Hostname: page-component-76fb5796d-qxdb6 Total loading time: 0 Render date: 2024-04-26T09:16:21.212Z Has data issue: false hasContentIssue false
  • English
  • Français

Non-Symmetric Ornstein-Uhlenbeck Processes in Banach Space Via Dirichlet Forms

Published online by Cambridge University Press:  20 November 2018

Byron Schmuland
Byron Schmuland*
Affiliation:
Department of Statistics and Applied Probability University of Alberta Edmonton, Alberta T6G 2G1
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We use recent advances in the theory of non-symmetric Dirichlet forms to study a class of Banach space valued Ornstein-Uhlenbeck processes. As an example, we look at Walsh's stochastic model of neural response and show that it is a continuous process in any Sobolev space Hα(α < ½), and that it takes values only among functions with unbounded variation.

Keywords

60H1560G1760G15
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 45 , Issue 6 , 01 December 1993 , pp. 1324 - 1338
Copyright
Copyright © Canadian Mathematical Society 1993

References

1. Albeverio, S. and Rockner, M., Stochastic differential equations in infinite dimensions: solutions via Dirichletforms, Probab. Th. Rel. Fields 89(1991), 347–386.Google Scholar
2. Bouleau, N. and Hirsch, F., Dirichlet forms and analysis on Wiener space, Walter de Gruyter, Berlin New York, 1991.Google Scholar
3. Fukushima, M., Dirichlet forms and Markov processes, North Holland, Amsterdam, 1980.Google Scholar
4. Gross, L., Abstract Wiener Spaces. In: Proc. 5th Berkeley Symp. Math. Stat. Prob., Part 1, 2(1965), 31–42.Google Scholar
5. Katznelson, Y., An introduction to harmonic analysis, Dover, New York, 1976.Google Scholar
6. Ma, Z.M. and Rockner, M., An introduction to the theory of (non-symmetric) Dirichlet form.;, Springer- Verlag, Berlin-Heidelberg-New York, 1992.Google Scholar
7. Silverstein, M.L., Symmetric Markov Processes, Lecture Notes in Mathematics 426, Springer-Verlag, Berlin-Heidelberg-New York, 1974.Google Scholar
8. Rockner, M. and Schmuland, B., Tightness of general C1 p capacities on Banach space, J. Funct. Anal. 108(1992), 1–12.Google Scholar
9. Schmuland, B., Sample path properties of ℓp-valued Ornstein-Uhlenbeckprocesses, Canadian Math. Bull. (3) 33(1990), 358–366.Google Scholar
10. Schmuland, B., Dirichlet forms with polynomial domain, Mathematica Japonica 37(1992), 1015.1024.Google Scholar
11. Schmuland, B., Tightness of Gaussian capacities on subspaces, C.R. Math. Rep. Acad. Sci. Can. 14(1992), 125— 130.Google Scholar
12. Walsh, J., A stochastic model of neural response, Adv. Appl. Probab. 33(1981), 231–281.Google Scholar