Introduction to Infinite Dimensional Analysis

Prof. B Zegarlinski

Imperial HXLY 6M42

Friday's 10 am-12noon (January 22 – March 11)

1. Introduction to C_0-semigroups : Linear and Nonlinear Semigroups in Banach Spaces, Markov Semigroups, Generators, Hille-Yosida Theorem; Examples in continuous and discrete setup, Diffusion & Jump type.

2. Coercive Inequalties & Smoothing and Ergodicity of Semigroups: Sobolev and Nash Inequalities, Ultracontractivity in Lp spaces; Poincare and Weak Poincare Inequalities; Spectral properties of the generators. Coercive inequalities of Log-Sobolev type; stability with respect to tensorisation, perturbation property.

3. Techniques of proving coercive inequalities in finite dimensions: U-bounds, coercive inequalities, isoperimetric inequalities; Riemannian and Sub-Riemannian setup, Analysis on Lie groups (Nilpotent Lie Groups). In Lp spaces associated to probability measures with quickly/slowly decaying tails.

4. Semigroups with Hoermander type generators: Constructive techniques for smoothing, short and long time quantitative estimates.

5. [Analysis on extended groups: Reflection operators associated to Hoermander fields, generalised 1st order operators, associated Markov generators and semigroups, ultracontractivity and heat kernel bounds.]

6. Probability measures and Markov Semigroups in infinite dimensions: Local specification and Gibbs measures, Coercive and Isoperimetric Inequalities for Gibbs measures; Finite speed of propagation of information and construction of Markov semigroups in infinite dimensions.

7. Constructive Techniques for Ergodicity: Generalised Gradient Bounds and Strong Ergodicity; Techniques based on Log Sobolev and Weak Poincare Inequalities in Infinite Dimensions.

8. Applications of Log Sobolev to Nonlinear PDEs: Analysis of Reaction-Diffusion Systems.

Course Material :

Lec1 ; Lec2 ; Lec3 ; Lec4 ; Lec5 ; Lec6 ; Lec7 ; Lec8 ; Lec9 ;

PS.1; PS.2; PS.3; PS.4; PS.5; PS.6 ; PS.7;

Lecture Notes

Guionnet A, Zegarlinski B, Lectures on logarithmic Sobolev inequalities, Lecture Notes Mmath, 2003, Vol:1801,Pages:1-134,

Books

Jerome A. Goldstein, Semigroups of linear operators and applications, Oxford University Press, Clarendon Press, 1985.

Dominique Bakry, Ivan Gentil & Michel Ledoux, Analysis and Geometry of Markov Diffusion Operators

Sur Les Inegalites de Sobolev Logarithmiques - S. Blanchere, D. Chafai, P. Fougeres, I. Gentil, F. Malrieu, C. Roberto, and G. Scheffer, Societe Mathematique de France, 2000.

Viorel Barbu, Nonlinear semigroups and differential equations in Banach spaces. Noordhoff International Publishing 1976.

---------------------------------------------------------------------------------------------------------