Invariant measure of duplicated diffusions and application to Richardson–Romberg extrapolation

Lemaire, Vincent; Pagès, Gilles; Panloup, Fabien

Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 4, pp. 1562-1596.

Initialement motivés par des problèmes numériques d’analyse en temps long d’une diffusion ergodique, nous abordons ici la question suivante : si une diffusion Brownienne ergodique a une unique probabilité invariante, quelles sont les probabilités invariantes associées à sa dupliquée, i.e. au système formé par deux copies de la diffusion initiale. Nous nous focalisons notamment sur le cas où ces deux copies sont dirigées par le même mouvement Brownien ($2$-point motion). Sous cette hypothèse, nous montrons que l’unicité de la probabilité invariante relative à la diffusion dupliquée (confluence faible) est essentiellement toujours vraie en dimension $1$. En dimension supérieure, après avoir exhibé un contre-exemple, nous proposons une série de critères de confluence faible (de type intégral) mais aussi de confluence trajectorielle presque sûre, lisibles sur les coefficients de la diffusion au travers d’un exposant de Lyapounov non-infinitésimal. Ces critères permettent en particulier de traiter des cas non triviaux comme certaines classes de systèmes-gradients à potentiel (sur-quadratique) non convexe ou, à l’inverse, des systèmes pour lesquels la confluence est induite par le coefficient de diffusion. Nous montrons enfin que la propriété de confluence faible est associée à un problème de transport optimal. Dans un second temps, nous appliquons nos résultats pour optimiser la mise en œuvre de l’extrapolation Richardson–Romberg pour l’approximation par simulation de mesures invariantes.

With a view to numerical applications we address the following question: given an ergodic Brownian diffusion with a unique invariant distribution, what are the invariant distributions of the duplicated system consisting of two trajectories? We mainly focus on the interesting case where the two trajectories are driven by the same Brownian path. Under this assumption, we first show that uniqueness of the invariant distribution (weak confluence) of the duplicated system is essentially always true in the one-dimensional case. In the multidimensional case, we begin by exhibiting explicit counter-examples. Then, we provide a series of weak confluence criterions (of integral type) and also of a.s. pathwise confluence, depending on the drift and diffusion coefficients through a non-infinitesimal Lyapunov exponent. As examples, we apply our criterions to some non-trivially confluent settings such as classes of gradient systems with non-convex potentials or diffusions where the confluence is generated by the diffusive component. We finally establish that the weak confluence property is connected with an optimal transport problem. As a main application, we apply our results to the optimization of the Richardson–Romberg extrapolation for the numerical approximation of the invariant measure of the initial ergodic Brownian diffusion.

Mots clés : invariant measure, ergodic diffusion, two-point motion, Lyapunov exponent, asymptotic flatness, confluence, gradient system, central limit theorem, Euler scheme, Richardson–Romberg extrapolation, hypoellipticity, optimal transport

[1] G. K. Basak and R. N. Bhattacharya. Stability in distribution for a class of singular diffusions. Ann. Probab. 20 (1) (1992) 312–321. | MR | Zbl

[2] P. H. Baxendale. Statistical equilibrium and two-point motion for a stochastic flow of diffeomorphisms. In Spatial Stochastic Processes 189–218. Progress in Probability 19. Birkhäuser, Boston, MA, 1991. | MR | Zbl

[3] P. H. Baxendale and D. W. Stroock. Large deviations and stochastic flows of diffeomorphisms. Probab. Theory Related Fields 80 (2) (1988) 169–215. | MR | Zbl

[4] R. N. Bhattacharya. On the functional central limit theorem and the law of the iterated logarithm for Markov processes. Z. Wahrsch. Verw. Gebiete 60 (2) (1982) 185–201. | MR | Zbl

[5] A. Carverhill. Flows of stochastic dynamical systems: Ergodic theory. Stochastics 14 (4) (1985) 273–317. | MR | Zbl

