\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Hypocoercive estimates on foliations and velocity spherical Brownian motion

  • * Corresponding author: Fabrice Baudoin

    * Corresponding author: Fabrice Baudoin 

The first author is supported in part by Grant NSF-DMS 15-11-328

Abstract / Introduction Full Text(HTML) Related Papers Cited by
  • By further developing the generalized $Γ$-calculus for hypoelliptic operators, we prove hypocoercive estimates for a large class of Kolmogorov type operators which are defined on non necessarily totally geodesic Riemannian foliations. We study then in detail the example of the velocity spherical Brownian motion, whose generator is a step-3 generating hypoelliptic Hörmander's type operator. To prove hypocoercivity in that case, the key point is to show the existence of a convenient Riemannian foliation associated to the diffusion. We will then deduce, under suitable geometric conditions, the convergence to equilibrium of the diffusion in $H^1$ and in $L^2$.

    Mathematics Subject Classification: Primary:60J60, 58J65, 35H10, 35B40;Secondary:55R25.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  •   I. Bailleul , J. Angst  and  C. Tardif , Kinetic Brownian motion on Riemannian manifolds, Electronic Journal of Probability, 20 (2015) , 40pp.  doi: 10.1214/EJP.v20-4054.
      D. Bakry , P. Cattiaux  and  A. Guillin , Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincaré, J. Funct. Anal., 254 (2008) , 727-759.  doi: 10.1016/j.jfa.2007.11.002.
      D. Bakry and M. Emery, Diffusions hypercontractives, Sémin. de probabilités, XIX, 1983/84, 177–206, Lecture Notes in Math. , 1123, Springer, Berlin, 1985.
      F. Baudoin, Bakry-Emery meet Villani, J. Funct. Anal. 273 (2017), no. 7,2275-2291
      F. Baudoin, Sub-Laplacians and Hypoelliptic Operators on Totally Geodesic Riemannian Foliations Course of the Institute Henri Poincaré, 2014.
      F. Baudoin, Wasserstein contraction properties for hypoelliptic diffusions, preprint, arXiv: 1602. 04177,2016.
      F. Baudoin  and  N. Garofalo , Curvature-dimension inequalities and Ricci lower bounds for sub-Riemannian manifolds with transverse symmetries, J. Eur. Math. Soc. (JEMS), 19 (2017) , 151-219.  doi: 10.4171/JEMS/663.
      L. Bérard-Bergery  and  J. P. Bourguignon , Laplacians and Riemannian submersions with totally geodesic fibres, Illinois J. Math., 26 (1982) , 181-200. 
      J.-M. Bismut , The hypoelliptic Laplacian on the cotangent bundle, J. Eur. Math. Soc. (JEMS), 19 (2017) , 151-219.  doi: 10.1090/S0894-0347-05-00479-0.
      J. Dolbeault , C. Mouhot  and  C. Schmeiser , Hypocoercivity for linear kinetic equations conserving mass, Trans. Amer. Math. Soc., 367 (2015) , 3807-3828.  doi: 10.1090/S0002-9947-2015-06012-7.
      J.-P. Eckmann  and  M. Hairer , Spectral properties of hypoelliptic operators, Communications in Mathematical Physics, 235 (2003) , 233-253.  doi: 10.1007/s00220-003-0805-9.
      K. D. Elworthy, Decompositions of diffusion operators and related couplings, Stochastic analysis and applications 2014, Springer Proc. Math. Stat. , 100 (2014), 283–306.
      J. Franchi  and  Y. Le Jan , Relativistic diffusions and Schwarzschild geometry, Comm. Pure Appl. Math., 60 (2007) , 187-251.  doi: 10.1002/cpa.20140.
      S. Gadat  and  L. Miclo , Spectral decompositions and $L^2$-operator norms of toy hypocoercive semi-groups, Kinetic and Related Models, 6 (2013) , 317-372.  doi: 10.3934/krm.2013.6.317.
      M. Grothaus  and  P. Stilgenbauer , Hypocoercivity for Kolmogorov backward evolution equations and applications, J. Funct. Anal., 267 (2014) , 3515-3556.  doi: 10.1016/j.jfa.2014.08.019.
      M. Grothaus, A. Klar, J. Maringer, P. Stilgenbauer and R. Wegener, Application of a three-dimensional fiber lay-down model to non-woven production processes J. Math. Ind 4 (2014), Art. 4, 19 pp.
      M. Grothaus and P. Stilgenbauer, Geometric Langevin equations on submanifolds and applications to the stochastic melt-spinning process of nonwovens and biology Stoch. Dyn. 13 (2013), 1350001, 34 pp.
      B. Helffer and F. Nier, Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators and Witten Laplacians, Lecture Notes in Mathematics, 1862. Springer-Verlag, Berlin, 2005.
      F. Hérau , Short and long time behavior of the Fokker-Planck equation in a confining potential and applications, J. Funct. Anal., 244 (2007) , 95-118.  doi: 10.1016/j.jfa.2006.11.013.
      F. Hérau  and  F. Nier , Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with high degree potential, Arch. Ration. Mech. Anal., 171 (2004) , 151-218.  doi: 10.1007/s00205-003-0276-3.
      R. Hladky , Connections and Curvature in sub-Riemannian geometry, Houston J. Math, 38 (2012) , 1107-1134. 
      X.-M. Li , Random perturbation to the geodesic equation, Annal of Prob., 44 (2016) , 544-566.  doi: 10.1214/14-AOP981.
      J. C. Mattingly , A. M. Stuart  and  D. J. Higham , Ergodicity for SDEs and approximations: Locally Lipschitz vector fields and degenerate noise, Stochastic Processes and their Applications, 101 (2002) , 185-232.  doi: 10.1016/S0304-4149(02)00150-3.
      P. Monmarché, Generalized $Γ$ calculus and application to interacting particles on a graph, Arxiv preprint, arXiv: 1510. 05936v2
      D. Talay , Stochastic Hamiltonian systems: Exponential convergence to the invariant measure, and discretization by the implicit Euler scheme, Markov Process. Related Fields, 8 (2002) , 163-198. 
      P. Tondeur, Foliations on Riemannian Manifolds Universitext. Springer-Verlag, New York, 1988. ⅹⅱ+247 pp.
      C. Villani, Hypocoercivity Mem. Amer. Math. Soc. 202 (2009), ⅳ+141 pp.
      F. -Y. Wang, Generalized Curvature Condition for Subelliptic Diffusion Processes, arXiv: 1202. 0778v2
      F. -Y. Wang, Analysis for Diffusion Processes on Riemannian Manifolds Advanced Series on Statistical Science and Applied Probability, Vol. 18. World Scientific, 2014.
      L. Wu , Large and moderate deviations and exponential convergence for stochastic damping Hamiltonian systems, Stochastic Process. Appl., 91 (2001) , 205-238.  doi: 10.1016/S0304-4149(00)00061-2.
  • 加载中
SHARE

Article Metrics

HTML views(2083) PDF downloads(257) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return