2018, 11(1): 1-23. doi: 10.3934/krm.2018001

Hypocoercive estimates on foliations and velocity spherical Brownian motion

1. 

Department of Mathematics, University of Connecticut, 341 Mansfield Road Storrs, CT 06269-1009, USA

2. 

LPMA, Université Pierre et Marie Curie, 4, Place Jussieu 75005 Paris, France

* Corresponding author: Fabrice Baudoin

Received  April 2016 Revised  April 2017 Published  August 2017

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

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 H1 and in L2.

Citation: Fabrice Baudoin, Camille Tardif. Hypocoercive estimates on foliations and velocity spherical Brownian motion. Kinetic & Related Models, 2018, 11 (1) : 1-23. doi: 10.3934/krm.2018001
References:
[1]

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.

[2]

D. Bakry, P. Cattiaux, 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.

[3]

D. Bakry and M. Emery, Diffusions hypercontractives, Sémin. de probabilités, XIX, 1983/84, 177-206, Lecture Notes in Math. , 1123, Springer, Berlin, 1985. doi: 10.1007/BFb0075847.

[4]

F. Baudoin, Bakry-Emery meet Villani, J. Funct. Anal. 273 (2017), no. 7, 2275-2291

[5]

F. Baudoin, Sub-Laplacians and Hypoelliptic Operators on Totally Geodesic Riemannian Foliations Course of the Institute Henri Poincaré, 2014.

[6]

F. Baudoin, Wasserstein contraction properties for hypoelliptic diffusions, preprint, arXiv: 1602.04177, 2016.

[7]

F. Baudoin, 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.

[8]

L. Bérard-Bergery, J. P. Bourguignon, Laplacians and Riemannian submersions with totally geodesic fibres, Illinois J. Math., 26 (1982), 181-200.

[9]

J.-M. Bismut, The hypoelliptic Laplacian on the cotangent bundle, J. Amer. Math. Soc, 18 (2005), 379-476. doi: 10.1090/S0894-0347-05-00479-0.

[10]

J. Dolbeault, C. Mouhot, 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.

[11]

J.-P. Eckmann, M. Hairer, Spectral properties of hypoelliptic operators, Communications in Mathematical Physics, 235 (2003), 233-253. doi: 10.1007/s00220-003-0805-9.

[12]

K. D. Elworthy, Decompositions of diffusion operators and related couplings, Stochastic analysis and applications 2014, 100 (2014), 283-306. doi: 10.1007/978-3-319-11292-3_10.

[13]

J. Franchi, Y. Le Jan, Relativistic diffusions and Schwarzschild geometry, Comm. Pure Appl. Math., 60 (2007), 187-251. doi: 10.1002/cpa.20140.

[14]

S. Gadat, 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.

[15]

M. Grothaus, 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.

[16]

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. doi: 10.1186/2190-5983-4-4.

[17]

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. doi: 10.1142/S0219493713500019.

[18]

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. doi: 10.1007/b104762.

[19]

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.

[20]

F. Hérau, 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.

[21]

R. Hladky, Connections and Curvature in sub-Riemannian geometry, Houston J. Math, 38 (2012), 1107-1134.

[22]

X.-M. Li, Random perturbation to the geodesic equation, Annal of Prob., 44 (2016), 544-566. doi: 10.1214/14-AOP981.

[23]

J. C. Mattingly, A. M. Stuart, 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.

[24]

P. Monmarché, Generalized Γ calculus and application to interacting particles on a graph, Arxiv preprint, arXiv: 1510.05936v2

[25]

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.

[26]

P. Tondeur, Foliations on Riemannian Manifolds Universitext. Springer-Verlag, New York, 1988. xii+247 pp. doi: 10.1007/978-1-4613-8780-0.

