We consider an ordinary differential equation with a unique hyperbolic attractor at the origin, to which we add a small random perturbation. It is known that under general conditions, the solution of this stochastic differential equation converges exponentially fast to an equilibrium distribution. We show that the convergence occurs abruptly: in a time window of small size compared to the natural time scale of the process, the distance to equilibrium drops from its maximal possible value to near zero, and only after this time window the convergence is exponentially fast. This is what is known as the cut-off phenomenon in the context of Markov chains of increasing complexity. In addition, we are able to give general conditions to decide whether the distance to equilibrium converges in this time window to a universal function, a fact known as profile cut-off.
References
[1] Aldous, D. and Diaconis, P. (1986). Shuffling cards and stopping times. Amer. Math. Monthly 93 333–348. 0603.60006 10.1080/00029890.1986.11971821[1] Aldous, D. and Diaconis, P. (1986). Shuffling cards and stopping times. Amer. Math. Monthly 93 333–348. 0603.60006 10.1080/00029890.1986.11971821
[2] Aldous, D. and Diaconis, P. (1987). Strong uniform times and finite random walks. Adv. in Appl. Math. 8 69–97. 0631.60065 10.1016/0196-8858(87)90006-6[2] Aldous, D. and Diaconis, P. (1987). Strong uniform times and finite random walks. Adv. in Appl. Math. 8 69–97. 0631.60065 10.1016/0196-8858(87)90006-6
[3] Arnold, A., Carrillo, J. A. and Manzini, C. (2010). Refined long-time asymptotics for some polymeric fluid flow models. Commun. Math. Sci. 8 763–782. 1213.35094 10.4310/CMS.2010.v8.n3.a8 euclid.cms/1282747139[3] Arnold, A., Carrillo, J. A. and Manzini, C. (2010). Refined long-time asymptotics for some polymeric fluid flow models. Commun. Math. Sci. 8 763–782. 1213.35094 10.4310/CMS.2010.v8.n3.a8 euclid.cms/1282747139
[4] Arnold, L. (2003). Linear and nonlinear diffusion approximation of the slow motion in systems with two time scales. In IUTAM Symposium on Nonlinear Stochastic Dynamics. Solid Mech. Appl. 110 5–18. Kluwer Academic, Dordrecht. 1079.34046[4] Arnold, L. (2003). Linear and nonlinear diffusion approximation of the slow motion in systems with two time scales. In IUTAM Symposium on Nonlinear Stochastic Dynamics. Solid Mech. Appl. 110 5–18. Kluwer Academic, Dordrecht. 1079.34046
[5] Athreya, K. B. and Hwang, C.-R. (2010). Gibbs measures asymptotics. Sankhya A 72 191–207. 1209.60004 10.1007/s13171-010-0006-5[5] Athreya, K. B. and Hwang, C.-R. (2010). Gibbs measures asymptotics. Sankhya A 72 191–207. 1209.60004 10.1007/s13171-010-0006-5
[7] Bakhtin, Y. (2010). Small noise limit for diffusions near heteroclinic networks. Dyn. Syst. 25 413–431. 1281.34105 10.1080/14689367.2010.482520[7] Bakhtin, Y. (2010). Small noise limit for diffusions near heteroclinic networks. Dyn. Syst. 25 413–431. 1281.34105 10.1080/14689367.2010.482520
[8] Bakhtin, Y. (2011). Noisy heteroclinic networks. Probab. Theory Related Fields 150 1–42. 1231.34100 10.1007/s00440-010-0264-0[8] Bakhtin, Y. (2011). Noisy heteroclinic networks. Probab. Theory Related Fields 150 1–42. 1231.34100 10.1007/s00440-010-0264-0
[9] Barrera, G. (2018). Abrupt convergence for a family of Ornstein–Uhlenbeck processes. Braz. J. Probab. Stat. 32 188–199. 1404.60109 10.1214/16-BJPS337 euclid.bjps/1520046140[9] Barrera, G. (2018). Abrupt convergence for a family of Ornstein–Uhlenbeck processes. Braz. J. Probab. Stat. 32 188–199. 1404.60109 10.1214/16-BJPS337 euclid.bjps/1520046140
