February  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. Google Scholar

[2]

D. BakryP. 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. Google Scholar

[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. Google Scholar

[4]

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

[5]

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

[6]

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

[7]

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. Google Scholar

[8]

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

[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. Google Scholar

[10]

J. DolbeaultC. 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. Google Scholar

[11]

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. Google Scholar

[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. Google Scholar

[13]

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

[14]

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. Google Scholar

[15]

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[20]

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. Google Scholar

[21]

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

[22]

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

[23]

J. C. MattinglyA. 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. Google Scholar

[24]

P. Monmarché, Generalized Γ calculus and application to interacting particles on a graph, Arxiv preprint, arXiv: 1510.05936v2Google Scholar

[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. Google Scholar

[26]

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

[27]

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

[28]

F. -Y. Wang, Generalized Curvature Condition for Subelliptic Diffusion Processes, arXiv: 1202.0778v2Google Scholar

[29]

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

[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. Google Scholar

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. Google Scholar

[2]

D. BakryP. 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. Google Scholar

[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. Google Scholar

[4]

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

[5]

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

[6]

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

[7]

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. Google Scholar

[8]

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

[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. Google Scholar

[10]

J. DolbeaultC. 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. Google Scholar

[11]

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. Google Scholar

[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. Google Scholar

[13]

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

[14]

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. Google Scholar

[15]

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[20]

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. Google Scholar

[21]

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

[22]

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

[23]

J. C. MattinglyA. 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. Google Scholar

[24]

P. Monmarché, Generalized Γ calculus and application to interacting particles on a graph, Arxiv preprint, arXiv: 1510.05936v2Google Scholar

[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. Google Scholar

[26]

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

[27]

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

[28]

F. -Y. Wang, Generalized Curvature Condition for Subelliptic Diffusion Processes, arXiv: 1202.0778v2Google Scholar

[29]

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

[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. Google Scholar

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