Stationary distributions for diffusions with inert drift

  • Richard F. Bass
  • Krzysztof Burdzy
  • Zhen-Qing Chen
  • Martin Hairer
Article

Abstract

Consider reflecting Brownian motion in a bounded domain in \({\mathbb R^d}\) that acquires drift in proportion to the amount of local time spent on the boundary of the domain. We show that the stationary distribution for the joint law of the position of the reflecting Brownian motion and the value of the drift vector has a product form. Moreover, the first component is uniformly distributed on the domain, and the second component has a Gaussian distribution. We also consider more general reflecting diffusions with inert drift as well as processes where the drift is given in terms of the gradient of a potential.

Mathematics Subject Classification (2000)

Primary: 60H10 Secondary: 60J55 60J60 

References

  1. 1.
    Bass R.F.: Diffusions and Elliptic Operators. Springer, Berlin (1997)Google Scholar
  2. 2.
    Bass R.F., Burdzy K., Chen Z.-Q.: Uniqueness for reflecting Brownian motion in lip domains. Ann. Inst. Henri Poincare Probab. Statist. 41, 197–235 (2005)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bass R.F., Burdzy K., Chen Z.-Q.: Pathwise uniqueness for a degenerate stochastic differential equation. Ann. Probab. 35, 2385–2418 (2007)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Benaïm M., Ledoux M., Raimond O.: Self-interacting diffusions. Probab. Theory Related Fields 122, 1–41 (2002)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Benaïm M., Raimond O.: Self-interacting diffusions. II. Convergence in law. Ann. Inst. H. Poincaré Probab. Statist. 39, 1043–1055 (2003)MATHCrossRefGoogle Scholar
  6. 6.
    Benaïm M., Raimond O.: Self-interacting diffusions. III. Symmetric interactions. Ann. Probab. 33, 1717–1759 (2005)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Bou-Rabee N., Owhadi H.: Ergodicity of Langevin processes with degenerate diffusion in momentums. Int. J. Pure Appl. Math. 45(3), 475–490 (2008)MATHMathSciNetGoogle Scholar
  8. 8.
    Burdzy K.: Multidimensional Brownian Excursions and Potential Theory. Longman Sci. Tech., Harlow (1987)Google Scholar
  9. 9.
    Burdzy K., Hołyst R., Pruski Ł.: Brownian motion with inert drift, but without flux: a model. Physica A 384, 278–284 (2007)CrossRefGoogle Scholar
  10. 10.
    Burdzy K., White D.: A Gaussian oscillator. Electron. Comm. Probab. 9, 92–95 (2004)MATHMathSciNetGoogle Scholar
  11. 11.
    Chen Z.-Q.: On reflecting diffusion processes and Skorokhod decompositions. Probab. Theory Related Fields 94, 281–315 (1993)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Chen Z.-Q., Fitzsimmons P.J., Takeda M., Ying J., Zhang T.-S.: Absolute continuity of symmetric Markov processes. Ann. Probab. 32, 2067–2098 (2004)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Da Prato G., Zabczyk J.: Ergodicity for Infinite-Dimensional Systems. Cambridge University Press, Cambridge (1996)MATHGoogle Scholar
  14. 14.
    Dupuis P., Ishii H.: SDEs with oblique reflection on nonsmooth domains. Ann. Probab. 21, 554–580 (1993)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Dupuis P., Ishii H.: Correction: SDEs with oblique reflection on nonsmooth domains. Ann. Probab. 36, 1992–1997 (2008)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Ethier S.N., Kurtz T.G.: Markov Processes: Characterization and Convergence. Wiley, New York (1986)MATHGoogle Scholar
  17. 17.
    Fukushima M., Oshima Y., Takeda M.: Dirichlet Forms and Symmetric Markov Processes. De Gruyter, New York (1994)MATHGoogle Scholar
  18. 18.
    Hairer M., Mattingly J.C.: Ergodicity of the 2D Navier–Stokes equations with degenerate stochastic forcing. Ann. Math. (2) 164, 993–1032 (2006)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Hörmander L.: The Analysis of Linear Partial Differential Operators. Springer, Berlin (1985)Google Scholar
  20. 20.
    Hsu P.: On excursions of reflecting Brownian motion. Trans. Am. Math. Soc. 296, 239–264 (1986)MATHCrossRefGoogle Scholar
  21. 21.
    Ikeda N., Watanabe S.: Stochastic Differential Equations and Diffusion Processes. Kodansha, Tokyo (1981)MATHGoogle Scholar
  22. 22.
    Knight F.: On the path of an inert object impinged on one side by a Brownian particle. Probab. Theory Related Fields 121, 577–598 (2001)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Kurtz T., Protter P.: Weak limit theorems for stochastic integrals and stochastic differential equations. Ann. Probab. 19, 1035–1070 (1991)MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Liggett T.M.: Interacting Particle Systems. Springer, New York (1985)MATHGoogle Scholar
  25. 25.
    Lions P.L., Sznitman A.S.: Stochastic differential equations with reflecting boundary conditions. Comm. Pure Appl. Math. 37, 511–537 (1984)MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Maisonneuve B.: Exit systems. Ann. Probab. 3(3), 399–411 (1975)MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Meyn S.P., Tweedie R.L.: Markov Chains and Stochastic Stability. Springer, London (1993)MATHGoogle Scholar
  28. 28.
    Nualart D.: The Malliavin Calculus and Related Topics. Springer, Berlin (1995)MATHGoogle Scholar
  29. 29.
    Pardoux E., Williams R.J.: Symmetric reflected diffusions. Ann. Inst. H. Poincaré Probab Statist. 30, 13–62 (1994)MATHMathSciNetGoogle Scholar
  30. 30.
    Pemantle R.: A survey of random processes with reinforcement. Probab. Surv. 4, 1–79 (2007)CrossRefMathSciNetGoogle Scholar
  31. 31.
    Revuz D., Yor M.: Continuous Martingales and Brownian Motion, 3rd edn. Springer, Berlin (1999)MATHGoogle Scholar
  32. 32.
    Sinai Y.: Topics in Ergodic Theory. Princeton University Press, Princeton (1994)MATHGoogle Scholar
  33. 33.
    Stroock, D.W., Varadhan, S.R.S.: On the support of diffusion processes with applications to the strong maximum principle. In: Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Vol. III: Probability Theory, pp. 333–359. University of California Press, Berkeley (1972)Google Scholar
  34. 34.
    White, D.: Processes with Inert Drift. Ph.D. Thesis, University of Washington (2005)Google Scholar
  35. 35.
    White D.: Processes with inert drift. Electr. J. Probab. 12, 1509–1546 (2007)Google Scholar
  36. 36.
    Williams R.J., Zheng W.: On reflecting Brownian motion—a weak convergence approach. Ann. Inst. H. Poincaré Probab. Statist. 26, 461–488 (1990)MATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • Richard F. Bass
    • 1
  • Krzysztof Burdzy
    • 2
  • Zhen-Qing Chen
    • 2
  • Martin Hairer
    • 3
  1. 1.Department of MathematicsUniversity of ConnecticutStorrsUSA
  2. 2.Department of MathematicsUniversity of WashingtonSeattleUSA
  3. 3.Mathematics InstituteThe University of WarwickCoventryUK

Personalised recommendations