[10] Barrera, G. and Jara, M. (2016). Abrupt convergence for stochastic small perturbations of one dimensional dynamical systems. J. Stat. Phys. 163 113–138. 1364.37116 10.1007/s10955-016-1468-1[10] Barrera, G. and Jara, M. (2016). Abrupt convergence for stochastic small perturbations of one dimensional dynamical systems. J. Stat. Phys. 163 113–138. 1364.37116 10.1007/s10955-016-1468-1
[11] Barrera, J., Bertoncini, O. and Fernández, R. (2009). Abrupt convergence and escape behavior for birth and death chains. J. Stat. Phys. 137 595–623. 1195.60115 10.1007/s10955-009-9861-7[11] Barrera, J., Bertoncini, O. and Fernández, R. (2009). Abrupt convergence and escape behavior for birth and death chains. J. Stat. Phys. 137 595–623. 1195.60115 10.1007/s10955-009-9861-7
[12] Barrera, J. and Ycart, B. (2014). Bounds for left and right window cutoffs. ALEA Lat. Am. J. Probab. Math. Stat. 11 445–458. 1346.60040[12] Barrera, J. and Ycart, B. (2014). Bounds for left and right window cutoffs. ALEA Lat. Am. J. Probab. Math. Stat. 11 445–458. 1346.60040
[13] Bogachev, V. I., Röckner, M. and Shaposhnikov, S. V. (2016). Distances between transition probabilities of diffusions and applications to nonlinear Fokker–Planck–Kolmogorov equations. J. Funct. Anal. 271 1262–1300. 1345.35051 10.1016/j.jfa.2016.05.016[13] Bogachev, V. I., Röckner, M. and Shaposhnikov, S. V. (2016). Distances between transition probabilities of diffusions and applications to nonlinear Fokker–Planck–Kolmogorov equations. J. Funct. Anal. 271 1262–1300. 1345.35051 10.1016/j.jfa.2016.05.016
[14] Bouchet, F. and Reygner, J. (2016). Generalisation of the Eyring–Kramers transition rate formula to irreversible diffusion processes. Ann. Henri Poincaré 17 3499–3532. 1375.82081 10.1007/s00023-016-0507-4[14] Bouchet, F. and Reygner, J. (2016). Generalisation of the Eyring–Kramers transition rate formula to irreversible diffusion processes. Ann. Henri Poincaré 17 3499–3532. 1375.82081 10.1007/s00023-016-0507-4
[15] Bovier, A., Eckhoff, M., Gayrard, V. and Klein, M. (2004). Metastability in reversible diffusion processes. I. Sharp asymptotics for capacities and exit times. J. Eur. Math. Soc. (JEMS) 6 399–424. MR2094397 1076.82045 10.4171/JEMS/14[15] Bovier, A., Eckhoff, M., Gayrard, V. and Klein, M. (2004). Metastability in reversible diffusion processes. I. Sharp asymptotics for capacities and exit times. J. Eur. Math. Soc. (JEMS) 6 399–424. MR2094397 1076.82045 10.4171/JEMS/14
[16] Bovier, A., Gayrard, V. and Klein, M. (2005). Metastability in reversible diffusion processes. II. Precise asymptotics for small eigenvalues. J. Eur. Math. Soc. (JEMS) 7 69–99. 1105.82025 10.4171/JEMS/22[16] Bovier, A., Gayrard, V. and Klein, M. (2005). Metastability in reversible diffusion processes. II. Precise asymptotics for small eigenvalues. J. Eur. Math. Soc. (JEMS) 7 69–99. 1105.82025 10.4171/JEMS/22
[17] Chen, G.-Y. and Saloff-Coste, L. (2008). The cutoff phenomenon for ergodic Markov processes. Electron. J. Probab. 13 26–78. 1190.60007 10.1214/EJP.v13-474[17] Chen, G.-Y. and Saloff-Coste, L. (2008). The cutoff phenomenon for ergodic Markov processes. Electron. J. Probab. 13 26–78. 1190.60007 10.1214/EJP.v13-474
[18] Day, M. (1982). Exponential leveling for stochastically perturbed dynamical systems. SIAM J. Math. Anal. 13 532–540. 0513.60077 10.1137/0513035[18] Day, M. (1982). Exponential leveling for stochastically perturbed dynamical systems. SIAM J. Math. Anal. 13 532–540. 0513.60077 10.1137/0513035
