## The Annals of Probability

### Couplings and quantitative contraction rates for Langevin dynamics

#### Abstract

We introduce a new probabilistic approach to quantify convergence to equilibrium for (kinetic) Langevin processes. In contrast to previous analytic approaches that focus on the associated kinetic Fokker–Planck equation, our approach is based on a specific combination of reflection and synchronous coupling of two solutions of the Langevin equation. It yields contractions in a particular Wasserstein distance, and it provides rather precise bounds for convergence to equilibrium at the borderline between the overdamped and the underdamped regime. In particular, we are able to recover kinetic behaviour in terms of explicit lower bounds for the contraction rate. For example, for a rescaled double-well potential with local minima at distance $a$, we obtain a lower bound for the contraction rate of order $\Omega(a^{-1})$ provided the friction coefficient is of order $\Theta(a^{-1})$.

#### Article information

Source
Ann. Probab., Volume 47, Number 4 (2019), 1982-2010.

Dates
Revised: June 2018
First available in Project Euclid: 4 July 2019

https://projecteuclid.org/euclid.aop/1562205696

Digital Object Identifier
doi:10.1214/18-AOP1299

Mathematical Reviews number (MathSciNet)
MR3980913

Zentralblatt MATH identifier
07114709

#### Citation

Eberle, Andreas; Guillin, Arnaud; Zimmer, Raphael. Couplings and quantitative contraction rates for Langevin dynamics. Ann. Probab. 47 (2019), no. 4, 1982--2010. doi:10.1214/18-AOP1299. https://projecteuclid.org/euclid.aop/1562205696

