Long time behaviour and stationary regime of memory gradient diffusions

Sébastien Gadat; Fabien Panloup

Annales de l'I.H.P. Probabilités et statistiques (2014)

  • Volume: 50, Issue: 2, page 564-601
  • ISSN: 0246-0203

Abstract

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In this paper, we are interested in a diffusion process based on a gradient descent. The process is non Markov and has a memory term which is built as a weighted average of the drift term all along the past of the trajectory. For this type of diffusion, we study the long time behaviour of the process in terms of the memory. We exhibit some conditions for the long-time stability of the dynamical system and then provide, when stable, some convergence properties of the occupation measures and of the marginal distribution, to the associated steady regimes. When the memory is too long, we show that in general, the dynamical system has a tendency to explode, and in the particular Gaussian case, we explicitly obtain the rate of divergence.

How to cite

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Gadat, Sébastien, and Panloup, Fabien. "Long time behaviour and stationary regime of memory gradient diffusions." Annales de l'I.H.P. Probabilités et statistiques 50.2 (2014): 564-601. <http://eudml.org/doc/271993>.

@article{Gadat2014,
abstract = {In this paper, we are interested in a diffusion process based on a gradient descent. The process is non Markov and has a memory term which is built as a weighted average of the drift term all along the past of the trajectory. For this type of diffusion, we study the long time behaviour of the process in terms of the memory. We exhibit some conditions for the long-time stability of the dynamical system and then provide, when stable, some convergence properties of the occupation measures and of the marginal distribution, to the associated steady regimes. When the memory is too long, we show that in general, the dynamical system has a tendency to explode, and in the particular Gaussian case, we explicitly obtain the rate of divergence.},
author = {Gadat, Sébastien, Panloup, Fabien},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {stochastic differential equation; memory diffusions; ergodic processes; Lyapunov function; Non-Markovian diffusions; long-time behavior; Lyapunov conditions},
language = {eng},
number = {2},
pages = {564-601},
publisher = {Gauthier-Villars},
title = {Long time behaviour and stationary regime of memory gradient diffusions},
url = {http://eudml.org/doc/271993},
volume = {50},
year = {2014},
}

TY - JOUR
AU - Gadat, Sébastien
AU - Panloup, Fabien
TI - Long time behaviour and stationary regime of memory gradient diffusions
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2014
PB - Gauthier-Villars
VL - 50
IS - 2
SP - 564
EP - 601
AB - In this paper, we are interested in a diffusion process based on a gradient descent. The process is non Markov and has a memory term which is built as a weighted average of the drift term all along the past of the trajectory. For this type of diffusion, we study the long time behaviour of the process in terms of the memory. We exhibit some conditions for the long-time stability of the dynamical system and then provide, when stable, some convergence properties of the occupation measures and of the marginal distribution, to the associated steady regimes. When the memory is too long, we show that in general, the dynamical system has a tendency to explode, and in the particular Gaussian case, we explicitly obtain the rate of divergence.
LA - eng
KW - stochastic differential equation; memory diffusions; ergodic processes; Lyapunov function; Non-Markovian diffusions; long-time behavior; Lyapunov conditions
UR - http://eudml.org/doc/271993
ER -

