Michel Bonnefont: Research

I am currently "maître de conférences" in the analysis team at the Institut de Mathématiques de Bordeaux since september 2010.
My main research interest are:

• Heat kernel and heat semi-group in Riemannian and sub-Riemannian geometry.
• Convergence to equilibrium and functional inequalities (Poincaré, Log-Sobolev).
• Curvature-dimension criterions. 
• Couplings for  sub elliptic diffusion.
• Intertwinning between gradien and semi-groups and semi-group on forms.
• Spectrum of discrete Laplacians.

I am currently a member of the ANR RAGE and of the ANR QuAMProcs. Before I was a member of the ANR HAB.

Here is the web page of the séminaire d'analyse.

PHD Student

With Marc Arnaudon, I co-supervised the PHD of Baptiste Huguet (december 2020 at Bordeaux).
With Marc Arnaudon, I co-supervised the PHD of Magalie Bénéfice (starting in september 2021).

Ph-D thesis

I defend my Ph-D thesis at the Institut de Mathématiques de Toulouse. My advisors were Dominique Bakry and Fabrice Baudoin. My thesis is entitled: "Functional Inequalities for Subelliptic Heat Kernels".

In this thesis, I study the heat kernel and the heat semigroup associated with three subelliptic structures that are actually model spaces of subriemannian geometry. Their study is therefore interesting in itself. But in fact, the main motivation of this work was to develop new techniques and new methods that could extend to more general situations in subelliptic geometry. The real goal is to understand a notion of Ricci curvature bounded below by a constant in subelliptic geometry.

You can find a short summary of the thesis here and get the entire version here.
 

HDR

I defend my HDR at the Institut de Mathématiques de Bordeaux in november 2019: HDR

 

Summer school notes:

• Global Sensitivity Analysis and Poincaré inequalities (Toulouse 2022): Poincaré inequality with explicit constant in dimension d >1 notes

 

Prépublications

•   Covariance inequalities for convex and log-concave functions.  https://arxiv.org/abs/2302.05208
  (With Erwan Hillion and Adrien Saumard)
   
•   A note on the spectral gap for log-concave probability measures on convex bodies. https://arxiv.org/abs/2302.05207
  (With Aldéric Joulin) 
   
• Sharp large deviations and concentration inequalities for the number of descents in a random permutation. https://arxiv.org/abs/2210.10382
  (with Bernard Bercu and Adrien Richou)
   
• Lower bounds for the spectral gap and an extension of the Bonnet Myers theorem.  Arxiv preprint
  (with El Maati Ouhabaz)
   
• Eigenvalue asymptotics  and unique continuation of eigenfunctions on planar graphs Arxiv preprint
  (with Sylvain Golenia, Matthias Keller)
   
• Convergence to equilibrium for hypoelliptic non-symmetric Ornstein-Uhlenbeck type operators. Arxiv preprint
  (with Fabrice Baudoin and Li Chen) 
  
   

Publications

• A note on eigenvalues estimates for one-dimensional diffusion operators. Bernoulli 28 (2022), no. 1, 64–86. Arxiv preprint
  (with Aldéric Joulin )
   
• Intertwinings, second-order Brascamp-Lieb inequalities and spectral estimates. Studia Math. 260 (2021), no. 3, 285–316.   Preprint
  (with A. Joulin).
   
• Magnetic sparseness and Schrödinger operators on graphs. Ann. Henri Poincaré 21 (2020), no. 5, 1489–1516. no. 1, 543–565. Arxiv preprint
  (with Sylvain Golénia, Matthias Keller, Shiping Liu, Florentin Münch).

• Couplings in L^p distance of two Brownian motions and their Lévy area. Ann. Inst. Henri Poincaré Probab. Stat. 56 (2020), no. 1, 543–565. Arxiv preprint
  (with Nicolas Juillet )
   
• On logarithmic Sobolev inequalities for the heat kernel on the Heisenberg group. Ann. Fac. Sci. Toulouse Math. (6) 29 (2020), no. 2, 335–355. Arxiv preprint
  (with D. Chafaï and R. Herry).
   
