## Electronic Journal of Probability

### Hypoelliptic multiscale Langevin diffusions: large deviations, invariant measures and small mass asymptotics

#### Abstract

We consider a general class of hypoelliptic Langevin diffusions and study two related questions. The first question is large deviations for hypoelliptic multiscale diffusions as the noise and the scale separation parameter go to zero. The second question is small mass asymptotics of (a) the invariant measure corresponding to the hypoelliptic Langevin operator and of (b) related hypoelliptic Poisson equations. The invariant measure corresponding to the hypoelliptic problem and appropriate hypoelliptic Poisson equations enter the large deviations rate function due to the multiscale effects. Based on the small mass asymptotics we derive that the large deviations behavior of the multiscale hypoelliptic diffusion is consistent with the large deviations behavior of its overdamped counterpart. Additionally, we rigorously obtain an asymptotic expansion of the solution to relevant hypoelliptic Poisson equations with respect to the mass parameter, characterizing the order of convergence as the mass parameter goes to zero. The proof of convergence of invariant measures is of independent interest, as it involves an improvement of the hypocoercivity result for the kinetic Fokker-Planck equation. We do not restrict attention to gradient drifts and our proof provides explicit information on the dependence of the bounds of interest in terms of the mass parameter.

#### Article information

Source
Electron. J. Probab., Volume 22 (2017), paper no. 55, 38 pp.

Dates
Accepted: 26 May 2017
First available in Project Euclid: 30 June 2017

https://projecteuclid.org/euclid.ejp/1498809677

Digital Object Identifier
doi:10.1214/17-EJP72

Mathematical Reviews number (MathSciNet)
MR3672831

Zentralblatt MATH identifier
1368.60027

#### Citation

Hu, Wenqing; Spiliopoulos, Konstantinos. Hypoelliptic multiscale Langevin diffusions: large deviations, invariant measures and small mass asymptotics. Electron. J. Probab. 22 (2017), paper no. 55, 38 pp. doi:10.1214/17-EJP72. https://projecteuclid.org/euclid.ejp/1498809677

