July  2020, 40(7): 4093-4112. doi: 10.3934/dcds.2020173

Accelerating planar Ornstein-Uhlenbeck diffusion with suitable drift

1. 

Laboratoire d'Algèbre, Géométrie, Théorie Spectrale LR 11 ES 53, Faculté des Sciences de Sfax, Univerité de Sfax, 3.5 km, Route de la Soukra, B.P. 1171, Sfax 3000, Tunisia

2. 

Laboratoire de Mathématiques de Bretagne Atlantique, UMR CNRS 6205, Université de Brest, UFR Sciences et Techniques, 6 Avenue Le Gorgeu, 29200 Brest, France

3. 

ESPRIT - Ecole Supérieure Privée d'Ingénierie et de Technologies, Pôle Technologique El Ghazela, Ariana 2083, Tunisia

* Corresponding author: Brice Franke

Received  June 2018 Revised  February 2019 Published  April 2020

Fund Project: This research was supported by the french tunisian research grant PHC Utique CMCU16G1505

The principal aim of this paper is to construct an explicit sequence of weighted divergence free vector fields which accelerates the rate of convergence of planar Ornstein-Uhlenbeck diffusion to its equilibrium state. The rate of convergence is expressed in terms of the spectral gap of the diffusion generator. We construct an explicit sequence of vector fields which pushes the spectral gap to infinity. The acceleration of the diffusion results from the strong oscillation of the flow lines generated by the vector field.

Citation: Mondher Damak, Brice Franke, Nejib Yaakoubi. Accelerating planar Ornstein-Uhlenbeck diffusion with suitable drift. Discrete & Continuous Dynamical Systems - A, 2020, 40 (7) : 4093-4112. doi: 10.3934/dcds.2020173
References:
[1] R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, 65, Academic Press, New York-London, 1975.   Google Scholar
[2]

P. Bérard, Spectral Geometry: Direct and Inverse Problems, Lecture Notes in Mathematics, 1207, Springer-Verlag, Berlin, 1986. doi: 10.1007/BFb0076330.  Google Scholar

[3] I. Chavel, Eigenvalues in Riemannian Geometry, Pure and Applied Mathematics, 115, Academic Press, Inc., Orlando, FL, 1984.   Google Scholar
[4]

P. ConstantinA. KiselevL. Ryzhik and A. Zlatos, Diffusion and mixing in fluid flow, Ann. of Math. (2), 168 (2008), 643-674.  doi: 10.4007/annals.2008.168.643.  Google Scholar

[5]

B. Franke, Integral inequalities for the fundamental solutions of diffusions on manifolds with divergence-free drift, Math. Z., 246 (2004), 373-403.  doi: 10.1007/s00209-003-0604-1.  Google Scholar

[6]

B. FrankeC.-R. HwangH.-M. Pai and S.-J. Sheu, The behavior of the spectral gap under growing drift, Trans. Amer. Math. Soc., 362 (2010), 1325-1350.  doi: 10.1090/S0002-9947-09-04939-3.  Google Scholar

[7]

B. Franke and N. Yaakoubi, On how to use drift to push the spectral gap of a diffusion on $S^{2}$ to infinity, Quart. Appl. Math., 74 (2016), 321-335.  doi: 10.1090/qam/1426.  Google Scholar

[8]

B. Franke and N. Yaakoubi, Accelerating diffusion on compact Riemannian surfaces by incompressible drift, Anal. Appl. (Singap.), 15 (2017), 653-666.  doi: 10.1142/S0219530516500184.  Google Scholar

[9]

S. Geman and C.-R. Hwang, Diffusion for global optimization, SIAM J. Control Optim., 24 (1986), 1031-1043.  doi: 10.1137/0324060.  Google Scholar

[10]

P. Hartman, Ordinary Differential Equations, Birkhäuser, Boston, MA, 1982.  Google Scholar

[11]

C.-R. HwangS.-Y. Hwang-Ma and S.-J. Sheu, Accelerating Gaussian diffusions, Ann. Appl. Probab., 3 (1993), 897-913.  doi: 10.1214/aoap/1177005371.  Google Scholar

[12]

C.-R. HwangS.-Y. Hwang-Ma and S.-J. Sheu, Accelerating diffusions, Ann. Appl. Probab., 15 (2005), 1433-1444.  doi: 10.1214/105051605000000025.  Google Scholar

