Pierre Fougères "Research"

Mathematical interests:

My research is about Markov semigroups and the study of their ergodic properties by means of some functional inequalities that bound some kind of "Entropy" by means of the Dirichlet form associated with the semigroup: Sobolev, Poincaré, log-Sobolev or other types of coercive inequalities. I'm also interested in their isoperimetric counterparts: Cheeger and Gaussian isoperimetric inequalities for example. Some of these inequalities satisfy a striking product property which makes them useful for analysis on infinite dimensional models. In particular, for spins systems in Statistical Mechanics.

In finite dimension, Sobolev inequalities for elliptic second order operators on manifolds provides precise informations about the underlying model (diameter control, bounds on the heat kernel, regularization properties of the associated semigroup, ...). Part of my research concerns geometric criteria (related with Bakry and Emery curvature-dimension criterion) to establish such Sobolev inequalities.

I'm also working on some nonlinear evolution equations and their associated semigroups: degenerate parabolic equations (evolution equation for p-Laplacian) and (non lipschitz) semilinear equations. Functional inequalities provide information on ergodicity properties, existence or regularity of solutions. And some kind of nonlinear analysis can be performed in infinite dimensional setting.

Published papers:

  • Fougères, P.
    Spectral gap for log-concave probability measures on the real line,
    "Séminaire de Probabilités", Lecture Notes in Mathematics 1857, Springer (2005).

  • Fougères, P. and Zegarlinski, B.
    Semi-Linear Problems in Infinite Dimensions, Journal of Functional Analysis 228 (2005), 39-88.


  • Fougères, P.
    Conformal liftings for Dirichlet forms, curvature-dimension and Sobolev type inequalities.

My PHD thesis:

I was a PHD student in the Laboratoire de
Statistique et Probabilités at Paul Sabatier University in Toulouse,
from Septembre 1998 to June 2002.
My supervisor was Dominique Bakry .

The viva took place on Friday 18th October 2002
at Paul Sabatier University:

Functional inequalities involving Dirichlet forms.
From isoperimetry to Sobolev inequalities.

manuscript in postscript ps.gz
manuscript pdf.gz

A few links:

Updated on 30 octobre 2007