Geometric and Functional Analysis

, Volume 22, Issue 3, pp 558–587 | Cite as

The rate of convergence of the Walk on Spheres Algorithm

Article

Abstract

In this paper we examine the rate of convergence of one of the standard algorithms for emulating exit probabilities of Brownian motion, the Walk on Spheres (WoS) algorithm. We obtain a complete characterization of the rate of convergence of WoS in terms of the local geometry of a domain.

Keywords and phrases

Walk on spheres algorithm harmonic measure potential theory 

Mathematics Subject Classification (2000)

60G42 65C05 31B25 31B05 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. BB07.
    Binder I., Braverman M.: Derandomization of Euclidean random walks. LNCS 4627, 353–365 (2007)Google Scholar
  2. BC06.
    Braverman M., Cook S.: Computing over the reals: Foundations for scientific computing. Notices of the AMS 53(3), 318–329 (2006)MathSciNetMATHGoogle Scholar
  3. BW99.
    Brattka V., Weihrauch K.: Computability of subsets of Euclidean space I: Closed and compact subsets. Theoretical Computer Science 219, 65–93 (1999)MathSciNetMATHCrossRefGoogle Scholar
  4. Car67.
    L. Carleson. Selected Problems on Exceptional Sets. Van Nostrand Mathematical Studies, No. 13. Nostrand, Princeton (1967).Google Scholar
  5. EKM97.
    Embrechts P., Klüppelberg C., Mikosch T.: Modelling Extremal Events for Insurance and Finance. Springer, New York (1997)MATHGoogle Scholar
  6. EKMS80.
    B.S. Elepov, A.A. Kronberg, G.A. Mihaĭlov, and K.K. Sabel’fel’d. Reshenie kraevykh zadach metodom Monte-Karlo. “Nauka” Sibirsk. Otdel., Novosibirsk (1980).Google Scholar
  7. Fal03.
    Falconer K.: Fractal Geometry. Mathematical Foundations and Applications. 2nd edn. Wiley, Hoboken (2003)MATHCrossRefGoogle Scholar
  8. GM04.
    Garnett J.B., Marshall D.E.: Harmonic Measure. Cambridge University Press, Cambridge (2004)Google Scholar
  9. Kak44.
    Kakutani S.: Two-dimensional Brownian motion and harmonic functions. Proceedings of the Imperial Academy of Japan Tokyo 20, 706–714 (1944)MathSciNetMATHCrossRefGoogle Scholar
  10. KS91.
    Karatzas I., Shreve S.E.: Brownian Motion and Stochastic Calculus, 2nd edn. Springer, New York (1991)MATHCrossRefGoogle Scholar
  11. KS98.
    Karatzas I., Shreve S.E.: Methods of Mathematical Finance. Springer, Berlin (1998)MATHGoogle Scholar
  12. Lan72.
    N.S. Landkof. Foundations of Modern Potential Theory. Translated from the Russian by AP Doohovskoy, vol. 180 (1972).Google Scholar
  13. Maz02.
    Mazo R.M.: Brownian Motion: Fluctuations, Dynamics, and Applications. Oxford University Press, Oxford (2002)MATHGoogle Scholar
  14. Mih79.
    G.A. Mihaĭlov. Estimation of the difficulty of simulating the process of “random walk on spheres” for some types of regions. Zh. Vychisl. Mat. i Mat. Fiz., (2)19 (1979), 510–515, 558–559.Google Scholar
  15. Mil95.
    Milstein G.N.: Numerical Integration of Stochastic Differential Equations. Kluwer Academic Publishers, Dordrecht (1995)Google Scholar
  16. Mot59.
    Motoo M.: Some evaluations for continuous Monte Carlo method by using Brownian hitting process. Annals of the Institute of Statistical Mathematics Tokyo 11, 49–54 (1959)MathSciNetMATHCrossRefGoogle Scholar
  17. Mul56.
    Muller M.E.: Some continuous Monte Carlo methods for the Dirichlet problem. Annals of Mathematical Statistics 27, 569–589 (1956)MathSciNetMATHCrossRefGoogle Scholar
  18. Nel67.
    Nelson E.: Dynamical Theories of Brownian Motion. Princeton University Press, Princeton (1967)MATHGoogle Scholar
  19. Szn98.
    Sznitman A.S.: Brownian Motion, Obstacles and Random Media. Springer, Berlin (1998)MATHGoogle Scholar
  20. Wei00.
    Weihrauch K.: Computable Analysis. Springer, Berlin (2000)MATHCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TorontoTorontoCanada
  2. 2.Department of Computer SciencePrinceton UniversityPrincetonUSA

Personalised recommendations