[13]

C.-R. HwangR. Normand and S.-J. Wu, Variance reduction for diffusions, Stochastic Process. Appl., 125 (2015), 3522-3540.  doi: 10.1016/j.spa.2015.03.006.  Google Scholar

[14]

C.-R. Hwang and H.-M. Pai, Accelerating Brownian motion on $N$-torus, Statist. Probab. Lett., 83 (2013), 1443-1447.  doi: 10.1016/j.spl.2013.02.009.  Google Scholar

[15]

N. Yaakoubi, Accélération Explicite des Diffusions par des Flots Incompressibles sur des Espaces Bidimensionnels, Ph.D thesis, University of Sfax in Tunisia, 2016. Google Scholar

[16]

W. P. Ziemer, Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation, Graduate Texts in Mathematics, 120, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4612-1015-3.  Google Scholar

show all references

References:
[1] R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, 65, Academic Press, New York-London, 1975.   Google Scholar
[2]

P. Bérard, Spectral Geometry: Direct and Inverse Problems, Lecture Notes in Mathematics, 1207, Springer-Verlag, Berlin, 1986. doi: 10.1007/BFb0076330.  Google Scholar

[3] I. Chavel, Eigenvalues in Riemannian Geometry, Pure and Applied Mathematics, 115, Academic Press, Inc., Orlando, FL, 1984.   Google Scholar
[4]

P. ConstantinA. KiselevL. Ryzhik and A. Zlatos, Diffusion and mixing in fluid flow, Ann. of Math. (2), 168 (2008), 643-674.  doi: 10.4007/annals.2008.168.643.  Google Scholar

[5]

B. Franke, Integral inequalities for the fundamental solutions of diffusions on manifolds with divergence-free drift, Math. Z., 246 (2004), 373-403.  doi: 10.1007/s00209-003-0604-1.  Google Scholar

[6]

B. FrankeC.-R. HwangH.-M. Pai and S.-J. Sheu, The behavior of the spectral gap under growing drift, Trans. Amer. Math. Soc., 362 (2010), 1325-1350.  doi: 10.1090/S0002-9947-09-04939-3.  Google Scholar

[7]

B. Franke and N. Yaakoubi, On how to use drift to push the spectral gap of a diffusion on $S^{2}$ to infinity, Quart. Appl. Math., 74 (2016), 321-335.  doi: 10.1090/qam/1426.  Google Scholar

[8]

B. Franke and N. Yaakoubi, Accelerating diffusion on compact Riemannian surfaces by incompressible drift, Anal. Appl. (Singap.), 15 (2017), 653-666.  doi: 10.1142/S0219530516500184.  Google Scholar

[9]

S. Geman and C.-R. Hwang, Diffusion for global optimization, SIAM J. Control Optim., 24 (1986), 1031-1043.  doi: 10.1137/0324060.  Google Scholar

[10]

P. Hartman, Ordinary Differential Equations, Birkhäuser, Boston, MA, 1982.  Google Scholar

[11]

C.-R. HwangS.-Y. Hwang-Ma and S.-J. Sheu, Accelerating Gaussian diffusions, Ann. Appl. Probab., 3 (1993), 897-913.  doi: 10.1214/aoap/1177005371.  Google Scholar

[12]

C.-R. HwangS.-Y. Hwang-Ma and S.-J. Sheu, Accelerating diffusions, Ann. Appl. Probab., 15 (2005), 1433-1444.  doi: 10.1214/105051605000000025.  Google Scholar

[13]

C.-R. HwangR. Normand and S.-J. Wu, Variance reduction for diffusions, Stochastic Process. Appl., 125 (2015), 3522-3540.  doi: 10.1016/j.spa.2015.03.006.  Google Scholar

[14]

C.-R. Hwang and H.-M. Pai, Accelerating Brownian motion on $N$-torus, Statist. Probab. Lett., 83 (2013), 1443-1447.  doi: 10.1016/j.spl.2013.02.009.  Google Scholar

[15]

N. Yaakoubi, Accélération Explicite des Diffusions par des Flots Incompressibles sur des Espaces Bidimensionnels, Ph.D thesis, University of Sfax in Tunisia, 2016. Google Scholar

[16]

W. P. Ziemer, Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation, Graduate Texts in Mathematics, 120, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4612-1015-3.  Google Scholar

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