163
L. Accardi
{"title":"Wilhelm von Waldenfels (2-3-1932 - 12-3-2021), a pioneer of quantum probability","authors":"L. Accardi","doi":"10.1142/s0219025722500163","DOIUrl":"https://doi.org/10.1142/s0219025722500163","url":null,"abstract":"","PeriodicalId":50366,"journal":{"name":"Infinite Dimensional Analysis Quantum Probability and Related Topics","volume":"64 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79313881","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-08-30DOI: 10.1142/s0219025722500151
L. Accardi, B. Choi, U. Ji
{"title":"Ergodic Theorems for Higher Order Cesaro Means","authors":"L. Accardi, B. Choi, U. Ji","doi":"10.1142/s0219025722500151","DOIUrl":"https://doi.org/10.1142/s0219025722500151","url":null,"abstract":"","PeriodicalId":50366,"journal":{"name":"Infinite Dimensional Analysis Quantum Probability and Related Topics","volume":"23 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77427596","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-07-12DOI: 10.1142/s0219025722400094
M. Schurmann
This article is on the research of Wilhelm von Waldenfels in the mathematical field of quantum (or non-commutative) probability theory. Wilhelm von Waldenfels cer-tainly was one of the pioneers of this field. His idea was to work with moments and to replace polynomials in commuting variables by free algebras which play the role of algebras of polynomials in non-commuting quantities. Before he contributed to quantum probability he already worked with free algebras and free Lie algebras. One can imagine that this helped to create his own special algebraic method which proved to be so very fruitful. He came from physics. His PhD thesis, supervised by Heinz K¨onig, was in probability theory, in the more modern and more algebraic branch of probability theory on groups. Maybe the three, physics, abstract algebra and probability, must have been the best prerequisites to become a pioneer, even one of the founders, of quantum probability. We concentrate on a small part of the scientific work of Wilhelm von Waldenfels. The aspects of physics are practically not mentioned at all. There is nothing on his results in classical probability on groups (Waldenfels operators). This is an attempt to show how the concepts of non-commutative notions of independence and of L´evy processes on structures like Hopf algebras developed from the ideas of Wilhelm von Waldenfels.
{"title":"Non-commutative stochastic processes with independent increments","authors":"M. Schurmann","doi":"10.1142/s0219025722400094","DOIUrl":"https://doi.org/10.1142/s0219025722400094","url":null,"abstract":"This article is on the research of Wilhelm von Waldenfels in the mathematical field of quantum (or non-commutative) probability theory. Wilhelm von Waldenfels cer-tainly was one of the pioneers of this field. His idea was to work with moments and to replace polynomials in commuting variables by free algebras which play the role of algebras of polynomials in non-commuting quantities. Before he contributed to quantum probability he already worked with free algebras and free Lie algebras. One can imagine that this helped to create his own special algebraic method which proved to be so very fruitful. He came from physics. His PhD thesis, supervised by Heinz K¨onig, was in probability theory, in the more modern and more algebraic branch of probability theory on groups. Maybe the three, physics, abstract algebra and probability, must have been the best prerequisites to become a pioneer, even one of the founders, of quantum probability. We concentrate on a small part of the scientific work of Wilhelm von Waldenfels. The aspects of physics are practically not mentioned at all. There is nothing on his results in classical probability on groups (Waldenfels operators). This is an attempt to show how the concepts of non-commutative notions of independence and of L´evy processes on structures like Hopf algebras developed from the ideas of Wilhelm von Waldenfels.","PeriodicalId":50366,"journal":{"name":"Infinite Dimensional Analysis Quantum Probability and Related Topics","volume":"11 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84161013","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-07-08DOI: 10.1142/s021902572250014x
D. Poletti
{"title":"Characterization of Gaussian Quantum Markov Semigroups","authors":"D. Poletti","doi":"10.1142/s021902572250014x","DOIUrl":"https://doi.org/10.1142/s021902572250014x","url":null,"abstract":"","PeriodicalId":50366,"journal":{"name":"Infinite Dimensional Analysis Quantum Probability and Related Topics","volume":"153 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88135115","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-07-08DOI: 10.1142/s0219025722400082
R. Speicher
to Wilhelm for showing me mathematics
感谢威廉教我数学
{"title":"Some thoughts on Wilhelm von Waldenfels and on universal second order constructions","authors":"R. Speicher","doi":"10.1142/s0219025722400082","DOIUrl":"https://doi.org/10.1142/s0219025722400082","url":null,"abstract":"to Wilhelm for showing me mathematics","PeriodicalId":50366,"journal":{"name":"Infinite Dimensional Analysis Quantum Probability and Related Topics","volume":"69 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74295815","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-05-26DOI: 10.1142/s0219025722500291
R. Balan, Wangjun Yuan
In this article, we investigate the asymptotic behaviour of the spatial integral of the solution to the parabolic Anderson model with time independent noise in dimension d ≥ 1, as the domain of the integral becomes large. We consider 3 cases: (a) the case when the noise has an integrable covariance function; (b) the case when the covariance of the noise is given by the Riesz kernel; (c) the case of the rough noise, i.e. fractional noise with index H ∈ ( 14 , 12 ) in dimension d = 1. In each case, we identify the order of magnitude of the variance of the spatial integral, we prove a quantitative central limit theorem for the normalized spatial integral by estimating its total variation distance to a standard normal distribution, and we give the corresponding functional limit result.