• Intertwinings and Generalized Brascamp-Lieb Inequalities. Rev. Mat. Iberoam. 34 (2018), no. 3, 1021–1054. Arxiv preprint
  (with M. Arnaudon and A. Joulin).
   
• Spectral gap for spherically symmetric log-concave probability measures, and beyond. J. Funct. Anal. 270 (2016), no. 7, 2456–2482. Arxiv preprint
  (with A. Joulin and Y. Ma).
   
• A note on spectral gap and weighted Poincaré inequalities for some one-dimensional diffusions ESAIM Probability & Statistics, vol. 20 (2016), 18-29. Arxiv preprint
  (with A. Joulin and Y. Ma).
   
• Reverse Poincare inequalities, Isoperimetry and Riesz transforms in Carnot groups. Nonlinear Anal. 131 (2016), 48–59. Arxiv preprint
  (with F. Baudoin).
   
• Eigenvalue asymptotics for Schrödinger operators on sparse graphs. Ann. Inst. Fourier (Grenoble) 65 (2015), no. 5, 1969–1998. Arxiv preprint
  (with S. Golénia and M. Keller).
   
• Essential spectrum and Weyl asymptotics for discrete Laplacians. Ann. Fac. Sci. Toulouse Math. (6) 24 (2015), no. 3, 563–624. Arxiv preprint
  (with S. Golénia).
   
• Curvature-dimension estimates for the Laplace-Beltrami operator of a totally geodesic foliation . Nonlinear Anal. 126 (2015), 159–169. Arxiv preprint
  (with F. Baudoin).
   
• Volume and distance comparison theorems for sub-Riemannian manifolds. J. Funct. Anal. 267 (2014), no. 7, 2005–2027. Arxiv preprint
  (with F. Baudoin N. Garofalo and I. Munive).
   
• Intertwining relations for one-dimensional diffusions and application to functional inequalities Potential Anal. 41 (2014), no. 4, 1005–1031. last version
  (with A. Joulin).
   
• A sub-Riemannian curvature-dimension inequality, volume doubling property and the Poincaré inequality. Mathematische Annalen 358 (2014), no. 3-4, 833–860. Arxiv preprint
  (with F. Baudoin and N. Garofalo).
   
• Subriemannian balls in CR Sasakian manifolds. Proceedings of the American Mathematical Society. 141 (2013), no. 11, 3919–3924. Arxiv preprint
  (with F. Baudoin).
   
• Log-Sobolev inequalities for subelliptic operators satisfying a generalized curvature dimension inequality. Journal of Functional Analysis. 262 (2012), no. 6, 2646–2676. Arxiv preprint
  (With F. Baudoin)
   
• The subelliptic heat kernel on SL(2,R) and on its universal covering: integral representations and some functional inequalities. Potential analysis. 36 (2012), no. 2, 275–300. Arxiv preprint
  
   
• A discrete version of Brunn Minkowski inequality and its stability. Annales Mathématiques Blaise Pascal. 16 (2009), no. 2, 245–257. Arxiv preprint
  
   
• Subelliptic Li-Yau estimates on three dimensional model spaces. Potential theory and stochastics in Albac, 1–10, Theta Ser. Adv. Math., 11, Theta, Bucharest, 2009. Arxiv preprint
  (with D. Bakry, F. Baudoin and B.Qian).
   
• The subelliptic heat kernel on SU(2): Representations, Asymptotics and Gradient bounds. Mathematische Zeitschritt. 263 (2009), no. 3, 647–672. Arxiv preprint
  (with F. Baudoin).
   
• On gradient bounds for the heat kernel on the Heisenberg group. Journal of Functional Analysis. 255 (2008), no. 8, 1905–1938. Arxiv preprint
  (with D. Bakry, F. Baudoin and D. Chafaï).

   

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