References

top
  1. [1] F. Alvarez. On the minimizing property of a second order dissipative system in Hilbert spaces. SIAM J. Control Optim. 38(4) (2000) 1102–1119. Zbl0954.34053MR1760062
  2. [2] F. Alvarez, H. Attouch, J. Bolte and P. Redont. A second-order gradient-like dissipative dynamical system with Hessian-driven damping. Application to optimization and mechanics. Journal des Mathématiques Pures et Appliquées 81(8) (2002) 747–779. Zbl1036.34072MR1930878
  3. [3] A. S. Antipin. Minimization of convex functions on convex sets by means of differential equations (in Russian). Differ. Eq. 30(9) (1994) 1365–1375. Zbl0852.49021MR1347800
  4. [4] Y. Bakhtin. Existence and uniqueness of stationary solution of nonlinear stochastic differential equation with memory. Theory Probab. Appl. 47(4) (2002) 684–688. Zbl1054.60062MR2001790
  5. [5] Y. Bakhtin. Lyapunov exponents for stochastic differential equations with infinite memory and application to stochastic Navier–Stokes equations. Discrete Contin. Dyn. Syst. Ser. B 6(4) (2006) 697–709. Zbl1154.34043MR2223903
  6. [6] Y. Bakhtin and J. Mattingly. Stationary solutions of stochastic differential equations with memory and stochastic partial differential equations. Commun. Contemp. Math. 7(5) (2005) 553–582. Zbl1098.34063MR2175090
  7. [7] D. Bakry, P. Cattiaux and A. Guillin. Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincare. J. Funct. Anal. 254(3) (2008) 727–759. Zbl1146.60058MR2381160
  8. [8] V. Bally and A. Kohatsu-Higa. Lower bounds for densities of Asian type stochastic differential equations. J. Funct. Anal. 258(9) (2010) 3134–3164. Zbl1196.60105MR2595738
  9. [9] M. Benaïm and M. W. Hirsch. Asymptotic pseudotrajectories and chain recurrent flows, with applications. J. Dynam. Differ. Eq. 8(1) (1996) 141–176. Zbl0878.58053MR1388167
  10. [10] M. Benaïm, M. Ledoux and O. Raimond. Self-interacting diffusions. Probab. Theory Related Fields 122(1) (2002) 1–41. Zbl1042.60060MR1883716
  11. [11] M. Benaïm and O. Raimond. Self-interacting diffusions III: Symmetric interactions. Ann. Probab. 33(5) (2003) 1716–1759. Zbl1085.60073MR2165577
  12. [12] F. Bolley, A. Guillin and F. Malrieu. Trend to equilibrium and particle approximation for a weakly selfconsistent Vlasov–Fokker–Planck equation. Mathematical Modelling and Numerical Analysis 44(5) (2010) 867–884. Zbl1201.82029MR2731396
  13. [13] A. Cabot. Asymptotics for a gradient system with memory term. Proc. Amer. Math. Soc. 137(9) (2009) 3013–3024. Zbl1177.34065MR2506460
  14. [14] A. Cabot, H. Engler and S. Gadat. On the long time behavior of second order differential equations with asymptotically small dissipation. Trans. Amer. Math. Soc. 361(11) (2009) 5983–6017. Zbl1191.34078MR2529922
  15. [15] A. Cabot, H. Engler and S. Gadat. Second-order differential equations with asymptotically small dissipation and piecewise flat potentials. Electron. J. Differential Equations17 (2009) 33–38. Zbl1171.34323MR2605582
  16. [16] P. Cattiaux and L. Mesnager. Hypoelliptic non-homogeneous diffusions. Probab. Theory Related Fields 123(4) (2002) 453–483. Zbl1009.60058MR1921010
  17. [17] M. Chaleyat-Maurel and D. Michel. Hypoellipticity theorems and conditionnal laws. Z. Wahrsch. verw. Gebiete 65(4) (1984) 573–597. Zbl0524.35028MR736147
  18. [18] S. Chambeu and A. Kurtzmann. Some particular self-interacting diffusions: Ergodic behaviour and almost sure convergence. Bernoulli 17(4) (2011) 1248–1267. Zbl1242.60101MR2854771
  19. [19] D. Coppersmith and P. Diaconis. Random walk with reinforcement. Preprint, 1987. 
