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Quasi-stationary distribution for the Langevin process in cylindrical domains. II: Overdamped limit. (English) Zbl 1491.35017

Summary: Consider the Langevin process, described by a vector (positions and momenta) in \({\mathbb{R}^d}\times{\mathbb{R}^d} \). Let \(\mathcal{O}\) be a \(({\mathcal{C}^2}\) open bounded and connected set of \({\mathbb{R}^d} \). Recent works showed the existence of a unique quasi-stationary distribution (QSD) of the Langevin process on the domain \(D:=\mathcal{O}\times{\mathbb{R}^d}\). In this article, we study the overdamped limit of this QSD, i.e. when the friction coefficient goes to infinity. In particular, we show that the marginal law in position of the overdamped limit is the QSD of the overdamped Langevin process on the domain \(\mathcal{O}\). For Part I, see [T. Lelièvre, Stochastic Processes Appl. 144, 173–201 (2022; Zbl 1481.35294)].

MSC:

35B25 Singular perturbations in context of PDEs
35R60 PDEs with randomness, stochastic partial differential equations
47B07 Linear operators defined by compactness properties
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics

Citations:

Zbl 1481.35294
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References:

[1] N. Champagnat, K. A. Coulibaly-Pasquier, and D. Villemonais. Criteria for exponential convergence to quasi-stationary distributions and applications to multi-dimensional diffusions. In Séminaire de Probabilités XLIX, volume 2215 of Lecture Notes in Math., pages 165-182. Springer, Cham, 2018. · Zbl 1452.60050
[2] N. Champagnat and D. Villemonais. General criteria for the study of quasi-stationarity. arXiv e-prints, page 1712.08092, Dec 2017. · Zbl 1361.60067
[3] P. Collet, S. Martínez, and J. San Martín. Quasi-stationary distributions. Probability and its Applications (New York). Springer, Heidelberg, 2013. Markov chains, diffusions and dynamical systems. · Zbl 1261.60002
[4] M. Freidlin. Some remarks on the Smoluchowski-Kramers approximation. Journal of Statistical Physics, 117(3-4):617-634, 2004. · Zbl 1113.82055
[5] A. Friedman. Stochastic differential equations and applications. Vol. 1. Academic Press, New York-London, 1975. Probability and Mathematical Statistics, Vol. 28. · Zbl 0323.60056
[6] A. Friedman. Stochastic differential equations and applications. Vol. 2. Academic Press, New York-London, 1976. Probability and Mathematical Statistics, Vol. 28. · Zbl 0323.60057
[7] G. L. Gong, M. P. Qian, and Z. X. Zhao. Killed diffusions and their conditioning. Probab. Theory Related Fields, 80:151-167, 1988. · Zbl 0631.60073
[8] S. E. Graversen and G. Peskir. Maximal inequalities for the Ornstein-Uhlenbeck process. Proc. Amer. Math. Soc., 128(10):3035-3041, 2000. · Zbl 0954.60062
[9] A. Guillin, B. Nectoux, and L. Wu. Quasi-stationary distribution for strongly Feller Markov processes by Lyapunov functions and applications to hypoelliptic Hamiltonian systems. https://hal.archives-ouvertes.fr/hal-03068461/, 2020.
[10] I. Karatzas and S. E. Shreve. Brownian motion. In Brownian Motion and Stochastic Calculus, pages 47-127. Springer, 1998.
[11] R. Knobloch and L. Partzsch. Uniform conditional ergodicity and intrinsic ultracontractivity. Potential Anal., 33(2):107-136, 2010. · Zbl 1197.47057
[12] H. A. Kramers. Brownian motion in a field of force and the diffusion model of chemical reactions. Physica, 7(4):284-304, 1940. · Zbl 0061.46405
[13] C. Le Bris, T. Lelièvre, M. Luskin, and D. Perez. A mathematical formalization of the parallel replica dynamics. Monte Carlo Methods Appl., 18(2):119-146, 2012. · Zbl 1243.82045
[14] T. Lelièvre, M. Rousset, and G. Stoltz. Free energy computations. Imperial College Press, London, 2010. A mathematical perspective. · Zbl 1227.82002
[15] P. Monmarche, and M. Ramil. Overdamped limit at stationarity for non-equilibrium Langevin diffusions. Electronic Communications in Probability, 27:1-8, 2022. · Zbl 1487.60146
[16] T. Lelièvre and G. Stoltz. Partial differential equations and stochastic methods in molecular dynamics. Acta Numer., 25:681-880, 2016. · Zbl 1348.82065
[17] T. Lelièvre, M. Ramil, and J. Reygner. A probabilistic study of the kinetic Fokker-Planck equation in cylindrical domains. arXiv e-prints, page 2010.10157, Jan 2021. · Zbl 1489.82060
[18] T. Lelièvre, M. Ramil, and J. Reygner. Quasi-stationary distribution for the Langevin process in cylindrical domains, part I: Existence, uniqueness and long time convergence. arXiv e-prints, page 2101.11999, Jan 2021. · Zbl 1481.35294
[19] S. Méléard and D. Villemonais. Quasi-stationary distributions and population processes. Probab. Surv., 9:340-410, 2012. · Zbl 1261.92056
[20] M. Ramil. Processus cinétiques dans des domaines à bord et quasi-stationnarité. PhD thesis, Ecole des Ponts ParisTech, 2020.
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