[19] Day, M. V. (1983). On the exponential exit law in the small parameter exit problem. Stochastics 8 297–323. 0504.60032 10.1080/17442508308833244[19] Day, M. V. (1983). On the exponential exit law in the small parameter exit problem. Stochastics 8 297–323. 0504.60032 10.1080/17442508308833244
[20] Diaconis, P. (1996). The cutoff phenomenon in finite Markov chains. Proc. Natl. Acad. Sci. USA 93 1659–1664. 0849.60070 10.1073/pnas.93.4.1659[20] Diaconis, P. (1996). The cutoff phenomenon in finite Markov chains. Proc. Natl. Acad. Sci. USA 93 1659–1664. 0849.60070 10.1073/pnas.93.4.1659
[21] Eberle, A. and Zimmer, R. Sticky couplings of multidimensional diffusions with different drifts. Ann. Inst. Henri Poincaré B, Calc. Probab. Stat.. To appear. (N.S.) Available at arXiv:1612.06125. 1612.06125 1434.60213 10.1214/18-AIHP951 euclid.aihp/1573203632[21] Eberle, A. and Zimmer, R. Sticky couplings of multidimensional diffusions with different drifts. Ann. Inst. Henri Poincaré B, Calc. Probab. Stat.. To appear. (N.S.) Available at arXiv:1612.06125. 1612.06125 1434.60213 10.1214/18-AIHP951 euclid.aihp/1573203632
[22] Freidlin, M. and Wentzell, A. (1970). On small random perturbations of dynamical systems. Russian Math. Surveys 25 1–55. 0297.34053 10.1070/RM1970v025n01ABEH001254[22] Freidlin, M. and Wentzell, A. (1970). On small random perturbations of dynamical systems. Russian Math. Surveys 25 1–55. 0297.34053 10.1070/RM1970v025n01ABEH001254
[23] Freidlin, M. and Wentzell, A. (1972). Some problems concerning stability under small random perturbations. Theory Probab. Appl. 17 269–283. 0268.93032[23] Freidlin, M. and Wentzell, A. (1972). Some problems concerning stability under small random perturbations. Theory Probab. Appl. 17 269–283. 0268.93032
[24] Freidlin, M. I. and Wentzell, A. D. (2012). Random Perturbations of Dynamical Systems, 3rd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 260. Springer, Heidelberg. Translated from the 1979 Russian original by Joseph Szücs. 0922.60006[24] Freidlin, M. I. and Wentzell, A. D. (2012). Random Perturbations of Dynamical Systems, 3rd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 260. Springer, Heidelberg. Translated from the 1979 Russian original by Joseph Szücs. 0922.60006
[25] Galves, A., Olivieri, E. and Vares, M. E. (1987). Metastability for a class of dynamical systems subject to small random perturbations. Ann. Probab. 15 1288–1305. 0709.60058 10.1214/aop/1176991977 euclid.aop/1176991977[25] Galves, A., Olivieri, E. and Vares, M. E. (1987). Metastability for a class of dynamical systems subject to small random perturbations. Ann. Probab. 15 1288–1305. 0709.60058 10.1214/aop/1176991977 euclid.aop/1176991977
[26] Hartman, P. (1960). On local homeomorphisms of Euclidean spaces. Bol. Soc. Mat. Mexicana (2) 5 220–241. 0127.30202[26] Hartman, P. (1960). On local homeomorphisms of Euclidean spaces. Bol. Soc. Mat. Mexicana (2) 5 220–241. 0127.30202
[27] Hwang, C.-R. (1980). Laplace’s method revisited: Weak convergence of probability measures. Ann. Probab. 8 1177–1182. 0452.60007 10.1214/aop/1176994579 euclid.aop/1176994579[27] Hwang, C.-R. (1980). Laplace’s method revisited: Weak convergence of probability measures. Ann. Probab. 8 1177–1182. 0452.60007 10.1214/aop/1176994579 euclid.aop/1176994579
[28] Hwang, C.-R., Hwang-Ma, S.-Y. and Sheu, S. J. (1993). Accelerating Gaussian diffusions. Ann. Appl. Probab. 3 897–913. 0780.60074 10.1214/aoap/1177005371 euclid.aoap/1177005371[28] Hwang, C.-R., Hwang-Ma, S.-Y. and Sheu, S. J. (1993). Accelerating Gaussian diffusions. Ann. Appl. Probab. 3 897–913. 0780.60074 10.1214/aoap/1177005371 euclid.aoap/1177005371
