Speaker: Aurel Stan
Title: Hölder-Young inequalities for norms of Wick products
Abstract: The Wick product is a natural product that can be defined for every probability measure, on the number line, having finite moments of all orders. This product is used in the definition of some stochastic integrals, and using Wick product antiderivatives instead of point-wise antiderivatives, formulas like Itô's Lemma become the classic Leibniz-Newton Fundamental Theorem of Calculus. Using second quantization operators, we can bound the norms of the Wick products. The inequalities governing the norms of Wick products are in general connected to sharp inequalities from classic Harmonic Analysis. In our talk, we define first the Wick product using the orthogonal structure given by the orthogonal polynomials generated by the probability measure. Then, in the particular case of Gaussian, Poisson, and Gamma distributions, we present some integral representations of the Wick product. Finally, we use these integral representations and some sharp inequalities from classic Harmonic Analysis to prove some Hölder-Young inequalities for the norms of Wick products.
Note: This is part of the Invitations to Mathematics lecture series given each year in Autumn and Spring semesters. Pre-candidacy students can sign up for this lecture series by registering for one or two credit hours of Math 6193, class #16988 (with Prof H. Moscovici).