Potential Analysis

, Volume 50, Issue 2, pp 245–277 | Cite as

Stopping Time Convergence for Processes Associated with Dirichlet Forms

  • J. R. BaxterEmail author
  • M. Nielsen Hernandez


Convergence is proved for solutions un of Dirichlet problems in regions with many small excluded sets (holes), as the holes become smaller and more numerous. The problem is formulated probabilistically in the context of general Dirichlet forms, for random and nonrandom excluded sets. Sufficient conditions are given in Theorem 2.1 under which the sequence of entrance times or hitting times of the excluded sets converges in the stable topology. Convergence in the stable topology is a strengthened form of convergence in distribution, introduced by Rényi. Stable convergence of the entrance times implies convergence of the solutions un of the corresponding Dirichlet problems. Theorem 2.1 applies to Dirichlet forms such that the Markov process associated with the form has continuous paths and satisfies an absolute continuity condition for occupation time measures (Eq. 2.4). Conditions for convergence are formulated in terms of the sum of the expectations of the equilibrium measures for the excluded sets. The proof of convergence uses the fact that any martingale with respect to the natural filtration of the process must be continuous. In the case that the excluded sets are iid random, Theorem 2.1 strengthens previous results for the classical Brownian motion setting.


Dirichlet problems Regions with holes 

Mathematics Subject Classification (2010)

60J45 60K37 35J25 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Balzano, M.: Random relaxed Dirichlet problems. Annali di Matematica Pura Applicata 153, 133–174 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Balzano, M., Notarantonio, L: On the asymptotic behaviour of Dirichlet problems in a Riemannian manifold less random holes. Rend. Sem. Mat. Univ. Padova 100, 249–282 (1998)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Baxter, J.R., Chacon, R.V.: Compactness of stopping times. Theor. Probab. 40, 169–181 (1977)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Baxter, J.R., Chacon, R.V., Jain, N.C.: Weak limits of stopped diffusions. Trans. Am. Math. Soc. 293, 767–792 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Baxter, J.R., Dal Maso, G., Mosco, U.: Stopping times and Γ-convergence. Trans. Am. Math. Soc. 303, 1–38 (1987)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Baxter, J.R., Jain, N.C.: Asymptotic capacities for finely diffused bodies and stopped diffusions. Ill. J. Math. 31, 469–495 (1987)zbMATHGoogle Scholar
  7. 7.
    Biroli, M., Mosco, U.: A Saint-Venant type principle for dirichlet forms on discontinuous media. Annali di Matematica pura et applicata (IV) 169, 121–181 (1995)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Biroli, M., Tchou, N.A.: Asymptotic behaviour of relaxed Dirichlet problems involving a dirichlet-poincaré form. Zeitschrift für Analysis und ihre Anwendung 16, 281–309 (1997)CrossRefzbMATHGoogle Scholar
  9. 9.
    Chen, Z.-Q., Fukushima, M.: Symmetric Markov Processes, Time Change, and Boundary Theory. Princeton University Press, Princeton (2012)zbMATHGoogle Scholar
  10. 10.
    Dal Maso, G.: Γ-convergence and μ-capacities. Annali della Scuola Normale Superiore di Pisa - Cl. Sci. 14, 423–464 (1987)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Dal Maso, G., De Cicco, V., Notarantonio, L., Tchou, N.A.: Limits of variational problems for Dirichlet forms in varying domains. J. Math. Pures Appl. 77, 89–116 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Dal Maso, G., Garroni, A: The capacity method for asymptotic Dirichlet problems. Asymptot. Anal. 15, 299–324 (1997)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Dal Maso, G., Mosco, U.: Wiener’s criterion and Γ-convergence. Appl. Math. Optim. 15, 15–63 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Edgar, G.A., Millet, A., Sucheston, L: On compactness and optimality of stopping times. In: Martingale Theory in Harmonic Analysis and Banach Spaces, Lecture Notes in Math, vol. 939, pp. 36–61. Springer, Berlin-New York (1982)Google Scholar
  15. 15.
    Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet Forms and Symmetric Markov Processes, 2nd edn. de Gruyter, Berlin (2011)zbMATHGoogle Scholar
  16. 16.
    Getoor, R.K.: Measures not charging semipolars and equations of Schrödinger type. Potential Anal. 4, 79–100 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Häusler, E., Luschgy, H.: Stable Convergence and Stable Limit Theorems. Springer International Publishing, Switzerland (2015)CrossRefzbMATHGoogle Scholar
  18. 18.
    Kac, M.: Probabilistic methods in some problems of scattering theory. Rocky Mountain J. of Mathematics 4, 511–538 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Khruslov, E.: The method of orthogonal projections and the Dirichlet problems in domains with a fine-grained boundary. Math. USSS-Sb. 17, 37–59 (1972)CrossRefzbMATHGoogle Scholar
  20. 20.
    Liptser, R.S., Shiryaev, A.N.: Statistics of Random Processes I, 2nd edn. Springer, Berlin (2001)CrossRefGoogle Scholar
  21. 21.
    Ma, Z.-M., Röckner, M.: Introduction to the Theory of (Non-symmetric) Dirichlet Forms. Springer, Berlin (1992)CrossRefzbMATHGoogle Scholar
  22. 22.
    Mataloni, S., Tchou, N.A.: Limits of Relaxed Dirichlet problems involving a non symmetric Dirichlet form. Annali di Matematica 179, 65–93 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Meyer, P.-A.: Convergence faible et compacité des temps d’ârret D’àpres Baxter Et Chacon. In: Sáminaire De Probabilités, XII, Univ. De Strasbourg, Lecture Notes in Mathematics, vol. 649. Springer, New York (1978)Google Scholar
  24. 24.
    Nielsen, M.: Stable Convergence and Markov Processes, University of Minnesota, ProQuest, UMI Dissertations Publishing (2010)Google Scholar
  25. 25.
    Oshima, Y.: Semi-Dirichlet Forms and Markov Processes. De Gruyter, Berlin (2013)CrossRefzbMATHGoogle Scholar
  26. 26.
    Papanicolaou, G.C., Varadhan, S.R.S.: Diffusion in regions with many small holes. In: Grigelionis, B. (ed.) Stochastic Differential Systems-Filtering and Control, Lecture Notes in Control and Information Sciences, vol. 25, pp. 190–206. Springer, New York (1980)Google Scholar
  27. 27.
    Rauch, J., Taylor, M.: Potential and scattering theory on wildly perturbed domains. J. Funct. Anal. 18, 27–59 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Rényi, A.: On stable sequences of events. Sankhyā Ser. A 25, 293–302 (1963)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Stroock, D.W.: Diffusion semigroups corresponding to uniformly elliptic divergence form operators. Séminaire de probabilités de Strasbourg 22, 316–347 (1988)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.MathematicsUniversity of MinnesotaMinneapolisUSA
  2. 2.MathematicsCalifornia Baptist UniversityRiversideUSA

Personalised recommendations