[29] Jacquot, S. (1992). Comportement asymptotique de la seconde valeur propre des processus de Kolmogorov. J. Multivariate Anal. 40 335–347. 0749.60026 10.1016/0047-259X(92)90030-J[29] Jacquot, S. (1992). Comportement asymptotique de la seconde valeur propre des processus de Kolmogorov. J. Multivariate Anal. 40 335–347. 0749.60026 10.1016/0047-259X(92)90030-J
[30] Jourdain, B., Le Bris, C., Lelièvre, T. and Otto, F. (2006). Long-time asymptotics of a multiscale model for polymeric fluid flows. Arch. Ration. Mech. Anal. 181 97–148. 1089.76006 10.1007/s00205-005-0411-4[30] Jourdain, B., Le Bris, C., Lelièvre, T. and Otto, F. (2006). Long-time asymptotics of a multiscale model for polymeric fluid flows. Arch. Ration. Mech. Anal. 181 97–148. 1089.76006 10.1007/s00205-005-0411-4
[31] Kabanov, Yu. M., Liptser, R. Sh. and Shiryaev, A. N. (1986). On the variation distance for probability measures defined on a filtered space. Probab. Theory Related Fields 71 19–35. 0554.60006 10.1007/BF00366270[31] Kabanov, Yu. M., Liptser, R. Sh. and Shiryaev, A. N. (1986). On the variation distance for probability measures defined on a filtered space. Probab. Theory Related Fields 71 19–35. 0554.60006 10.1007/BF00366270
[32] Kipnis, C. and Newman, C. M. (1985). The metastable behavior of infrequently observed, weakly random, one-dimensional diffusion processes. SIAM J. Appl. Math. 45 972–982. 0592.60063 10.1137/0145059[32] Kipnis, C. and Newman, C. M. (1985). The metastable behavior of infrequently observed, weakly random, one-dimensional diffusion processes. SIAM J. Appl. Math. 45 972–982. 0592.60063 10.1137/0145059
[33] Kulik, A. (2018). Ergodic Behavior of Markov Processes: With Applications to Limit Theorems. De Gruyter Studies in Mathematics 67. de Gruyter, Berlin. 1386.60006[33] Kulik, A. (2018). Ergodic Behavior of Markov Processes: With Applications to Limit Theorems. De Gruyter Studies in Mathematics 67. de Gruyter, Berlin. 1386.60006
[34] Lachaud, B. (2005). Cut-off and hitting times of a sample of Ornstein–Uhlenbeck processes and its average. J. Appl. Probab. 42 1069–1080. 1092.60031 10.1239/jap/1134587817 euclid.jap/1134587817[34] Lachaud, B. (2005). Cut-off and hitting times of a sample of Ornstein–Uhlenbeck processes and its average. J. Appl. Probab. 42 1069–1080. 1092.60031 10.1239/jap/1134587817 euclid.jap/1134587817
[35] Lancaster, P. and Tismenetsky, M. (1985). The Theory of Matrices, 2nd ed. Computer Science and Applied Mathematics. Academic Press, Orlando, FL. 0558.15001[35] Lancaster, P. and Tismenetsky, M. (1985). The Theory of Matrices, 2nd ed. Computer Science and Applied Mathematics. Academic Press, Orlando, FL. 0558.15001
[36] Lelièvre, T., Nier, F. and Pavliotis, G. A. (2013). Optimal non-reversible linear drift for the convergence to equilibrium of a diffusion. J. Stat. Phys. 152 237–274. 1276.82042 10.1007/s10955-013-0769-x[36] Lelièvre, T., Nier, F. and Pavliotis, G. A. (2013). Optimal non-reversible linear drift for the convergence to equilibrium of a diffusion. J. Stat. Phys. 152 237–274. 1276.82042 10.1007/s10955-013-0769-x
[37] Levin, D. A., Peres, Y. and Wilmer, E. L. (2009). Markov Chains and Mixing Times. Amer. Math. Soc., Providence, RI. With a chapter by James G. Propp and David B. Wilson. 1160.60001[37] Levin, D. A., Peres, Y. and Wilmer, E. L. (2009). Markov Chains and Mixing Times. Amer. Math. Soc., Providence, RI. With a chapter by James G. Propp and David B. Wilson. 1160.60001
