Limit and Law Characterizations for Chaotic Random Variables

SCHEME: INTER Mobility

CALL: 2017

DOMAIN: MT - Mathematics

FIRST NAME: Simon

LAST NAME: Campese

INDUSTRY PARTNERSHIP / PPP: No

INDUSTRY / PPP PARTNER:

HOST INSTITUTION: University of Luxembourg

KEYWORDS: Malliavin calculus, Stochastic Analysis, Lie algebras, Markov triples, Markov generators, multiple Wiener integrals, limit theorems, chaotic random variables

START: 2017-09-01

END: 2018-06-01

WEBSITE: https://www.uni.lu

Submitted Abstract

Multiple Wiener-Itô integrals are the building blocks of square integrablerandom functionals and therefore central objects in the theory ofinfinite-dimensional stochastic analysis and non-linear stochasticapproximation. They are sucessfully being used to model real-world phenomena,for example in physics, cosmology, biology or finance. As recently discovered bythe principal investigator, such integrals are particular instances of chaoticrandom variables, which provide a much more flexible modelling framework, alsocovering situations where the Gaussian nature of multiple Wiener-Itô integralsis not an optimal modelling choice. Moreover, in the general framework ofchaotic random variables very powerful mathematical techniques become availablewhich have the potential to in turn prove results about multiple Wiener-Itôintegrals.The principal investigator intends to setup a long-term collaboration betweenthe Department of Mathematics and Statistics of Boston University and theMathematics Research Unit of Luxembourg University in order to fully developthe underlying theory and mathematical modelling framework provided by chaoticrandom variables and then apply this to tackle several important open problems.To start this collaboration, the principal investigator proposes to visit theDepartment of Mathematics and Statistics of Boston University, which is home ofa leading research group in probability theory and statistics,in order two work on two fundamental problems at the heart of the theory: Thecharacterization of laws of chaotic random variables and the characterization oflimits a sequence of chaotic random variables can converge to indistribution. Both of these problems are of very high importance inapplications, as their solution would provide a complete understanding of theapproximating objects as well as the limits which can be attained. Even in the special case of multiple Wiener-Itô integrals, solutions to theseproblems (not considering trivial cases) are only available for integrals oforder two. The proof is specific to this order as it makes heavy use ofHilbert-Schmidt techniques and therefore leaves little hope for a generalizationto higher orders. For generic chaotic random variables, both problems arecompletely open.Our approach relies on a novel combination of Markov generator techniques, inparticular the associated Gamma-calculus, and Lie algebra methods. We stronglybelieve that this will not only lead to major advances, if not completesolutions to the two problems outlined above, but also open the door to newapplications, followup projects and collaborations.Three of such followup projects are already foreseeable and planned to be workedon after the proposed research stay of the principal investigator.It is to be expected that the proposed research will be published inhigh-caliber international peer-reviewed mathematical journals such asStochastic Processes and Related Fields, Geometric and Functional Analysis, theAnnals of Probability or the Journal of Functional Analysis. It will be ofsignificant and sustainable value to the research program of the MathematicsResearch Unit of Luxembourg University and capture the interest of a large andinternational scientific audience.

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