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)

Abstract

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)