Propagating Lyapunov functions to prove noise-induced stabilization

Avanti Athreya (The Johns Hopkins University)
Tiffany Kolba (Valparaiso University)
Jonathan C Mattingly (Duke University)

Abstract


We investigate an example of noise-induced stabilization in the plane that was also considered in (Gawedzki, Herzog, Wehr 2010) and (Birrell, Herzog, Wehr 2011). We show that despite the deterministic system not being globally stable, the addition of additive noise in the vertical direction leads to a unique invariant probability measure to which the system converges at a uniform, exponential rate. These facts are established primarily through the construction of a Lyapunov function which we generate as the solution to a sequence of Poisson equations. Unlike a number of other works, however, our Lyapunov function is constructed in a systematic way, and we present a meta-algorithm we hope will be applicable to other problems. We conclude by proving positivity properties of the transition density by using Malliavin calculus via some unusually explicit calculations.

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Pages: 1-38

Publication Date: November 2, 2012

DOI: 10.1214/EJP.v17-2410

References

  • Handbook of mathematical functions with formulas, graphs, and mathematical tables. Edited by Milton Abramowitz and Irene A. Stegun. Reprint of the 1972 edition. Dover Publications, Inc., New York, 1992. xiv+1046 pp. ISBN: 0-486-61272-4 MR1225604
  • Arnold, Ludwig; Kleimann, Wolfgang. Qualitative theory of stochastic systems. Probabilistic analysis and related topics, Vol. 3, 1--79, Academic Press, New York, 1983. MR0748853
  • Barlow, M. T.; Nualart, D. Lectures on probability theory and statistics. Lectures from the 25th Saint-Flour Summer School held July 10–26, 1995. Edited by P. Bernard. Lecture Notes in Mathematics, 1690. Springer-Verlag, Berlin, 1998. viii+227 pp. ISBN: 3-540-64620-5 MR1668107
  • Bass, Richard F. Diffusions and elliptic operators. Probability and its Applications (New York). Springer-Verlag, New York, 1998. xiv+232 pp. ISBN: 0-387-98315-5 MR1483890
  • Bell, Denis R. Degenerate stochastic differential equations and hypoellipticity. Pitman Monographs and Surveys in Pure and Applied Mathematics, 79. Longman, Harlow, 1995. xii+114 pp. ISBN: 0-582-24689-X MR1471702
  • Bell, Denis R. Stochastic differential equations and hypoelliptic operators. Real and stochastic analysis, 9--42, Trends Math., Birkhäuser Boston, Boston, MA, 2004. MR2090751
  • Ben Arous, G.; Léandre, R. Décroissance exponentielle du noyau de la chaleur sur la diagonale. I. (French) [Exponential decay of the heat kernel on the diagonal. I] Probab. Theory Related Fields 90 (1991), no. 2, 175--202. MR1128069
  • Ben Arous, G.; Léandre, R. Décroissance exponentielle du noyau de la chaleur sur la diagonale. II. (French) [Exponential decay of the heat kernel on the diagonal. II] Probab. Theory Related Fields 90 (1991), no. 3, 377--402. MR1133372
  • Bender, Carl M.; Orszag, Steven A. Advanced mathematical methods for scientists and engineers. International Series in Pure and Applied Mathematics. McGraw-Hill Book Co., New York, 1978. xiv+593 pp. ISBN: 0-07-004452-X MR0538168
  • Birrell, Jeremiah; Herzog, David P.; Wehr, Jan. The transition from ergodic to explosive behavior in a family of stochastic differential equations. Stochastic Process. Appl. 122 (2012), no. 4, 1519--1539. MR2914761
  • Boďová, Katarína; Doering, Charles R. Noise-induced statistically stable oscillations in a deterministically divergent nonlinear dynamical system. Commun. Math. Sci. 10 (2012), no. 1, 137--157. MR2901305
  • B. Cooke, J. C. Mattingly, S. A. McKinley, and S. C. Schmidler, phGeometric Ergodicity of Two--dimensional Hamiltonian systems with a Lennard--Jones--like Repulsive Potential, ArXiv e-prints (2011).
