• Bernoulli
  • Volume 22, Number 2 (2016), 681-710.

Long time behavior of stochastic hard ball systems

Patrick Cattiaux, Myriam Fradon, Alexei M. Kulik, and Sylvie Roelly

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We study the long time behavior of a system of $n=2,3$ Brownian hard balls, living in $\mathbb{R}^{d}$ for $d\ge2$, submitted to a mutual attraction and to elastic collisions.

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Bernoulli, Volume 22, Number 2 (2016), 681-710.

Received: February 2014
Revised: July 2014
First available in Project Euclid: 9 November 2015

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Zentralblatt MATH identifier

hard core interaction local time Lyapunov function normal reflection Poincaré inequality reversible measure stochastic differential equations


Cattiaux, Patrick; Fradon, Myriam; Kulik, Alexei M.; Roelly, Sylvie. Long time behavior of stochastic hard ball systems. Bernoulli 22 (2016), no. 2, 681--710. doi:10.3150/14-BEJ672.

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  • [1] Bakry, D., Cattiaux, P. and Guillin, A. (2008). Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincaré. J. Funct. Anal. 254 727–759.
  • [2] Bass, R.F. and Hsu, P. (1990). The semimartingale structure of reflecting Brownian motion. Proc. Amer. Math. Soc. 108 1007–1010.
  • [3] Bass, R.F. and Hsu, P. (1991). Some potential theory for reflecting Brownian motion in Hölder and Lipschitz domains. Ann. Probab. 19 486–508.
  • [4] Bobkov, S.G. (2003). Spectral gap and concentration for some spherically symmetric probability measures. In Geometric Aspects of Functional Analysis. Lecture Notes in Math. 1807 37–43. Berlin: Springer.
  • [5] Boissard, E., Cattiaux, P., Guillin, A. and Miclo, L. (2013). Ornstein–Uhlenbeck pinball: I. Poincaré Inequalities in a punctured domain. Preprint. Available at
  • [6] Böröczky, K. Jr. (2004). Finite Packing and Covering. Cambridge Tracts in Mathematics 154. Cambridge: Cambridge Univ. Press.
  • [7] Borodin, A.N. and Salminen, P. (2002). Handbook of Brownian Motion – Facts and Formulae, 2nd ed. Probability and Its Applications. Basel: Birkhäuser.
  • [8] Cattiaux, P. (1986). Hypoellipticité et hypoellipticité partielle pour les diffusions avec une condition frontière. Ann. Inst. Henri Poincaré Probab. Stat. 22 67–112.
  • [9] Cattiaux, P. (1987). Régularité au bord pour les densités et les densités conditionnelles d’une diffusion réfléchie hypoelliptique. Stochastics 20 309–340.
  • [10] Cattiaux, P. (1992). Stochastic calculus and degenerate boundary value problems. Ann. Inst. Fourier (Grenoble) 42 541–624.
  • [11] Cattiaux, P., Guillin, A. and Zitt, P.A. (2013). Poincaré inequalities and hitting times. Ann. Inst. Henri Poincaré Probab. Stat. 49 95–118.
  • [12] Chen, Z.-Q., Fitzsimmons, P.J., Takeda, M., Ying, J. and Zhang, T.-S. (2004). Absolute continuity of symmetric Markov processes. Ann. Probab. 32 2067–2098.
  • [13] Chen, Z.Q., Fitzsimmons, P.J. and Williams, R.J. (1993). Reflecting Brownian motions: Quasimartingales and strong Caccioppoli sets. Potential Anal. 2 219–243.
  • [14] Chow, T.Y. (1995). Penny-packings with minimal second moments. Combinatorica 15 151–158.
  • [15] Conway, J.H. and Sloane, N.J.A. (1993). Sphere Packings, Lattices and Groups, 2nd ed. Grundlehren der Mathematischen Wissenschaften 290. New York: Springer.
  • [16] Diaconis, P., Lebeau, G. and Michel, L. (2011). Geometric analysis for the Metropolis algorithm on Lipschitz domains. Invent. Math. 185 239–281.