[27]

C. Villani, Hypocoercivity Mem. Amer. Math. Soc. 202 (2009), iv+141 pp. doi: 10.1090/S0065-9266-09-00567-5.

[28]

F. -Y. Wang, Generalized Curvature Condition for Subelliptic Diffusion Processes, arXiv: 1202.0778v2

[29]

F. -Y. Wang, Analysis for Diffusion Processes on Riemannian Manifolds Advanced Series on Statistical Science and Applied Probability, Vol. 18. World Scientific, 2014.

[30]

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.

show all references

References:
[1]

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.

[2]

D. Bakry, P. Cattiaux, 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.

[3]

D. Bakry and M. Emery, Diffusions hypercontractives, Sémin. de probabilités, XIX, 1983/84, 177-206, Lecture Notes in Math. , 1123, Springer, Berlin, 1985. doi: 10.1007/BFb0075847.

[4]

F. Baudoin, Bakry-Emery meet Villani, J. Funct. Anal. 273 (2017), no. 7, 2275-2291

[5]

F. Baudoin, Sub-Laplacians and Hypoelliptic Operators on Totally Geodesic Riemannian Foliations Course of the Institute Henri Poincaré, 2014.

[6]

F. Baudoin, Wasserstein contraction properties for hypoelliptic diffusions, preprint, arXiv: 1602.04177, 2016.

[7]

F. Baudoin, 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.

[8]

L. Bérard-Bergery, J. P. Bourguignon, Laplacians and Riemannian submersions with totally geodesic fibres, Illinois J. Math., 26 (1982), 181-200.

[9]

J.-M. Bismut, The hypoelliptic Laplacian on the cotangent bundle, J. Amer. Math. Soc, 18 (2005), 379-476. doi: 10.1090/S0894-0347-05-00479-0.

[10]

J. Dolbeault, C. Mouhot, 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.

[11]

J.-P. Eckmann, M. Hairer, Spectral properties of hypoelliptic operators, Communications in Mathematical Physics, 235 (2003), 233-253. doi: 10.1007/s00220-003-0805-9.

[12]

K. D. Elworthy, Decompositions of diffusion operators and related couplings, Stochastic analysis and applications 2014, 100 (2014), 283-306. doi: 10.1007/978-3-319-11292-3_10.

[13]

J. Franchi, Y. Le Jan, Relativistic diffusions and Schwarzschild geometry, Comm. Pure Appl. Math., 60 (2007), 187-251. doi: 10.1002/cpa.20140.

[14]

S. Gadat, 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.

[15]

M. Grothaus, 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.

[16]

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. doi: 10.1186/2190-5983-4-4.

[17]

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. doi: 10.1142/S0219493713500019.

[18]

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. doi: 10.1007/b104762.

[19]

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.

[20]

F. Hérau, 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.

[21]

R. Hladky, Connections and Curvature in sub-Riemannian geometry, Houston J. Math, 38 (2012), 1107-1134.

[22]

X.-M. Li, Random perturbation to the geodesic equation, Annal of Prob., 44 (2016), 544-566. doi: 10.1214/14-AOP981.

[23]

J. C. Mattingly, A. M. Stuart, 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.

[24]

P. Monmarché, Generalized Γ calculus and application to interacting particles on a graph, Arxiv preprint, arXiv: 1510.05936v2

[25]

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.

[26]

P. Tondeur, Foliations on Riemannian Manifolds Universitext. Springer-Verlag, New York, 1988. xii+247 pp. doi: 10.1007/978-1-4613-8780-0.

[27]

C. Villani, Hypocoercivity Mem. Amer. Math. Soc. 202 (2009), iv+141 pp. doi: 10.1090/S0065-9266-09-00567-5.

[28]

F. -Y. Wang, Generalized Curvature Condition for Subelliptic Diffusion Processes, arXiv: 1202.0778v2

[29]

F. -Y. Wang, Analysis for Diffusion Processes on Riemannian Manifolds Advanced Series on Statistical Science and Applied Probability, Vol. 18. World Scientific, 2014.

[30]

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.

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