[6] P. Cattiaux. Stochastic calculus and degenerate boundary value problems. Ann. Inst. Fourier (Grenoble) 42 (3) (1992) 541–624. | Numdam | MR | Zbl

[7] M. F. Chen and S. F. Li. Coupling methods for multidimensional diffusion processes. Ann. Probab. 17 (1) (1989) 151–177. | MR | Zbl

[8] G. Da Prato and J. Zabczyk. Ergodicity for Infinite-Dimensional Systems. London Mathematical Society Lecture Note Series 229. Cambridge Univ. Press, Cambridge, 1996. | DOI | MR | Zbl

[9] D. Dolgopyat, V. Kaloshin and L. Koralov. Sample path properties of the stochastic flows. Ann. Probab. 32 (1A) (2004) 1–27. | MR | Zbl

[10] T. E. Harris. Brownian motions on the homeomorphisms of the plane. Ann. Probab. 9 (2) (1981) 232–254. | MR | Zbl

[11] R. Z. Has’Minskiĭ. Stochastic Stability of Differential Equations. Monographs and Textbooks on Mechanics of Solids and Fluids: Mechanics and Analysis 7. Sijthoff & Noordhoff, Alphen aan den Rijn, 1980. | MR | Zbl

[12] L. Hörmander. Hypoelliptic second order differential equations. Acta Math. 119 (1967) 147–171. | DOI | MR | Zbl

[13] L. Hörmander. The Analysis of Linear Partial Differential Operators 1–4. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 275. Springer, Berlin, 1985. | MR | Zbl

[14] N. Ikeda and S. Watanabe. A comparison theorem for solutions of stochastic differential equations and its applications. Osaka J. Math. 14 (3) (1977) 619–633. | MR | Zbl

[15] S. Karlin and H. M. Taylor. A Second Course in Stochastic Processes. Academic Press, New York, 1981. | MR | Zbl

[16] P. E. Kloeden and E. Platen. Numerical Solution of Stochastic Differential Equations. Applications of Mathematics (New York) 23. Springer, Berlin, 1992. | DOI | MR | Zbl

[17] H. Kunita. Stochastic Flows and Stochastic Differential Equations. Cambridge Studies in Advanced Mathematics 24. Cambridge Univ. Press, Cambridge, 1990. | MR | Zbl

[18] D. Lamberton and G. Pagès. Recursive computation of the invariant distribution of a diffusion. Bernoulli 8 (3) (2002) 367–405. | MR | Zbl

[19] 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. | MR | Zbl

[20] V. Lemaire. Estimation récursive de la mesure invariante d’un processus de diffusion. Thèse de doctorat, Univ. Marne-la-Vallée, 2005.

[21] V. Lemaire, G. Pagès and F. Panloup. Invariant distribution of duplicated diffusions and application to Richardson–Romberg extrapolation, 2014. Extended version available at arXiv:1302.1651. | Numdam | MR | Zbl

[22] G. Pagès. Multi-step Richardson–Romberg extrapolation: Remarks on variance control and complexity. Monte Carlo Methods Appl. 13 (1) (2007) 37–70. | MR | Zbl

[23] F. Panloup. Approximation récursive du régime stationnaire d’une Équation Différentielle Stochastique avec sauts. Thèse de doctorat, Univ. Pierre et Marie Curie, 2006.

[24] E. Pardoux and A. Yu. Veretennikov. On the Poisson equation and diffusion approximation. I. Ann. Probab. 29 (3) (2001) 1061–1085. | MR | Zbl

[25] D. Talay and L. Tubaro. Expansion of the global error for numerical schemes solving stochastic differential equations. Stoch. Anal. Appl. 8 (4) (1990) 483–509. | MR | Zbl

[26] O. M. Tearne. Collapse of attractors for ODEs under small random perturbations. Probab. Theory Related Fields 141 (1–2) (2008) 1–18. | MR | Zbl

[27] C. Villani. Optimal Transport: Old and New. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 338. Springer, Berlin, 2009. | MR | Zbl