Snapping Out Walsh’s Brownian Motion and Related Stiff Problem

  • Liping Li
  • Wenjie SunEmail author


Firstly, we shall introduce the so-called snapping out Walsh’s Brownian motion and present its relation with Walsh’s Brownian motion. Then the stiff problem related to Walsh’s Brownian motion will be described and we shall build a phase transition for it. The snapping out Walsh’s Brownian motion corresponds to the so-called semi-permeable pattern of this stiff problem.


Stiff problems Dirichlet forms Mosco convergences Walsh’s Brownian motion 

Mathematics Subject Classification (2010)

31C25 60J25 60J45 60J50 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



The first named author is partially supported by NSFC (No. 11688101 and 11801546) and Key Laboratory of Random Complex Structures and Data Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences (No. 2008DP173182). The second named author is partially supported by China Scholarship Council (No. 201706100095).


  1. 1.
    Barlow, M., Pitman, J., Yor, M.: On Walsh’s Brownian motions. In: Séminaire de Probabilités, XXIII, Lecture Notes in Math., vol. 1372, pp. 275–293. Springer, Berlin (1989),
  2. 2.
    Chen, Z.Q., Fukushima, M.: One-point extensions of Markov processes by darning. Probab. Theory Relat. Fields 141(1–2), 61–112 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chen, Z.Q., Fukushima, M.: Symmetric Markov processes, time change, and boundary theory. In: London Mathematical Society Monographs Series, vol. 35. Princeton University Press, Princeton (2012)Google Scholar
  4. 4.
    Chen, Z.Q., Fukushima, M.: One-point reflection. Stochastic Process. Appl. 125(4), 1368–1393 (2015). MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chen, Z.Q., Peng, J.: Markov processes with darning and their approximations. Stochastic Process. Appl. 128(9), 3030–3053 (2018). MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chen, Z.Q., Fukushima, M., Ying, J.: Traces of symmetric Markov processes and their characterizations. Ann. Probab. 34 (3), 1052–1102 (2006). MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet Forms and Symmetric Markov Processes, De Gruyter Studies in Mathematics, vol. 19, extended edn. Walter de Gruyter & Co., Berlin (2011)Google Scholar
  8. 8.
    Itô, K., McKean, H.P. Jr.: Diffusion Processes and Their Sample Paths. Springer, Berlin-New York (1974). Second printing, corrected, Die Grundlehren der mathematischen Wissenschaften, Band 125zbMATHGoogle Scholar
  9. 9.
    Lejay, A.: The snapping out Brownian motion. Ann. Appl. Probab. 26(3), 1727–1742 (2016). MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Li, L., Sun, W.: On stiff problems via dirichlet forms. arXiv:1804.02634v3
  11. 11.
    Mosco, U.: Composite media and asymptotic Dirichlet forms. J. Funct. Anal. 123(2), 368–421 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Sanchez-Palencia, E.: Non-homogeneous Media and Vibration Theory. Lecture Notes in Physics, vol. 127. Springer, Berlin (1980)Google Scholar
  13. 13.
    Walsh, J.B.: A diffusion with a discontinuous local time. Astérisque 52–53, 37–45 (1978)Google Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.RCSDS, HCMSAcademy of Mathematics and Systems Science, Chinese Academy of SciencesBeijingChina
  2. 2.Shanghai Center for Mathematical SciencesFudan UniversityShanghaiChina

Personalised recommendations