September  2020, 40(9): 5131-5148. doi: 10.3934/dcds.2020222

Exponential convergence in the Wasserstein metric $ W_1 $ for one dimensional diffusions

1. 

School of Science, Nanjing University of Science and Technology, Nanjing 210094, Jiangsu, China

2. 

School of Statistics and Information, Shanghai University of International Business and Economics, Shanghai 201620, China

3. 

Laboratoire de Math. CNRS-UMR 6620, Université Clermont-Auvergne, Clermont Ferrand, 63177 Aubière, France

* Corresponding author: Ruinan Li

Received  April 2019 Revised  March 2020 Published  June 2020

Fund Project: The first author is supported by "the Fundamental Research Funds for the Central Universities'' grant 30920021145

In this paper, we find some general and efficient sufficient conditions for the exponential convergence $ W_{1,d}(P_t(x,\cdot), P_t(y,\cdot) )\le Ke^{-\delta t}d(x,y) $ for the semigroup $ (P_t) $ of one-dimensional diffusion. Moreover, some sharp estimates of the involved constants $ K\ge 1, \delta>0 $ are provided. Those general results are illustrated by a series of examples.

Citation: Lingyan Cheng, Ruinan Li, Liming Wu. Exponential convergence in the Wasserstein metric $ W_1 $ for one dimensional diffusions. Discrete and Continuous Dynamical Systems, 2020, 40 (9) : 5131-5148. doi: 10.3934/dcds.2020222
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F. Barthe and C. Roberto, Sobolev inequalities for probability measures on the real line, Studia Math., 159 (2003), 481-497.  doi: 10.4064/sm159-3-9.

[2]

M. F. Chen, From Markov Chains to Nonequilibrium Partcile Systems, World Scientific Publishing Co., Inc., River Edge, NJ, 1992. doi: 10.1142/1389.

[3]

M. F. Chen, Analytic proof of dual variational formula for the first eigenvalue in dimension one, Sci. Sin. A, 42 (1999), 805-815.  doi: 10.1007/BF02884267.

[4]

M. F. Chen, Eigenvalues, Inequalities and Ergodic Theory, Springer-Verlag London, Ltd., London, 2005.

[5]

M. F. Chen and F.-Y. Wang, Estimation of the first eigenvalue of second order elliptic operators, J. Funct. Anal., 131 (1995), 345-363.  doi: 10.1006/jfan.1995.1092.

[6]

M. F. Chen and F.-Y. Wang, Estimation of spectral gap for elliptic operators, Trans. Am. Math. Soc., 349 (1997), 1239-1267.  doi: 10.1090/S0002-9947-97-01812-6.

[7]

L. Y. Cheng and L. M. Wu, Centered Sobolev inequality and exponential convergence in $\Phi$-entropy, Statistics and Probability Letters, 148 (2019), 101-111.  doi: 10.1016/j.spl.2019.01.002.

[8]

H. Djellout, $L^p$-Uniqueness for One-Dimensional Diffusions, Mémoire de D.E.A Université Blaise Pascal, Clermont-Ferrand, 1997.

[9]

H. Djellout and L. M. Wu, Lipschitzian norm estimate of one-dimention Poisson equations and applications, Ann. Inst. Henri Poincaré Probab. Stat., 47 (2011), 450-465.  doi: 10.1214/10-AIHP360.

[10]

A. Eberle, Uniqueness and Non-Uniqueness of Semigroups Generated by Sigular Diffusion Operators, Lecture Notes in Mathmatics, 1718. Springer-Verlag, Berlin, 1999. doi: 10.1007/BFb0103045.

[11]

A. Eberle, Reflection couplings and contraction rates for diffusions, Probability Theory and Related Fields, 166 (2016), 851-886.  doi: 10.1007/s00440-015-0673-1.

[12]

A. EberleA. Guillin and R. Zimmer, Quantitative Harris theorem for diffusions and Mckean-Vlasov processes, Trans. Amer. Math. Soc., 371 (2019), 7135-7137.  doi: 10.1090/tran/7576.

[13]

A. EberleA. Guillin and R. Zimmer, Couplings and quantitative contraction rates for Langevin dynamics, The Annals of Probability, 47 (2019), 1982-2010.  doi: 10.1214/18-AOP1299.

[14]

A. GuillinC. LéonardL. M. Wu and N. Yao, Transportation-information inequalities for Markov processes, Probab. Theory Relat. Fields., 144 (2009), 669-695.  doi: 10.1007/s00440-008-0159-5.

[15]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, Second edition, North-Holland Mathematical Library, 24. North-Holland Publishing Co., Amsterdam, Kodansha, Ltd., Tokyo, 1989.

[16] K. ItȏH. P. Mckean and Jr ., Diffusion Processes and Their Sample Paths, Die Grundlehren der Mathematischen Wissenschaften, Band 125 Academic Press, Inc., Publishers, New York, Springer-Verlag, Berlin-New York, 1965. 
[17]

R. Latala and K. Oleszkiewicz, Between Sobolev and Poincaré, Geometric Aspects of Functional Analysis, Lect. Notes in Math., Springer, Berlin, 1745 (2000), 147-168.  doi: 10.1007/BFb0107213.

[18]

J. Lott and C. Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. (2), 169 (2009), 903-991.  doi: 10.4007/annals.2009.169.903.

[19]

D. J. Luo and J. Wang, Exponential convergence in Wasserstein distance for diffusion processes without uniform dissipativity, Math. Nachr., 289 (2016), 1909-1926. 

[20]

S. P. Meyn and R. L. Tweedie, Markov Chains and Stochastic Stability, Communications and Control Engineering Series, Springer-Verlag London, Ltd., London, 1993. doi: 10.1007/978-1-4471-3267-7.