#### References

• [1] Bakery, D., Gentil, I., Ledoux, M.: Analysis and Geometry of Markov diffusion operators, Grundlehren der mathematischen Wissenschaften, Springer, 348, 2014.
• [2] Benabou, G: Homogenization of Ornstein-Uhlenbeck process in random environment, Communications in Mathematical Physics, 266, (2006), 699–714.
• [3] Boué, M., and Dupuis, P.: A variational representation for certain functionals of Brownian motion, Annals of Probability, 26(4), (1998), 1641–1659.
• [4] Chen, Z., and Freidlin, M.: Smoluchowski-Kramers approximation and exit problems, Stochastics and Dynamics, 5(4), (2005), 569–585.
• [5] Dolbeault, J., Mouhot, C., and Schmeiser, C.: Hypocoercivity for linear kinetic equations conserving mass, Transactions of the American Mathematical Society, 367(6), (2015), 3807–3828.
• [6] Dupuis, P: and Ellis, R.S.: A Weak Conergence Approach to the Theory of Large Deviations, John Wiley & Sons, New York, 1997.
• [7] Dupuis, P., and Spiliopoulos, K.: Large deviations for multiscale problems via weak convergence methods, Stochastic Processes and their Applications, 122, (2012), 1947–1987.
• [8] Dupuis, P., Spiliopoulos, K., and Wang, H.: Importance sampling for multiscale diffusions, SIAM Multiscale Modeling and Simulation, 12(1), (2012), 1–27.
• [9] Dupuis, P., Spiliopoulos, K., and Wang, H.: Rare Event Simulation in Rough Energy Landscapes, Winter Simulation Conference, (IEEE, 2011), 504–515.
• [10] Eckmann, J–P., Hairer, M.: Non–equilibrium statistical mechanics of strongly anharmonic chains of oscillators. Communications in Mathematical Physics, 212, (2000), 105–164.
• [11] Einstein, A., Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen, Annalen der Physik, 17(8), (1905), 549–560.
• [12] Freidlin, M.: Some Remarks on the Smoluchowski-Kramers Approximation, Journal of Statistical Physics, 117(314), (2004), 617–634.
• [13] Freidlin, M., and Hu, W.: Smoluchowski-Kramers approximation in the case of variable friction, Journal of Mathematical Sciences, 79(1), (2011), 184–207.
• [14] Hairer, M., and Pavliotis, G.: Periodic Homogenization for Hypoelliptic Diffusions, Journal of Statistical Physics, 117(1/2), (2004), 261–279.
• [15] Hottovy, S., McDaniel, A., Volpe, G. and Wehr, J.: The Smoluchowski-Kramers limit of stochastic differential equations with arbitrary state-dependent friction, Communications in Mathematical Physics, 336(3), (2015), 1259–1283.
• [16] Komorowski, T., Landim, C., and Olla, S.: Fluctuations in Markov Processes: Time Symmetry and Martingale Approximation, Springer, 2012.
• [17] Kosygina, E., Rezakhanlou, F., and Varadhan, S. R. S.: Stochastic Homogenization of Hamilton-Jacobi-Bellman Equations, Communications on Pure and Applied Mathematics, LIX, (2006), 0001–0033.
• [18] Langevin, P.: Sur la théorie du mouvement brownien. C. R. Acad. Sci. (Paris), 146, (1908), 530–533.
• [19] Lifson, S., and Jackson, J.L.: On the self-diffusion of ions in a polyelectrolyte solution, Journal of Chemical Physics, 36, (1962), 2410–2414.
• [20] Lelievre, T., Stoltz, G., and Rousset, M.: Free Energy Computations: A Mathematical Perspective, London College Press, 2010.
• [21] Lions, P.-L., and Souganidis, P.E.: Homogenization of “viscous” Hamilton-Jacobi-Bellman equations in stationary ergodic media, Communications in Partial Differential Equations, 30(3), (2006), 335–375.
• [22] Lunardi, A., On the Ornstein–Uhlenbeck operator in $L^2$ spaces with respect to invariant measure, Transactions of the AMS, 349(1),(1997), 155–169.
• [23] Øksendal, B.: Stochastic differential equations: an introduction with applications, Springer, 5th edition, 2003.
• [24] Papanicolaou, G., and Varadhan, S.R.S.: Ornstein-Uhlebeck process in random potential, Communications in Pure and Applied Mathematics, 38, (1985), 819–834.
• [25] Pavliotis, G.A., and Stuart, A.M.: Periodic homogenization for inertial particles, Physica D, 204, (2005), 161–187.
• [26] Smoluchowski, M., Drei Vortrage über Diffusion Brownsche Bewegung and Koagulation von Kolloidteilchen, Phys. Z., 17, (1916), 557–585.
• [27] Spiliopoulos, K,: Large deviations and importance sampling for systems of slow-fast motion, Applied Mathematics and Optimization, 67, (2013), 123–161.
• [28] Spiliopoulos, K.: Quenched Large Deviations for Multiscale Diffusion Processes in Random Environments, Electronic Journal of Probability, 20(15), (2015), 1–29.
• [29] Spiliopoulos, K.: Rare event simulation for multiscale diffusions in random environments, SIAM Multiscale Modeling and Simulation, 13(4), (2015), 1290–1311.
• [30] C. Villani: Hypocoercivity, Memoirs of the American Mathematical Society, 202(950), 2009.
• [31] Zhikov, V.V., Kozlov, S.M., Oleinik, O.A., and Kha T’en Ngoan, Averaging and G-convergence of differential operators, Russian Math Surveys, 34(5),(1979), 65–133.
• [32] Zwanzig, R., Diffusion in a rough potential, Proc. Natl. Acad. Sci. USA, 85, (1988), 2029–2030.