## Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

### Long time behaviour and stationary regime of memory gradient diffusions

#### 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.

#### Résumé

Nous nous intéressons dans ce travail à une diffusion issue d’une descente de gradient, dont le terme de dérive utilise une mémoire sur le passé de la trajectoire. Le processus ainsi introduit est non-Markovien. Nous étudions les propriétés de stabilité et de convergence à l’équilibre des mesures d’occupation des trajectoires. Dans les situations stables, nous donnons des vitesses de convergence à la stationnarité alors que dans les cas où la mémoire possède une longue portée, nous prouvons l’explosion du système dynamique. Nous exhibons enfin des formules précises dans le cas gaussien.

#### Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 50, Number 2 (2014), 564-601.

Dates
First available in Project Euclid: 26 March 2014

https://projecteuclid.org/euclid.aihp/1395856141

Digital Object Identifier
doi:10.1214/12-AIHP536

Mathematical Reviews number (MathSciNet)
MR3189085

Zentralblatt MATH identifier
1299.60092

#### Citation

Gadat, Sébastien; Panloup, Fabien. Long time behaviour and stationary regime of memory gradient diffusions. Ann. Inst. H. Poincaré Probab. Statist. 50 (2014), no. 2, 564--601. doi:10.1214/12-AIHP536. https://projecteuclid.org/euclid.aihp/1395856141

#### References

• [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.
• [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.
• [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.
• [4] Y. Bakhtin. Existence and uniqueness of stationary solution of nonlinear stochastic differential equation with memory. Theory Probab. Appl. 47(4) (2002) 684–688.
• [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.
• [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.
• [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.
• [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.
• [9] M. Benaïm and M. W. Hirsch. Asymptotic pseudotrajectories and chain recurrent flows, with applications. J. Dynam. Differ. Eq. 8(1) (1996) 141–176.
• [10] M. Benaïm, M. Ledoux and O. Raimond. Self-interacting diffusions. Probab. Theory Related Fields 122(1) (2002) 1–41.
• [11] M. Benaïm and O. Raimond. Self-interacting diffusions III: Symmetric interactions. Ann. Probab. 33(5) (2003) 1716–1759.
• [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.
• [13] A. Cabot. Asymptotics for a gradient system with memory term. Proc. Amer. Math. Soc. 137(9) (2009) 3013–3024.
• [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.
• [15] A. Cabot, H. Engler and S. Gadat. Second-order differential equations with asymptotically small dissipation and piecewise flat potentials. Electron. J. Differential Equations 17 (2009) 33–38.
• [16] P. Cattiaux and L. Mesnager. Hypoelliptic non-homogeneous diffusions. Probab. Theory Related Fields 123(4) (2002) 453–483.
• [17] M. Chaleyat-Maurel and D. Michel. Hypoellipticity theorems and conditionnal laws. Z. Wahrsch. verw. Gebiete 65(4) (1984) 573–597.
• [18] S. Chambeu and A. Kurtzmann. Some particular self-interacting diffusions: Ergodic behaviour and almost sure convergence. Bernoulli 17(4) (2011) 1248–1267.
• [19] D. Coppersmith and P. Diaconis. Random walk with reinforcement. Preprint, 1987.
• [20] J. M. Coron. Control and Nonlinearity. Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2007.
• [21] M. Cranston and Y. Le Jan. Self-attracting diffusions: Two case studies. Math. Ann. 303(1) (1995) 87–93.
• [22] G. Da Prato and J. Zabczyk. Ergodicity for Infinite-Dimensional Systems. Mathematical Society Lecture Note Series. Cambridge Univ. Press, London, 1996.
• [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.
• [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.
• [25] D. Down, S. P. Meyn and R. L. Tweedie. Exponential and uniform ergodicity of Markov processes. Ann. Probab. 23(4) (1995) 1671–1691.
• [26] R. T. Durrett and L. C. G. Rogers. Asymptotic behavior of Brownian polymers. Probab. Theory Related Fields 92(3) (1992) 337–349.
• [27] S. N. Ethier and T. G. Kurtz. Markov Processes. Wiley, New York, 1986.
• [28] M. Hairer. On Malliavin’s proof of Hörmander’s theorem. Bull. Sci. Math. 165(6–7) (2011) 650–666.
• [29] A. Haraux. Systèmes dynamiques dissipatifs et applications. R.M.A. Masson, Paris, 1991.
• [30] R. Z. Has’minskii. Stochastic Stability of Differential Equations. Sijthoff & Noordhoff, Alphen aan den Rijn, The Nederlands, 1980.
• [31] L. Hörmander. Hypoelliptic second order differential equations. Acta Math. 117(4) (1967) 147–171.
• [32] N. Ikeda and S. Watanabe. Stochastic Differential Equations and Diffusion Processes. North-Holland, Amsterdam, 1981.
• [33] J. J. Kohn. Pseudodifferential Operator with Applications. Lectures on Degenerate Elliptic Problems. Liguori, Naples, 1977.
• [34] A. Kurtzmann. The ODE method for some self-interacting diffusions on $\mathbb{R}^{d}$. Ann. Inst. Henri Poincaré Probab. Stat. 3 (2010) 618–643.
• [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.
• [36] R. Pemantle. Vertex-reinforced random walk. Probab. Theory Related Fields 1 (1992) 117–136.
• [37] B. T. Polyak. Introduction to Optimization. Optimization Software, New York, 1987.
• [38] O. Raimond. Self-attracting diffusions: Case of the constant interaction. Probab. Theory Related Fields 107(2) (1997) 177–196.
• [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.
• [40] F. Treves. Introduction to Pseudodifferential and Fourier Integral Operators. Vol. 1. Plenum Press, New York, 1980.
• [41] C. Villani. Hypocoercivity. Mem. Amer. Math. Soc. 202(950) (2009) iv+141.
• [42] L. Wu. Large and moderate deviations and exponential convergence for stochastic damping Hamilton systems. Stochastic Process. Appl. 91(2) (2001) 205–238.