[40] Maier, R. S. and Stein, D. L. (1997). Limiting exit location distributions in the stochastic exit problem. SIAM J. Appl. Math. 57 752–790. 0874.60072 10.1137/S0036139994271753[40] Maier, R. S. and Stein, D. L. (1997). Limiting exit location distributions in the stochastic exit problem. SIAM J. Appl. Math. 57 752–790. 0874.60072 10.1137/S0036139994271753
[42] Martínez, S. and Ycart, B. (2001). Decay rates and cutoff for convergence and hitting times of Markov chains with countably infinite state space. Adv. in Appl. Probab. 33 188–205. 0979.60066 10.1017/S0001867800010697 euclid.aap/999187903[42] Martínez, S. and Ycart, B. (2001). Decay rates and cutoff for convergence and hitting times of Markov chains with countably infinite state space. Adv. in Appl. Probab. 33 188–205. 0979.60066 10.1017/S0001867800010697 euclid.aap/999187903
[43] Matkowsky, B. J. and Schuss, Z. (1977). The exit problem for randomly perturbed dynamical systems. SIAM J. Appl. Math. 33 365–382. 0369.60071 10.1137/0133024[43] Matkowsky, B. J. and Schuss, Z. (1977). The exit problem for randomly perturbed dynamical systems. SIAM J. Appl. Math. 33 365–382. 0369.60071 10.1137/0133024
[44] Olivieri, E. and Vares, M. E. (2005). Large Deviations and Metastability. Encyclopedia of Mathematics and Its Applications 100. Cambridge Univ. Press, Cambridge. MR2123364[44] Olivieri, E. and Vares, M. E. (2005). Large Deviations and Metastability. Encyclopedia of Mathematics and Its Applications 100. Cambridge Univ. Press, Cambridge. MR2123364
[45] Perko, L. (2001). Differential Equations and Dynamical Systems, 3rd ed. Texts in Applied Mathematics 7. Springer, New York. 0973.34001[45] Perko, L. (2001). Differential Equations and Dynamical Systems, 3rd ed. Texts in Applied Mathematics 7. Springer, New York. 0973.34001
[46] Priola, E., Shirikyan, A., Xu, L. and Zabczyk, J. (2012). Exponential ergodicity and regularity for equations with Lévy noise. Stochastic Process. Appl. 122 106–133. 1239.60059 10.1016/j.spa.2011.10.003[46] Priola, E., Shirikyan, A., Xu, L. and Zabczyk, J. (2012). Exponential ergodicity and regularity for equations with Lévy noise. Stochastic Process. Appl. 122 106–133. 1239.60059 10.1016/j.spa.2011.10.003
[47] Saloff-Coste, L. (2004). Random walks on finite groups. In Probability on Discrete Structures. Encyclopaedia Math. Sci. 110 263–346. Springer, Berlin. 1049.60006[47] Saloff-Coste, L. (2004). Random walks on finite groups. In Probability on Discrete Structures. Encyclopaedia Math. Sci. 110 263–346. Springer, Berlin. 1049.60006
[48] Schuss, Z. and Matkowsky, B. J. (1979). The exit problem: A new approach to diffusion across potential barriers. SIAM J. Appl. Math. 36 604–623. 0406.60071 10.1137/0136043[48] Schuss, Z. and Matkowsky, B. J. (1979). The exit problem: A new approach to diffusion across potential barriers. SIAM J. Appl. Math. 36 604–623. 0406.60071 10.1137/0136043
[49] Siegert, W. (2009). Local Lyapunov Exponents: Sublimiting Growth Rates of Linear Random Differential Equations. Lecture Notes in Math. 1963. Springer, Berlin.[49] Siegert, W. (2009). Local Lyapunov Exponents: Sublimiting Growth Rates of Linear Random Differential Equations. Lecture Notes in Math. 1963. Springer, Berlin.
[50] Stroock, D. W. and Varadhan, S. R. S. (2006). Multidimensional Diffusion Processes. Classics in Mathematics. Springer, Berlin. Reprint of the 1997 edition. MR2190038[50] Stroock, D. W. and Varadhan, S. R. S. (2006). Multidimensional Diffusion Processes. Classics in Mathematics. Springer, Berlin. Reprint of the 1997 edition. MR2190038
