## Bernoulli

• Bernoulli
• Volume 24, Number 4A (2018), 2461-2498.

### Evolution of the Wasserstein distance between the marginals of two Markov processes

#### Abstract

In this paper, we are interested in the time derivative of the Wasserstein distance between the marginals of two Markov processes. As recalled in the introduction, the Kantorovich duality leads to a natural candidate for this derivative. Up to the sign, it is the sum of the integrals with respect to each of the two marginals of the corresponding generator applied to the corresponding Kantorovich potential. For pure jump processes with bounded intensity of jumps, we prove that the evolution of the Wasserstein distance is actually given by this candidate. In dimension one, we show that this remains true for Piecewise Deterministic Markov Processes. We apply the formula to estimate the exponential decrease rate of the Wasserstein distance between the marginals of two birth and death processes with the same generator in terms of the Wasserstein curvature.

#### Article information

Source
Bernoulli, Volume 24, Number 4A (2018), 2461-2498.

Dates
Revised: December 2016
First available in Project Euclid: 26 March 2018

Permanent link to this document
https://projecteuclid.org/euclid.bj/1522051215

Digital Object Identifier
doi:10.3150/17-BEJ934

Mathematical Reviews number (MathSciNet)
MR3779692

Zentralblatt MATH identifier
06853255

#### Citation

Alfonsi, Aurélien; Corbetta, Jacopo; Jourdain, Benjamin. Evolution of the Wasserstein distance between the marginals of two Markov processes. Bernoulli 24 (2018), no. 4A, 2461--2498. doi:10.3150/17-BEJ934. https://projecteuclid.org/euclid.bj/1522051215

#### References

• [1] Alfonsi, A., Jourdain, B. and Kohatsu-Higa, A. (2014). Pathwise optimal transport bounds between a one-dimensional diffusion and its Euler scheme. Ann. Appl. Probab. 24 1049–1080.
• [2] Alfonsi, A., Jourdain, B. and Kohatsu-Higa, A. (2015). Optimal transport bounds between the time-marginals of a multidimensional diffusion and its Euler scheme. Electron. J. Probab. 20 no. 70, 31.
• [3] Ambrosio, L., Gigli, N. and Savaré, G. (2008). Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd ed. Lectures in Mathematics ETH Zürich. Basel: Birkhäuser.
• [4] Bardet, J.-B., Christen, A., Guillin, A., Malrieu, F. and Zitt, P.-A. (2013). Total variation estimates for the TCP process. Electron. J. Probab. 18 no. 10, 21.
• [5] Bolley, F., Gentil, I. and Guillin, A. (2012). Convergence to equilibrium in Wasserstein distance for Fokker–Planck equations. J. Funct. Anal. 263 2430–2457.
• [6] Bolley, F., Gentil, Y. and Guillin, A. (2013). Uniform convergence to equilibrium for granular media. Arch. Ration. Mech. Anal. 208 429–445.
• [7] Chafaï, D. and Joulin, A. (2013). Intertwining and commutation relations for birth–death processes. Bernoulli 19 1855–1879.
• [8] Eberle, A. (2016). Reflection couplings and contraction rates for diffusions. Probab. Theory Related Fields 166 851–886.
• [9] Ikeda, N. and Watanabe, S. (1989). Stochastic Differential Equations and Diffusion Processes, 2nd ed. North-Holland Mathematical Library 24.
• [10] Jacod, J. and Shiryaev, A.N. (1987). Limit Theorems for Stochastic Processes. Fundamental Principles of Mathematical Sciences 288. Berlin: Springer.
• [11] Joulin, A. (2007). Poisson-type deviation inequalities for curved continuous-time Markov chains. Bernoulli 13 782–798.
• [12] Joulin, A. (2009). A new Poisson-type deviation inequality for Markov jump processes with positive Wasserstein curvature. Bernoulli 15 532–549.
• [13] Lepeltier, J.-P. and Marchal, B. (1976). Problème des martingales et équations différentielles stochastiques associées à un opérateur intégro-différentiel. Ann. Inst. Henri Poincaré B, Probab. Stat. 12 43–103.
• [14] Luo, D. and Wang, J. (2016). Exponential convergence in $L^{p}$-Wasserstein distance for diffusion processes without uniformly dissipative drift. Math. Nachr. 289 1090–1926.
• [15] Luo, D. and Wang, J. (2016). Refined basic couplings and Wasserstein-type distances for SDEs with Lévy noises. Preprint. Available at arXiv:1604.07206.
• [16] Rachev, S.T. and Rüschendorf, L. (1998). Mass Transportation Problems: Volume I: Theory. New York: Springer Science & Business Media.
• [17] Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 293. Berlin: Springer.
• [18] Villani, C. (2008). Optimal Transport: Old and New 338. New York: Springer Science & Business Media.
• [19] von Renesse, M.-K. and Sturm, K.-T. (2005). Transport inequalities, gradient estimates, entropy, and Ricci curvature. Comm. Pure Appl. Math. 58 923–940.
• [20] Wang, F.-Y. (2016). Exponential contraction in Wasserstein distance for diffusion semigroups with negative curvature. Preprint. Availabel at arXiv:1603.05749.
• [21] Wang, J. (2016). $L^{p}$-Wasserstein distance for stochastic differential equations driven by Lévy processes. Bernoulli 22 1598–1616.