The small noise limit of order-based diffusion processes

Benjamin Jourdain (Université Paris-Est ENPC)
Julien Reygner (Sorbonne Universités UPMC & Université Paris-Est ENPC)


In this article, we introduce and study order-based diffusion processes. They are the solutions to multidimensional stochastic differential equations with constant diffusion matrix, proportional to the identity, and drift coefficient depending only on the ordering of the coordinates of the process. These processes describe the evolution of a system of Brownian particles moving on the real line with piecewise constant drifts, and are the natural generalization of the rank-based diffusion processes introduced in stochastic portfolio theory or in the probabilistic interpretation of nonlinear evolution equations. Owing to the discontinuity of the drift coefficient, the corresponding ordinary differential equations are ill-posed. Therefore, the small noise limit of order-based diffusion processes is not covered by the classical Freidlin-Wentzell theory. The description of this limit is the purpose of this article.

We first give a complete analysis of the two-particle case. Despite its apparent simplicity, the small noise limit of such a system already exhibits various behaviours. In particular, depending on the drift coefficient, the particles can either stick into a cluster, the velocity of which is determined by elementary computations, or drift away from each other at constant velocity, in a random ordering. The persistence of randomness in the small noise limit is of the very same nature as in the pioneering works by Veretennikov (Mat. Zametki, 1983) and Bafico and Baldi (Stochastics, 1981) concerning the so-called Peano phenomenon.

In the case of rank-based processes, we use a simple convexity argument to prove that the small noise limit is described by the sticky particle dynamics introduced by Brenier and Grenier (SIAM J. Numer. Anal., 1998), where particles travel at constant velocity between collisions, at which they stick together. In the general case of order-based processes, we give a sufficient condition on the drift for all the particles to aggregate into a single cluster, and compute the velocity of this cluster. Our argument consists in turning the study of the small noise limit into the study of the long time behaviour of a suitably rescaled process, and then exhibiting a Lyapunov functional for this rescaled process.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1-36

