THE CLOSURE PROPERTY OF THE FOKKER-PLANCK EQUATION, GAUSSIAN HYPERCONTRACTIVTY, AND LOGARITHMIC SOBOLEV INEQUALITIES (Mathematical aspects of quantum fields and related topics)

Abstract

An importance of functional inequalities can be usually seen by being applied to analysis of differential equations. In this report, we explain an idea reversing such understanding, namely applying properties of differential equations to analyze functional inequalities. This idea is motivated from the work on the theory of Brascamp-Lieb inequality due to Bennett-Carbery-Christ- Tao [5] and Carlen-Lieb-Loss [9]. More precisely, we report that one can improve the best constant of Nelson's hypercontractivity inequality and Grass's logarithmic Sobolev inequality via the regularizing property of the Fokker-Planck equation, which is the main result in the work with Bez and Tsuji [7].