ICNCP · Independence and Convolutions in Noncommutative Probability
FP7 — People (Marie Curie Actions)
- Duration
- 2013-04-01 → 2015-03-31
- EU contribution
- €194,047
- Participants
- 1
- Scheme
- MC-IIF
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Results in brief
Independence and Convolutions in Noncommutative Probability
The research project is classified into three parts: I Analysis of infinitely divisible distributions in classical and free probability. Papers, preprints or work in progress (J4), (P1), (P2), (P5), (P6), (P8), (P9); Talks (Talk1), (Talk2),(Talk3), (Talk4), (Talk7), (Talk8), (Talk9), (Talk10). II Applications of monotone independence to free probability. (P4) III Application and development of Lenczewski’s matricial free independence. (P3), (P7);(Talk5), (Talk6). On all of these parts significant new results have been obtained, in particular concerning part I. I Many distributions were shown to be FID, including beta distributions of the first and second kinds, gamma distributions, inverse gamma distributions and scale mixtures of Boolean stable laws. Some of them are HCM, and some of them are completely monotone. Since now we know examples, the next step is to find a general theory abstracting these examples and relating classical theory of ID distributions. During one year, talks and participations in conferences have been mostly devoted to Project I. II We found that monotone increment processes are intimately connected to the theory of univalent functions. More direct contribution to univalent functions are expected. III It is known that the study of cumulants in noncommutative probability often involves combinatorics and graph theory. This project also finds new connections to these fields, involving trees and symmetric groups. We found that matricial free cumulants give us a formula connecting monotone cumulants and free cumulants using trees. Looking at this, further study of matricial freeness may yield more combinatorial and graph theoretical consequences. For more details see the enclosed PDF file
Data: CORDIS, © European Union
Project objective
Noncommutative probability, also called quantum probability or algebraicprobability theory, is an extension of classical probability theory where thealgebra of random variables is replaced by a possibly noncommutativealgebra. A surprising feature of noncommutative probability is the existenceof many very different notions of independence. The most prominent among themis freeness or free probability, which was introduced by Voiculescu to studyquestions in operator algebra theory. In the last twenty-five years, freeprobability has turned into a very active and very competitive research area,in which analogues for many important probabilistic notions like limittheorems, infinite divisibility, and L\'evy processes have been discovered. Italso turned out to be closely related to random matrix theory, which hasimportant applications in quantum physics and telecommunication.The current project proposes to study the mathematical theory of independencein noncommutative probability, and the associated convolution products. Wewill concentrate on the following topics:(1) Applications of monotone independence to free probability. Someapplications have been found already, but recent work indicates that much moreis possible.(2) Analysis of infinitely divisible distributions in classical and freeprobability. Common complex analysis methods will be used for both classes,and we expect more insight into their mutual relations.(3) Application and development of Lenczewski's matricial freeindependence. This concept introduces very new ideas, whose betterunderstanding will certainly lead to new interesting results.The methods we will use in this project come not only from noncommutativeprobability, but also from functional analysis, complex analysis, combinatorics, classical probability, random matrices, and graph theory.
Original text from CORDIS.
Participants
- UNIVERSITE DE FRANCHE-COMTE · BesanconCoordinatorFrance
Links
Data: CORDIS, © European Union