{"title":"Central limit theorems for heat equation with time-independent noise: the regular and rough cases","authors":"R. Balan, Wangjun Yuan","doi":"10.1142/s0219025722500291","DOIUrl":"https://doi.org/10.1142/s0219025722500291","url":null,"abstract":"In this article, we investigate the asymptotic behaviour of the spatial integral of the solution to the parabolic Anderson model with time independent noise in dimension d ≥ 1, as the domain of the integral becomes large. We consider 3 cases: (a) the case when the noise has an integrable covariance function; (b) the case when the covariance of the noise is given by the Riesz kernel; (c) the case of the rough noise, i.e. fractional noise with index H ∈ ( 14 , 12 ) in dimension d = 1. In each case, we identify the order of magnitude of the variance of the spatial integral, we prove a quantitative central limit theorem for the normalized spatial integral by estimating its total variation distance to a standard normal distribution, and we give the corresponding functional limit result.","PeriodicalId":50366,"journal":{"name":"Infinite Dimensional Analysis Quantum Probability and Related Topics","volume":"25 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83058785","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-05-25DOI: 10.1142/s0219025722500114
Yun-Ching Chang, Hsin-Hung Shih
{"title":"Analysis of Space Dependent Noise Functionals with an Application to Linearly Correlated Processes","authors":"Yun-Ching Chang, Hsin-Hung Shih","doi":"10.1142/s0219025722500114","DOIUrl":"https://doi.org/10.1142/s0219025722500114","url":null,"abstract":"","PeriodicalId":50366,"journal":{"name":"Infinite Dimensional Analysis Quantum Probability and Related Topics","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89535240","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-05-23DOI: 10.1142/s0219025722500138
Michael Skeide
Bringing forward the concept of convergence in moments from classical random variables to quantum random variables is what leads to what can be called algebraic central limit theorem for (classical and) quantum random variables. I reflect in a very personal way how such an idea is typical for the spirit of doing research in mathematics as I learned it in Wilhelm von Waldenfels’s research group in Heidelberg.
{"title":"Algebraic Central Limit Theorems: A Personal View on One of Wilhelm's Legacies","authors":"Michael Skeide","doi":"10.1142/s0219025722500138","DOIUrl":"https://doi.org/10.1142/s0219025722500138","url":null,"abstract":"Bringing forward the concept of convergence in moments from classical random variables to quantum random variables is what leads to what can be called algebraic central limit theorem for (classical and) quantum random variables. I reflect in a very personal way how such an idea is typical for the spirit of doing research in mathematics as I learned it in Wilhelm von Waldenfels’s research group in Heidelberg.","PeriodicalId":50366,"journal":{"name":"Infinite Dimensional Analysis Quantum Probability and Related Topics","volume":"27 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86999256","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Biorthogonal Approach to Infinite Dimensional Fractional Poisson Measure","authors":"Jerome B. Bendong, Sheila M. Menchavez, Jose Luis da Silva","doi":"10.1142/s0219025723500157","DOIUrl":"https://doi.org/10.1142/s0219025723500157","url":null,"abstract":"In this paper we use a biorthogonal approach to the analysis of the infinite dimensional fractional Poisson measure $pi_{sigma}^{beta}$, $0<betaleq1$, on the dual of Schwartz test function space $mathcal{D}'$. The Hilbert space $L^{2}(pi_{sigma}^{beta})$ of complex-valued functions is described in terms of a system of generalized Appell polynomials $mathbb{P}^{sigma,beta,alpha}$ associated to the measure $pi_{sigma}^{beta}$. The kernels $C_{n}^{sigma,beta}(cdot)$, $ninmathbb{N}_{0}$, of the monomials may be expressed in terms of the Stirling operators of the first and second kind as well as the falling factorials in infinite dimensions. Associated to the system $mathbb{P}^{sigma,beta,alpha}$, there is a generalized dual Appell system $mathbb{Q}^{sigma,beta,alpha}$ that is biorthogonal to $mathbb{P}^{sigma,beta,alpha}$. The test and generalized function spaces associated to the measure $pi_{sigma}^{beta}$ are completely characterized using an integral transform as entire functions.","PeriodicalId":50366,"journal":{"name":"Infinite Dimensional Analysis Quantum Probability and Related Topics","volume":"10 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87560546","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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