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NCTS Nonlinear PDE and Analysis Seminar
10:00 - 11:00, September 22, 2023 (Friday)
Room 509, Cosmology Building, NTU
(臺灣大學次震宇宙館 509研討室)
Sharp Transfer of Regularity for the Ornstein-Uhlenbeck Equation
Luboš Pick (Charles University)


The Ornstein--Uhlenbeck operator , defined by
is the natural counterpart of the Laplace operator when the Euclidean space endowed with the -dimensional Lebesgue measure is replaced by the probablility space , generated by the Gaussian measure
It arises naturally in some problems studied in physics and also in stochastic calculus of variations (the Malliavin calculus). It has applications in many branches of mathematics and physics including financial mathematics and the calculus of variations in infinitely many variables.
For a given function  normalized so that , there always exists a unique (up to additive constants) solution to the Ornstein--Uhlenbeck equation
in an appropriate weak sense. We study the question of transfer of regularity from to and show that an optimal (best possible) such transfer, in a certain sense, is possible. More precisely, given a rearrangement-invariant Banach function space , the smallest rearrangement-invariant Banach function space can be constructed so that
with a generic positive constant independent of and . We observe certain interesting dissimilarities to the Euclidean case, the perhaps most notable one being the fact that the gain of the degree of integrability inherited by from is not always guaranteed. Sharp form of Moser--Adams exponential inequalities is offered as well as the existence of its maximizers. 
This is a joint work with Andrea Cianchi (University of Florence) and Vít Musil (Masaryk University Brno).
Organizer: Daniel Spector (NTNU)


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