#### References

• [1] Bakry, D., Cattiaux, P. and Guillin, A. (2008). Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincaré. J. Funct. Anal. 254 727–759.
• [2] Banerjee, S. and Kendall, W. S. (2016). Coupling the Kolmogorov diffusion: Maximality and efficiency considerations. Adv. in Appl. Probab. 48 15–35.
• [3] Baudoin, F. (2016). Wasserstein contraction properties for hypoelliptic diffusions. Preprint. Available at arXiv:1602.04177.
• [4] Ben Arous, G., Cranston, M. and Kendall, W. S. (1995). Coupling constructions for hypoelliptic diffusions: Two examples. In Stochastic Analysis (Ithaca, NY, 1993). Proc. Sympos. Pure Math. 57 193–212. Amer. Math. Soc., Providence, RI.
• [5] Bolley, F., Guillin, A. and Malrieu, F. (2010). Trend to equilibrium and particle approximation for a weakly selfconsistent Vlasov–Fokker–Planck equation. M2AN Math. Model. Numer. Anal. 44 867–884.
• [6] Bou-Rabee, N. and Sanz-Serna, J. M. (2017). Randomized Hamiltonian Monte Carlo. Ann. Appl. Probab. 27 2159–2194.
• [7] Calogero, S. (2012). Exponential convergence to equilibrium for kinetic Fokker–Planck equations. Comm. Partial Differential Equations 37 1357–1390.
• [8] Chen, M.-F. and Wang, F.-Y. (1997). Estimation of spectral gap for elliptic operators. Trans. Amer. Math. Soc. 349 1239–1267.
• [9] Chen, M. F. and Li, S. F. (1989). Coupling methods for multidimensional diffusion processes. Ann. Probab. 17 151–177.
• [10] Desvillettes, L. and Villani, C. (2001). On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: The linear Fokker–Planck equation. Comm. Pure Appl. Math. 54 1–42.
• [11] Dolbeault, J., Mouhot, C. and Schmeiser, C. (2015). Hypocoercivity for linear kinetic equations conserving mass. Trans. Amer. Math. Soc. 367 3807–3828.
• [12] Duane, S., Kennedy, A. D., Pendleton, B. J. and Roweth, D. (1987). Hybrid Monte Carlo. Phys. Lett. B 195 216–222.
• [13] Eberle, A. (2011). Reflection coupling and Wasserstein contractivity without convexity. C. R. Math. Acad. Sci. Paris 349 1101–1104.
• [14] Eberle, A. (2016). Reflection couplings and contraction rates for diffusions. Probab. Theory Related Fields 166 851–886.
• [15] Eberle, A., Guillin, A. and Zimmer, R. (2019). Quantitative Harris type theorems for diffusions and McKean–Vlasov processes. Trans. Amer. Math. Soc. 371 7135–7173.
• [16] Eberle, A. and Zimmer, R. (2016). Sticky couplings of multidimensional diffusions with different drifts. Preprint. Available at arXiv:1612.06125.
• [17] Eckmann, J.-P. and Hairer, M. (2003). Spectral properties of hypoelliptic operators. Comm. Math. Phys. 235 233–253.
• [18] Einstein, A. (1905). Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen. Ann. Phys. 322 549–560.
• [19] Grothaus, M. and Stilgenbauer, P. (2014). Hypocoercivity for Kolmogorov backward evolution equations and applications. J. Funct. Anal. 267 3515–3556.
• [20] Grothaus, M. and Stilgenbauer, P. (2016). Hilbert space hypocoercivity for the Langevin dynamics revisited. Methods Funct. Anal. Topology 22 152–168.
• [21] Guillin, A. and Monmarché, P. (2016). Optimal linear drift for the speed of convergence of an hypoelliptic diffusion. Electron. Commun. Probab. 21 Paper No. 74, 14.
• [22] Hairer, M. (2002). Exponential mixing properties of stochastic PDEs through asymptotic coupling. Probab. Theory Related Fields 124 345–380.
• [23] Hairer, M. and Mattingly, J. C. (2011). Yet another look at Harris’ ergodic theorem for Markov chains. In Seminar on Stochastic Analysis, Random Fields and Applications VI. Progress in Probability 63 109–117. Birkhäuser/Springer Basel AG, Basel.
• [24] Hairer, M., Mattingly, J. C. and Scheutzow, M. (2011). Asymptotic coupling and a general form of Harris’ theorem with applications to stochastic delay equations. Probab. Theory Related Fields 149 223–259.
• [25] Helffer, B. and Nier, F. (2005). Hypoelliptic Estimates and Spectral Theory for Fokker–Planck Operators and Witten Laplacians. Lecture Notes in Math. 1862. Springer, Berlin.
• [26] Hérau, F. (2006). Hypocoercivity and exponential time decay for the linear inhomogeneous relaxation Boltzmann equation. Asymptot. Anal. 46 349–359.
• [27] Hérau, F. and Nier, F. (2004). Isotropic hypoellipticity and trend to equilibrium for the Fokker–Planck equation with a high-degree potential. Arch. Ration. Mech. Anal. 171 151–218.
• [28] Langevin, P. (1908). Sur la théorie du mouvement brownien. C. R. Math. Acad. Sci. Paris 146 530–533.
• [29] Lelièvre, T., Rousset, M. and Stoltz, G. (2010). Free Energy Computations: A Mathematical Perspective. Imperial College Press, London.
• [30] Lindvall, T. and Rogers, L. C. G. (1986). Coupling of multidimensional diffusions by reflection. Ann. Probab. 14 860–872.
• [31] Mattingly, J. C. (2002). Exponential convergence for the stochastically forced Navier–Stokes equations and other partially dissipative dynamics. Comm. Math. Phys. 230 421–462.
• [32] Mattingly, J. C., Stuart, A. M. and Higham, D. J. (2002). Ergodicity for SDEs and approximations: Locally Lipschitz vector fields and degenerate noise. Stochastic Process. Appl. 101 185–232.
• [33] Mischler, S. and Mouhot, C. (2016). Exponential stability of slowly decaying solutions to the kinetic-Fokker–Planck equation. Arch. Ration. Mech. Anal. 221 677–723.
• [34] Neal, R. M. (2011). MCMC using Hamiltonian dynamics. In Handbook of Markov Chain Monte Carlo. 113–162. CRC Press, Boca Raton, FL.
• [35] Nelson, E. (1967). Dynamical Theories of Brownian Motion, Vol. 2. Princeton Univ. Press, Princeton, NJ.
• [36] Pavliotis, G. A. (2014). Stochastic Processes and Applications: Diffusion Processes, the Fokker-Planck and Langevin Equations. Texts in Applied Mathematics 60. Springer, New York.
• [37] Rey-Bellet, L. and Thomas, L. E. (2002). Exponential convergence to non-equilibrium stationary states in classical statistical mechanics. Comm. Math. Phys. 225 305–329.
• [38] Schuss, Z. (2010). Theory and Applications of Stochastic Processes: An Analytical Approach. Applied Mathematical Sciences 170. Springer, New York.
• [39] Talay, D. (2002). Stochastic Hamiltonian systems: Exponential convergence to the invariant measure, and discretization by the implicit Euler scheme. Markov Process. Related Fields 8 163–198.
• [40] Villani, C. (2007). Hypocoercive diffusion operators. Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 10 257–275.
• [41] Villani, C. (2009). Optimal Transport: Old and New. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 338. Springer, Berlin.
• [42] Villani, C. (2009). Hypocoercivity. Mem. Amer. Math. Soc. 202 iv+141.
• [43] von Smoluchowski, M. (1906). Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen. Ann. Phys. 326 756–780.
• [44] Watanabe, S. (1971). On stochastic differential equations for multi-dimensional diffusion processes with boundary conditions. J. Math. Kyoto Univ. 11 169–180.
• [45] Watanabe, S. (1971). On stochastic differential equations for multi-dimensional diffusion processes with boundary conditions. II. J. Math. Kyoto Univ. 11 545–551.
• [46] Wu, L. (2001). Large and moderate deviations and exponential convergence for stochastic damping Hamiltonian systems. Stochastic Process. Appl. 91 205–238.
• [47] Zimmer, R. (2017). Explicit contraction rates for a class of degenerate and infinite-dimensional diffusions. Stoch. Partial Differ. Equ. Anal. Comput. 5 368–399.
• [48] Zimmer, R. (2017). Couplings and Kantorovich contractions with explicit rates for diffusions. Ph.D. thesis, Univ. Bonn.