[21]

M.-K. von Renesse and K.-T. Sturm, Transport inequalities, gradient estimates, entropy, and Ricci curvature, Comm. Pure Appl. Math., 58 (2005), 923-940.  doi: 10.1002/cpa.20060.

[22]

F. Y. Wang, Exponential contraction in Wasserstein distances for diffusion semigroups with negative curvature, preprint, arXiv: 1603.05749.

[23]

L. M. Wu, Essential spectral radius for Markov semigroups. I. Discrete time case, Probab. Theory Raleted Fields, 128 (2004), 255-321.  doi: 10.1007/s00440-003-0304-0.

show all references

References:
[1]

F. Barthe and C. Roberto, Sobolev inequalities for probability measures on the real line, Studia Math., 159 (2003), 481-497.  doi: 10.4064/sm159-3-9.

[2]

M. F. Chen, From Markov Chains to Nonequilibrium Partcile Systems, World Scientific Publishing Co., Inc., River Edge, NJ, 1992. doi: 10.1142/1389.

[3]

M. F. Chen, Analytic proof of dual variational formula for the first eigenvalue in dimension one, Sci. Sin. A, 42 (1999), 805-815.  doi: 10.1007/BF02884267.

[4]

M. F. Chen, Eigenvalues, Inequalities and Ergodic Theory, Springer-Verlag London, Ltd., London, 2005.

[5]

M. F. Chen and F.-Y. Wang, Estimation of the first eigenvalue of second order elliptic operators, J. Funct. Anal., 131 (1995), 345-363.  doi: 10.1006/jfan.1995.1092.

[6]

M. F. Chen and F.-Y. Wang, Estimation of spectral gap for elliptic operators, Trans. Am. Math. Soc., 349 (1997), 1239-1267.  doi: 10.1090/S0002-9947-97-01812-6.

[7]

L. Y. Cheng and L. M. Wu, Centered Sobolev inequality and exponential convergence in $\Phi$-entropy, Statistics and Probability Letters, 148 (2019), 101-111.  doi: 10.1016/j.spl.2019.01.002.

[8]

H. Djellout, $L^p$-Uniqueness for One-Dimensional Diffusions, Mémoire de D.E.A Université Blaise Pascal, Clermont-Ferrand, 1997.

[9]

H. Djellout and L. M. Wu, Lipschitzian norm estimate of one-dimention Poisson equations and applications, Ann. Inst. Henri Poincaré Probab. Stat., 47 (2011), 450-465.  doi: 10.1214/10-AIHP360.

[10]

A. Eberle, Uniqueness and Non-Uniqueness of Semigroups Generated by Sigular Diffusion Operators, Lecture Notes in Mathmatics, 1718. Springer-Verlag, Berlin, 1999. doi: 10.1007/BFb0103045.

[11]

A. Eberle, Reflection couplings and contraction rates for diffusions, Probability Theory and Related Fields, 166 (2016), 851-886.  doi: 10.1007/s00440-015-0673-1.

[12]

A. EberleA. Guillin and R. Zimmer, Quantitative Harris theorem for diffusions and Mckean-Vlasov processes, Trans. Amer. Math. Soc., 371 (2019), 7135-7137.  doi: 10.1090/tran/7576.

[13]

A. EberleA. Guillin and R. Zimmer, Couplings and quantitative contraction rates for Langevin dynamics, The Annals of Probability, 47 (2019), 1982-2010.  doi: 10.1214/18-AOP1299.

[14]

A. GuillinC. LéonardL. M. Wu and N. Yao, Transportation-information inequalities for Markov processes, Probab. Theory Relat. Fields., 144 (2009), 669-695.  doi: 10.1007/s00440-008-0159-5.

[15]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, Second edition, North-Holland Mathematical Library, 24. North-Holland Publishing Co., Amsterdam, Kodansha, Ltd., Tokyo, 1989.

[16] K. ItȏH. P. Mckean and Jr ., Diffusion Processes and Their Sample Paths, Die Grundlehren der Mathematischen Wissenschaften, Band 125 Academic Press, Inc., Publishers, New York, Springer-Verlag, Berlin-New York, 1965. 
[17]

R. Latala and K. Oleszkiewicz, Between Sobolev and Poincaré, Geometric Aspects of Functional Analysis, Lect. Notes in Math., Springer, Berlin, 1745 (2000), 147-168.  doi: 10.1007/BFb0107213.

[18]

J. Lott and C. Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. (2), 169 (2009), 903-991.  doi: 10.4007/annals.2009.169.903.

[19]

D. J. Luo and J. Wang, Exponential convergence in Wasserstein distance for diffusion processes without uniform dissipativity, Math. Nachr., 289 (2016), 1909-1926. 

[20]

S. P. Meyn and R. L. Tweedie, Markov Chains and Stochastic Stability, Communications and Control Engineering Series, Springer-Verlag London, Ltd., London, 1993. doi: 10.1007/978-1-4471-3267-7.

[21]

M.-K. von Renesse and K.-T. Sturm, Transport inequalities, gradient estimates, entropy, and Ricci curvature, Comm. Pure Appl. Math., 58 (2005), 923-940.  doi: 10.1002/cpa.20060.

[22]

F. Y. Wang, Exponential contraction in Wasserstein distances for diffusion semigroups with negative curvature, preprint, arXiv: 1603.05749.

[23]

L. M. Wu, Essential spectral radius for Markov semigroups. I. Discrete time case, Probab. Theory Raleted Fields, 128 (2004), 255-321.  doi: 10.1007/s00440-003-0304-0.

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