  20. [20] J. M. Coron. Control and Nonlinearity. Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2007. Zbl1140.93002MR2302744
  21. [21] M. Cranston and Y. Le Jan. Self-attracting diffusions: Two case studies. Math. Ann. 303(1) (1995) 87–93. Zbl0838.60052MR1348356
  22. [22] G. Da Prato and J. Zabczyk. Ergodicity for Infinite-Dimensional Systems. Mathematical Society Lecture Note Series. Cambridge Univ. Press, London, 1996. Zbl0849.60052MR1417491
  23. [23] F. Delarue and S. Menozzi. Density estimates for a random noise propagating through a chain of differential equations. J. Funct. Anal. 259(6) (2010) 1577–1630. Zbl1223.60037MR2659772
  24. [24] R. Douc, G. Fort and A. Guillin. Subgeometric rates of convergence of f-ergodic strong Markov processes. Stochastic Process. Appl. 119(3) (2009) 897–923. Zbl1163.60034MR2499863
  25. [25] D. Down, S. P. Meyn and R. L. Tweedie. Exponential and uniform ergodicity of Markov processes. Ann. Probab. 23(4) (1995) 1671–1691. Zbl0852.60075MR1379163
  26. [26] R. T. Durrett and L. C. G. Rogers. Asymptotic behavior of Brownian polymers. Probab. Theory Related Fields 92(3) (1992) 337–349. Zbl0767.60080MR1165516
  27. [27] S. N. Ethier and T. G. Kurtz. Markov Processes. Wiley, New York, 1986. Zbl0592.60049MR838085
  28. [28] M. Hairer. On Malliavin’s proof of Hörmander’s theorem. Bull. Sci. Math. 165(6–7) (2011) 650–666. Zbl1242.60085MR2838095
  29. [29] A. Haraux. Systèmes dynamiques dissipatifs et applications. R.M.A. Masson, Paris, 1991. Zbl0726.58001MR1084372
  30. [30] R. Z. Has’minskii. Stochastic Stability of Differential Equations. Sijthoff & Noordhoff, Alphen aan den Rijn, The Nederlands, 1980. Zbl0441.60060MR600653
  31. [31] L. Hörmander. Hypoelliptic second order differential equations. Acta Math. 117(4) (1967) 147–171. Zbl0156.10701MR222474
  32. [32] N. Ikeda and S. Watanabe. Stochastic Differential Equations and Diffusion Processes. North-Holland, Amsterdam, 1981. Zbl0684.60040MR1011252
  33. [33] J. J. Kohn. Pseudodifferential Operator with Applications. Lectures on Degenerate Elliptic Problems. Liguori, Naples, 1977. Zbl0448.35046MR660652
  34. [34] A. Kurtzmann. The ODE method for some self-interacting diffusions on ℝ d . Ann. Inst. Henri Poincaré Probab. Stat.3 (2010) 618–643. Zbl1215.60056MR2682260
  35. [35] D. Lamberton and G. Pagès. Recursive computation of the invariant distribution of a diffusion: The case of a weakly mean reverting drift. Stoch. Dyn. 3(4) (2003) 435–451. Zbl1044.60069MR2030742
  36. [36] R. Pemantle. Vertex-reinforced random walk. Probab. Theory Related Fields1 (1992) 117–136. Zbl0741.60029MR1156453
  37. [37] B. T. Polyak. Introduction to Optimization. Optimization Software, New York, 1987. Zbl0652.49002MR1099605
  38. [38] O. Raimond. Self-attracting diffusions: Case of the constant interaction. Probab. Theory Related Fields 107(2) (1997) 177–196. Zbl0881.60055MR1431218
  39. [39] D. W. Stroock and S. R. S. Varadhan. Diffusion processes. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Vol. III: Probability Theory 361–368. Univ. California Press, Berkeley, 1972. Zbl0255.60055MR397899
  40. [40] F. Treves. Introduction to Pseudodifferential and Fourier Integral Operators. Vol. 1. Plenum Press, New York, 1980. Zbl0453.47027MR597144
  41. [41] C. Villani. Hypocoercivity. Mem. Amer. Math. Soc. 202(950) (2009) iv+141. Zbl1197.35004MR2562709
  42. [42] L. Wu. Large and moderate deviations and exponential convergence for stochastic damping Hamilton systems. Stochastic Process. Appl. 91(2) (2001) 205–238. Zbl1047.60059MR1807683

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