  • Douc, Randal; Fort, Gersende; Guillin, Arnaud. Subgeometric rates of convergence of $f$-ergodic strong Markov processes. Stochastic Process. Appl. 119 (2009), no. 3, 897--923. MR2499863
  • Dupuis, Paul; Williams, Ruth J. Lyapunov functions for semimartingale reflecting Brownian motions. Ann. Probab. 22 (1994), no. 2, 680--702. MR1288127
  • Fort, Gersende; Meyn, Sean; Moulines, Eric; Priouret, Pierre. The ODE method for stability of skip-free Markov chains with applications to MCMC. Ann. Appl. Probab. 18 (2008), no. 2, 664--707. MR2399709
  • Gawȩdzki, Krzysztof; Herzog, David P.; Wehr, Jan. Ergodic properties of a model for turbulent dispersion of inertial particles. Comm. Math. Phys. 308 (2011), no. 1, 49--80. MR2842970
  • Gitterman, Moshe. The noisy oscillator. The first hundred years, from Einstein until now. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005. xiv+144 pp. ISBN: 981-256-512-4 MR2313138
  • Hairer, Martin; Mattingly, Jonathan C. Yet another look at Harris' ergodic theorem for Markov chains. Seminar on Stochastic Analysis, Random Fields and Applications VI, 109--117, Progr. Probab., 63, Birkhäuser/Springer Basel AG, Basel, 2011. MR2857021
  • Martin Hairer, phP@W course on the convergence of markov processes, http://www.hairer.org/notes/Convergence.pdf, 2010.
  • Hairer, Martin; Mattingly, Jonathan C. Slow energy dissipation in anharmonic oscillator chains. Comm. Pure Appl. Math. 62 (2009), no. 8, 999--1032. MR2531551
  • Harris, T. E. The existence of stationary measures for certain Markov processes. Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954–1955, vol. II, pp. 113--124. University of California Press, Berkeley and Los Angeles, 1956. MR0084889
  • Hasʹminskiĭ, R. Z. Stochastic stability of differential equations. Translated from the Russian by D. Louvish. Monographs and Textbooks on Mechanics of Solids and Fluids: Mechanics and Analysis, 7. Sijthoff & Noordhoff, Alphen aan den Rijn—Germantown, Md., 1980. xvi+344 pp. ISBN: 90-286-0100-7 MR0600653
  • Huang, Jianyi; Kontoyiannis, Ioannis; Meyn, Sean P. The ODE method and spectral theory of Markov operators. Stochastic theory and control (Lawrence, KS, 2001), 205--221, Lecture Notes in Control and Inform. Sci., 280, Springer, Berlin, 2002. MR1931654
  • Mattingly, Jonathan C.; McKinley, Scott A.; Pillai, Natesh S. Geometric ergodicity of a bead–spring pair with stochastic Stokes forcing. Stochastic Process. Appl. 122 (2012), no. 12, 3953--3979. MR2971721
  • Mattingly, J. C.; Stuart, A. M.; Higham, D. J. Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise. Stochastic Process. Appl. 101 (2002), no. 2, 185--232. MR1931266
  • Meyn, S. P.; Tweedie, R. L. Markov chains and stochastic stability. Communications and Control Engineering Series. Springer-Verlag London, Ltd., London, 1993. xvi+ 548 pp. ISBN: 3-540-19832-6 MR1287609
  • Meyn, Sean. Control techniques for complex networks. Cambridge University Press, Cambridge, 2008. xx+562 pp. ISBN: 978-0-521-88441-9 MR2372453
  • Scheutzow, M. Stabilization and destabilization by noise in the plane. Stochastic Anal. Appl. 11 (1993), no. 1, 97--113. MR1202675
  • Veretennikov, A. Yu. On polynomial mixing bounds for stochastic differential equations. Stochastic Process. Appl. 70 (1997), no. 1, 115--127. MR1472961
  • Veretennikov, A. Yu. On polynomial mixing and the rate of convergence for stochastic differential and difference equations. (Russian) Teor. Veroyatnost. i Primenen. 44 (1999), no. 2, 312--327; translation in Theory Probab. Appl. 44 (2000), no. 2, 361--374 MR1751475
  • White, Roscoe B. Asymptotic analysis of differential equations. Revised edition. Imperial College Press, London, 2010. xxiv+405 pp. ISBN: 978-1-84816-608-0; 1-84816-608-7 MR2724458
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