Publication Date: March 5, 2014

DOI: 10.1214/EJP.v19-2906


  • Attanasio, Stefano; Flandoli, Franco. Zero-noise solutions of linear transport equations without uniqueness: an example. C. R. Math. Acad. Sci. Paris 347 (2009), no. 13-14, 753--756. MR2543977
  • Bafico, R.; Baldi, P. Small random perturbations of Peano phenomena. Stochastics 6 (1981/82), no. 3-4, 279--292. MR0665404
  • Banner, Adrian D.; Fernholz, Robert; Karatzas, Ioannis. Atlas models of equity markets. Ann. Appl. Probab. 15 (2005), no. 4, 2296--2330. MR2187296
  • Billingsley, Patrick. Convergence of probability measures. Second edition. Wiley Series in Probability and Statistics: Probability and Statistics. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1999. x+277 pp. ISBN: 0-471-19745-9 MR1700749
  • Bossy, Mireille; Talay, Denis. Convergence rate for the approximation of the limit law of weakly interacting particles: application to the Burgers equation. Ann. Appl. Probab. 6 (1996), no. 3, 818--861. MR1410117
  • Brenier, Yann; Grenier, Emmanuel. Sticky particles and scalar conservation laws. SIAM J. Numer. Anal. 35 (1998), no. 6, 2317--2328 (electronic). MR1655848
  • Buckdahn, R.; Ouknine, Y.; Quincampoix, M. On limiting values of stochastic differential equations with small noise intensity tending to zero. Bull. Sci. Math. 133 (2009), no. 3, 229--237. MR2512827
  • F. Delarue, F. Flandoli, and D. Vincenzi, Noise prevents collapse of Vlasov-Poisson point charges, Communications on Pure and Applied Mathematics (2013).
  • A. Dembo, M. Shkolnikov, S.R.S. Varadhan, and O. Zeitouni, Large deviations for diffusions interacting through their ranks, Preprint available at arXiv:1211.5223, 2012.
  • E, Weinan; Vanden-Eijnden, Eric. A note on generalized flows. Phys. D 183 (2003), no. 3-4, 159--174. MR2006631
  • Ethier, Stewart N.; Kurtz, Thomas G. Markov processes. Characterization and convergence. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York, 1986. x+534 pp. ISBN: 0-471-08186-8 MR0838085
  • Fernholz, E. Robert. Stochastic portfolio theory. Applications of Mathematics (New York), 48. Stochastic Modelling and Applied Probability. Springer-Verlag, New York, 2002. xiv+177 pp. ISBN: 0-387-95405-8 MR1894767
  • Fernholz, E. Robert; Ichiba, Tomoyuki; Karatzas, Ioannis; Prokaj, Vilmos. Planar diffusions with rank-based characteristics and perturbed Tanaka equations. Probab. Theory Related Fields 156 (2013), no. 1-2, 343--374. MR3055262
  • E. R. Fernholz and I. Karatzas, Stochastic portfolio theory: A survey, In Handbook of Numerical Analysis. Mathematical Modeling and Numerical Methods in Finance, 2009.
  • Fernholz, Robert; Ichiba, Tomoyuki; Karatzas, Ioannis. A second-order stock market model. Ann. Finance 9 (2013), no. 3, 439--454. MR3082660
  • Fleming, Wendell H.; Soner, H. Mete. Controlled Markov processes and viscosity solutions. Second edition. Stochastic Modelling and Applied Probability, 25. Springer, New York, 2006. xviii+429 pp. ISBN: 978-0387-260457; 0-387-26045-5 MR2179357
  • Freidlin, M. I.; Wentzell, A. D. Random perturbations of dynamical systems. Translated from the 1979 Russian original by Joseph Szücs. Second edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 260. Springer-Verlag, New York, 1998. xii+430 pp. ISBN: 0-387-98362-7 MR1652127
  • Gradinaru, Mihai; Herrmann, Samuel; Roynette, Bernard. A singular large deviations phenomenon. Ann. Inst. H. Poincaré Probab. Statist. 37 (2001), no. 5, 555--580. MR1851715
  • Herrmann, Samuel. Phénomène de Peano et grandes déviations. (French) [Large deviations for the Peano phenomenon] C. R. Acad. Sci. Paris Sér. I Math. 332 (2001), no. 11, 1019--1024. MR1838131
  • Ichiba, Tomoyuki; Karatzas, Ioannis. On collisions of Brownian particles. Ann. Appl. Probab. 20 (2010), no. 3, 951--977. MR2680554
  • Ichiba, Tomoyuki; Karatzas, Ioannis; Shkolnikov, Mykhaylo. Strong solutions of stochastic equations with rank-based coefficients. Probab. Theory Related Fields 156 (2013), no. 1-2, 229--248. MR3055258
  • Ichiba, Tomoyuki; Pal, Soumik; Shkolnikov, Mykhaylo. Convergence rates for rank-based models with applications to portfolio theory. Probab. Theory Related Fields 156 (2013), no. 1-2, 415--448. MR3055264
  • Ichiba, Tomoyuki; Papathanakos, Vassilios; Banner, Adrian; Karatzas, Ioannis; Fernholz, Robert. Hybrid atlas models. Ann. Appl. Probab. 21 (2011), no. 2, 609--644. MR2807968
  • Jourdain, B. Probabilistic approximation for a porous medium equation. Stochastic Process. Appl. 89 (2000), no. 1, 81--99. MR1775228
  • Jourdain, Benjamin. Particules collantes signées et lois de conservation scalaires 1D. (French) [Signed sticky particles and 1D scalar conservation laws] C. R. Math. Acad. Sci. Paris 334 (2002), no. 3, 233--238. MR1891065
  • Jourdain, Benjamin; Malrieu, Florent. Propagation of chaos and Poincaré inequalities for a system of particles interacting through their CDF. Ann. Appl. Probab. 18 (2008), no. 5, 1706--1736. MR2462546
  • B. Jourdain and J. Reygner, Propagation of chaos for rank-based interacting diffusions and long time behaviour of a scalar quasilinear parabolic equation, Stochastic Partial Differential Equations: Analysis and Computations 1 (2013), no. 3, 455--506 (English).
  • Karatzas, Ioannis; Shreve, Steven E. Trivariate density of Brownian motion, its local and occupation times, with application to stochastic control. Ann. Probab. 12 (1984), no. 3, 819--828. MR0744236
  • Karatzas, Ioannis; Shreve, Steven E. Brownian motion and stochastic calculus. Second edition. Graduate Texts in Mathematics, 113. Springer-Verlag, New York, 1991. xxiv+470 pp. ISBN: 0-387-97655-8 MR1121940
  • Pagès, Gilles. Sur quelques algorithmes récursifs pour les probabilités numériques. (French) [On some recursive algorithms for numerical probabilities] ESAIM Probab. Statist. 5 (2001), 141--170 (electronic). MR1875668
  • Pal, Soumik; Pitman, Jim. One-dimensional Brownian particle systems with rank-dependent drifts. Ann. Appl. Probab. 18 (2008), no. 6, 2179--2207. MR2473654
  • Revuz, Daniel; Yor, Marc. Continuous martingales and Brownian motion. Third edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 293. Springer-Verlag, Berlin, 1999. xiv+602 pp. ISBN: 3-540-64325-7 MR1725357
  • Rey-Bellet, Luc. Ergodic properties of Markov processes. Open quantum systems. II, 1--39, Lecture Notes in Math., 1881, Springer, Berlin, 2006. MR2248986
  • Stroock, Daniel W.; Varadhan, S. R. Srinivasa. Multidimensional diffusion processes. Reprint of the 1997 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2006. xii+338 pp. ISBN: 978-3-540-28998-2; 3-540-28998-4 MR2190038
  • Tanaka, Hiroshi. Stochastic differential equations with reflecting boundary condition in convex regions. Hiroshima Math. J. 9 (1979), no. 1, 163--177. MR0529332
  • Veretennikov, A. Ju. Strong solutions and explicit formulas for solutions of stochastic integral equations. (Russian) Mat. Sb. (N.S.) 111(153) (1980), no. 3, 434--452, 480. MR0568986
  • Veretennikov, A. Yu. Approximation of ordinary differential equations by stochastic ones. (Russian) Mat. Zametki 33 (1983), no. 6, 929--932. MR0709231

Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.