  • [17] Fradon, M. (2010). Brownian dynamics of globules. Electron. J. Probab. 15 142–161.
  • [18] Fradon, M. and Rœlly, S. (2000). Infinite-dimensional diffusion processes with singular interaction. Bull. Sci. Math. 124 287–318.
  • [19] Fradon, M. and Rœlly, S. (2006). Infinite system of Brownian balls: Equilibrium measures are canonical Gibbs. Stoch. Dyn. 6 97–122.
  • [20] Fradon, M. and Rœlly, S. (2007). Infinite system of Brownian balls with interaction: The non-reversible case. ESAIM Probab. Stat. 11 55–79.
  • [21] Fradon, M. and Rœlly, S. (2010). Infinitely many Brownian globules with Brownian radii. Stoch. Dyn. 10 591–612.
  • [22] Fradon, M., Roelly, S. and Tanemura, H. (2000). An infinite system of Brownian balls with infinite range interaction. Stochastic Process. Appl. 90 43–66.
  • [23] Fukushima, M., Ōshima, Y. and Takeda, M. (1994). Dirichlet Forms and Symmetric Markov Processes. De Gruyter Studies in Mathematics 19. Berlin: de Gruyter.
  • [24] Fukushima, M. and Tomisaki, M. (1996). Construction and decomposition of reflecting diffusions on Lipschitz domains with Hölder cusps. Probab. Theory Related Fields 106 521–557.
  • [25] Kinkladze, G.N. (1982). A note on the structure of processes the measure of which is absolutely continuous with respect to the Wiener process modulus measure. Stochastics 8 39–44.
  • [26] Kulik, A.M. (2009). Exponential ergodicity of the solutions to SDEs with a jump noise. Stochastic Process. Appl. 119 602–632.
  • [27] Kulik, A.M. (2011). Asymptotic and spectral properties of exponentially ${\varphi}$-ergodic Markov processes. Stochastic Process. Appl. 121 1044–1075.
  • [28] Kulik, A.M. (2011). Poincaré inequality and exponential integrability of the hitting times of a Markov process. Theory Stoch. Process. 17 71–80.
  • [29] Linetsky, V. (2005). On the transition densities for reflected diffusions. Adv. in Appl. Probab. 37 435–460.
  • [30] Meyn, S.P. and Tweedie, R.L. (1993). Markov Chains and Stochastic Stability. Communications and Control Engineering Series. London: Springer.
  • [31] Osada, H. (1996). Dirichlet form approach to infinite-dimensional Wiener processes with singular interactions. Comm. Math. Phys. 176 117–131.
  • [32] Saisho, Y. and Tanaka, H. (1986). Stochastic differential equations for mutually reflecting Brownian balls. Osaka J. Math. 23 725–740.
  • [33] Saisho, Y. and Tanaka, H. (1987). On the symmetry of a reflecting Brownian motion defined by Skorohod’s equation for a multidimensional domain. Tokyo J. Math. 10 419–435.
  • [34] Sloane, N.J.A., Hardin, R.H., Duff, T.D.S. and Conway, J.H. (1995). Minimal-energy clusters of hard spheres. Discrete Comput. Geom. 14 237–259.
  • [35] Tanemura, H. (1996). A system of infinitely many mutually reflecting Brownian balls in ${\mathbf{R}}^{d}$. Probab. Theory Related Fields 104 399–426.
  • [36] Tanemura, H. (1997). Uniqueness of Dirichlet forms associated with systems of infinitely many Brownian balls in $\mathbf{R}^{d}$. Probab. Theory Related Fields 109 275–299.
  • [37] Temesvári, Á. (1974). On the extremum of power sums of distances. Mat. Lapok (N.S.) 25 329–342.
  • [38] Veretennikov, A.Y. (1987). Estimates of the mixing rate for stochastic equations. Teor. Veroyatn. Primen. 32 299–308.
  • [39] Ward, A.R. and Glynn, P.W. (2003). Properties of the reflected Ornstein–Uhlenbeck process. Queueing Syst. 44 109–123.