Sticky Particles and Stochastic Flows

Part of the Lecture Notes in Mathematics book series (LNM, volume 2137)

Abstract

Gawȩdzki and Horvai have studied a model for the motion of particles carried in a turbulent fluid and shown that in a limiting regime with low levels of viscosity and molecular diffusivity, pairs of particles exhibit the phenomena of stickiness when they meet. In this paper we characterise the motion of an arbitrary number of particles in a simplified version of their model.

Keywords

Brownian Motion Covariance Function Exit Time Stochastic Partial Differential Equation Point Motion 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

This work was started during a visit to Université Paris-Sud, and I would like to thanks the mathematics department there, and Yves Le Jan in particular, for their hospitality. I’d also like to thank Peter Windridge for his help with writing the R code for the simulations.

References

  1. 1.
    R. Bass, A stochastic differential equation with a sticky point. Electron. J. Probab. 19(32), 1–22 (2014)MathSciNetGoogle Scholar
  2. 2.
    R.J. Chitashvili, On the nonexistence of a strong solution in the boundary problem for a sticky Brownian motion. Centrum voor Wiskunde en Informatica (1989)Google Scholar
  3. 3.
    H. Engelbert, G. Peskir, Stochastic differential equations for sticky Brownian motion. Stochastics 86(6), 993–1021 (2014)MathSciNetGoogle Scholar
  4. 4.
    K. Gawȩdzki, P. Horvai, Sticky behavior of fluid particles in the compressible Kraichnan model. J. Stat. Phys. 116(5–6), 1247–1300 (2004)CrossRefGoogle Scholar
  5. 5.
    C.J. Howitt, J. Warren, Consistent families of Brownian motions and stochastic flows of kernels. Ann. Probab. 37(4), 1237–1272 (2009)MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    C.J. Howitt, J. Warren, Dynamics for the Brownian web and the Erosion flow. Stoch. Process. Appl. 119, 2028–2051 (2009)MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Y. Le Jan, S. Lemaire, Products of Beta matrices and sticky flows. Probab. Theory Relat. Fields 130(1), 109–134 (2004)MATHGoogle Scholar
  8. 8.
    Y. Le Jan, O. Raimond, Integration of Brownian vector fields. Ann. Probab. 30(2), 826–873 (2002)MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Y. Le Jan, O. Raimond, Flows, coalescence and noise. Ann. Probab. 32(2), 1247–1315 (2004)MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Y. Le Jan, O. Raimond, Sticky flows on the circle and their noises. Probab. Theory Relat. Fields 129(1), 63–82 (2004)MATHCrossRefGoogle Scholar
  11. 11.
    G. Peskir, On boundary behaviour of one-dimensional diffusions: from Brown to Feller and beyond (2014); Research Report No. 8 (2014); Probab. Statist. Group Manchester (14 pp.). To appear in Selected Works of William Feller (Springer)Google Scholar
  12. 12.
    E. Schertzer, R. Sun, J.M. Swart, Stochastic flows in the Brownian web and net. Mem. Am. Math. Soc. 227(1065) (2014)Google Scholar
  13. 13.
    J. Warren, Branching processes, the Ray-Knight theorem, and sticky Brownian motion, in Séminaire de Probabilités XXXI (Springer, Berlin, 1997), pp. 1–15Google Scholar
  14. 14.
    J. Warren, An elliptic pde with convex solutions (2014) [arXiv:1407.3985]Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of StatisticsUniversity of WarwickCoventryUK

Personalised recommendations