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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 |
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.
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. Eberle, A. 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. Eberle, A. 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. Guillin, C. Léonard, L. 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. Itȏ, H. 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. Eberle, A. 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. Eberle, A. 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. Guillin, C. Léonard, L. 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. Itȏ, H. 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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