Geodesic
Breaking a chain of interacting Brownian particles: a Gumbel limit theorem
Teoriâ veroâtnostej i ee primeneniâ, Tome 66 (2021) no. 2, pp. 231-260 Cet article a éte moissonné depuis la source Math-Net.Ru

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We investigate the behavior of a finite chain of Brownian particles interacting through a pairwise quadratic potential, with one end of the chain fixed and the other end pulled away at slow speed, in the limit of slow speed and small Brownian noise. We study the instant when the chain “breaks,” that is, the distance between two neighboring particles becomes larger than a certain limit. In the regime where both the pulling and the noise significantly influence the behavior of the chain, we prove weak limit theorems for the break time and the break position.
Keywords: interacting Brownian particles, stochastic differential equations, Ornstein–Uhlenbeck processes. 
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F. Aurzada; V. Betz; M. A. Lifshits. Breaking a chain of interacting Brownian particles: a Gumbel limit theorem. Teoriâ veroâtnostej i ee primeneniâ, Tome 66 (2021) no. 2, pp. 231-260. http://geodesic.mathdoc.fr/item/TVP_2021_66_2_a1/

[1] M. Allman, V. Betz, “Breaking the chain”, Stochastic Process. Appl., 119:8 (2009), 2645–2659 | DOI | MR | Zbl

[2] M. Allman, V. Betz, M. Hairer, “A chain of interacting particles under strain”, Stochastic Process. Appl., 121:9 (2011), 2014–2042 | DOI | MR | Zbl

[3] F. Aurzada, V. Betz, M. Lifshits, Breaking a chain of interacting Brownian particles, 2019, arXiv: 1912.05168

[4] F. Aurzada, V. Betz, M. Lifshits, “Breaking a chain of interacting Brownian particles”, Ann. Appl. Probab. (to appear) https://imstat.org/journals-and-publications/annals-of-applied-probability/annals-of-applied-probability-future-papers

[5] F. Aurzada, V. Betz, M. Lifshits, Universal break law for chains of Brownian particles with nearest neighbour interaction, 2020, arXiv: 2010.07706

[6] A. Bovier, F. den Hollander, Metastability. A potential-theoretic approach, Grundlehren Math. Wiss., 351, Springer, Cham, 2015, xxi+581 pp. | DOI | MR | Zbl

[7] D. S. Donchev, “Exit probability levels of diffusion processes”, Proc. Amer. Math. Soc., 145:5 (2017), 2241–2253 | DOI | MR | Zbl

[8] M. I. Freidlin, A. D. Wentzell, Random perturbations of dynamical systems, Grundlehren Math. Wiss., 260, 2nd ed., Springer-Verlag, New York, 1998, xii+430 pp. | DOI | MR | MR | Zbl | Zbl

[9] V. A. Malyshev, S. A. Muzychka, “Dynamical phase transition in the simplest molecular chain model”, Theoret. and Math. Phys., 179:1 (2014), 490–499 | DOI | DOI | MR | Zbl

[10] V. A. Malyshev, “One-dimensional mechanical networks and crystals”, Mosc. Math. J., 6:2 (2006), 353–358 | DOI | MR | Zbl

[11] S. A. Muzychka, “Mean exit time for a chain of $N=2,3,4$ oscillators”, Moscow Univ. Math. Bull., 68:4 (2013), 206–210 | DOI | MR | Zbl

[12] S. Noschese, L. Pasquini, L. Reichel, “Tridiagonal Toeplitz matrices: properties and novel applications”, Numer. Linear Algebra Appl., 20:2 (2013), 302–326 | DOI | MR | Zbl

[13] A. A. Novikov, “On estimates and the asymptotic behavior of nonexit probabilities of a Wiener process to a moving boundary”, Math. USSR-Sb., 38:4 (1981), 495–505 | DOI | MR | Zbl

[14] J. Pickands, III, “Upcrossing probabilities for stationary Gaussian processes”, Trans. Amer. Math. Soc., 145 (1969), 51–73 | DOI | MR | Zbl

[15] V. I. Piterbarg, Twenty lectures about Gaussian processes, Atlantic Financial Press, London, 2015, xi+167 pp. | Zbl

[16] C. Qualls, H. Watanabe, “Asymptotic properties of Gaussian processes”, Ann. Math. Statist., 43:2 (1972), 580–596 | DOI | MR | Zbl

[17] A. Sain, C. L. Dias, M. Grant, “Rupture of an extended object: a many-body Kramers calculation”, Phys. Rev. E, 74:4 (2006), 046111 | DOI

[18] P. Salminen, “On the first hitting time and the last exit time for a Brownian motion to/from a moving boundary”, Adv. in Appl. Probab., 20:2 (1988), 411–426 | DOI | MR | Zbl