Table of Contents
International Journal of Stochastic Analysis
Volume 2012, Article ID 163096, 17 pages
http://dx.doi.org/10.1155/2012/163096
Research Article

Probabilistic Solution of the General Robin Boundary Value Problem on Arbitrary Domains

Institut für Angewandte Analysis, Universität Ulm, 89069 Ulm, Germany

Received 13 April 2012; Revised 30 August 2012; Accepted 5 December 2012

Academic Editor: Donal O'Regan

Copyright © 2012 Khalid Akhlil. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. K. Sato and T. Ueno, “Multi-dimensional diffusion and the Markov process on the boundary,” Journal of Mathematics of Kyoto University, vol. 4, pp. 529–605, 1965. View at Google Scholar
  2. V. G. Papanicolaou, “The probabilistic solution of the third boundary value problem for second order elliptic equations,” Probability Theory and Related Fields, vol. 87, no. 1, pp. 27–77, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. S. Ramasubramanian, “Reflecting Brownian motion in a Lipschitz domain and a conditional gauge theorem,” Sankhyā A, vol. 63, no. 2, pp. 178–193, 2001. View at Google Scholar · View at Zentralblatt MATH
  4. S. Renming, “Probabilistic approach to the third boundary value problem,” Journal of Hebei University, vol. 9, no. 1, pp. 85–100, 1989. View at Google Scholar
  5. R. F. Bass, K. Burdzy, and Z.-Q. Chen, “Uniqueness for reflecting Brownian motion in lip domains,” Annales de l'Institut Henri Poincaré. Probabilités et Statistiques, vol. 41, no. 2, pp. 197–235, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. R. F. Bass and P. Hsu, “The semimartingale structure of reflecting Brownian motion,” Proceedings of the American Mathematical Society, vol. 108, no. 4, pp. 1007–1010, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. R. F. Bass and P. Hsu, “Some potential theory for reflecting Brownian motion in Hölder and Lipschitz domains,” The Annals of Probability, vol. 19, no. 2, pp. 486–508, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. Z. Q. Chen, P.J. Fitzsimmons, and R. J. Williams, “Quasimartingales and strong Caccioppoli set,” Potential Analysis, vol. 2, pp. 281–315, 1993. View at Google Scholar
  9. M. Fukushima, “A construction of reflecting barrier Brownian motions for bounded domains,” Osaka Journal of Mathematics, vol. 4, pp. 183–215, 1967. View at Google Scholar · View at Zentralblatt MATH
  10. E. P. Hsu, Reflecting Brownian motion, boundary local time and the Neumann problem [Thesis], Stanford University, 1984.
  11. W. Arendt and M. Warma, “The Laplacian with Robin boundary conditions on arbitrary domains,” Potential Analysis, vol. 19, no. 4, pp. 341–363, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. W. Arendt and M. Warma, “Dirichlet and Neumann boundary conditions: what is in between?” Journal of Evolution Equations, vol. 3, no. 1, pp. 119–135, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. M. Warma, The laplacian with general Robin boundary conditions [Ph.D. thesis], University of Ulm, 2002.
  14. S. Albeverio and Z. M. Ma, “Perturbation of Dirichlet forms—lower semiboundedness, closability, and form cores,” Journal of Functional Analysis, vol. 99, no. 2, pp. 332–356, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. S. Albeverio and Z. M. Ma, “Additive functionals, nowhere Radon and Kato class smooth measures associated with Dirichlet forms,” Osaka Journal of Mathematics, vol. 29, no. 2, pp. 247–265, 1992. View at Google Scholar · View at Zentralblatt MATH
  16. Ph. Blanchard and Z. M. Ma, “New results on the Schrödinger semigroups with potentials given by signed smooth measures,” in Stochastic Analysis and Related Topics, II (Silivri, 1988), vol. 1444, pp. 213–243, Springer, Berlin, Germany, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. M. Fukushima, Y. Ōshima, and M. Takeda, Dirichlet Forms and Symmetric Markov Processes, vol. 19, Walter de Gruyter, Berlin, Germany, 1994. View at Publisher · View at Google Scholar
  18. T. Kato, Perturbation theory for linear operators, Classics in Mathematics, Springer, Berlin, Germany, 1995.
  19. B. Simon, “Schrödinger semigroups,” Bulletin of American Mathematical Society, vol. 7, no. 3, pp. 447–526, 1982. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. P. Stollmann, “Smooth perturbations of regular Dirichlet forms,” Proceedings of the American Mathematical Society, vol. 116, no. 3, pp. 747–752, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  21. P. Stollmann and J. Voigt, “Perturbation of Dirichlet forms by measures,” Potential Analysis, vol. 5, no. 2, pp. 109–138, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. J. Voigt, “Absorption semigroups,” Journal of Operator Theory, vol. 20, no. 1, pp. 117–131, 1988. View at Google Scholar · View at Zentralblatt MATH
  23. J. Voigt, “Absorption semigroups, their generators, and Schrödinger semigroups,” Journal of Functional Analysis, vol. 67, no. 2, pp. 167–205, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  24. M. Fukushima, “Dirichlet forms, Caccioppoli sets and the Skorohod equation,” in Stochastic Differential and Difference Equations, vol. 23, pp. 59–66, Birkhäuser, Boston, Mass, USA, 1997. View at Google Scholar · View at Zentralblatt MATH
  25. D. S. Grebenkov, “Partially reflected Brownian motion: a stochastic approach to transport phenomena,” in Focus on Probability Theory, pp. 135–169, Nova Science Publishers, New York, NY, USA, 2006. View at Google Scholar