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July 2012 The stochastic reflection problem on an infinite dimensional convex set and BV functions in a Gelfand triple
Michael Röckner, Rong-Chan Zhu, Xiang-Chan Zhu
Ann. Probab. 40(4): 1759-1794 (July 2012). DOI: 10.1214/11-AOP661

Abstract

In this paper, we introduce a definition of BV functions in a Gelfand triple which is an extension of the definition of BV functions in [Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 21 (2010) 405–414] by using Dirichlet form theory. By this definition, we can consider the stochastic reflection problem associated with a self-adjoint operator $A$ and a cylindrical Wiener process on a convex set $\Gamma$ in a Hilbert space $H$. We prove the existence and uniqueness of a strong solution of this problem when $\Gamma$ is a regular convex set. The result is also extended to the nonsymmetric case. Finally, we extend our results to the case when $\Gamma=K_{\alpha}$, where $K_{\alpha}=\{f\in L^{2}(0,1)|f\geq-\alpha\}$, $\alpha\geq0$.

Citation

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Michael Röckner. Rong-Chan Zhu. Xiang-Chan Zhu. "The stochastic reflection problem on an infinite dimensional convex set and BV functions in a Gelfand triple." Ann. Probab. 40 (4) 1759 - 1794, July 2012. https://doi.org/10.1214/11-AOP661

Information

Published: July 2012
First available in Project Euclid: 4 July 2012

zbMATH: 1267.60074
MathSciNet: MR2978137
Digital Object Identifier: 10.1214/11-AOP661

Subjects:
Primary: 26A45, 31C25, 60G60, 60GXX

Rights: Copyright © 2012 Institute of Mathematical Statistics

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Vol.40 • No. 4 • July 2012
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