The stochastic reflection problem on an infinite dimensional convex set and BV functions in a Gelfand triple

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 $K_{\alpha}=\{f\in L^{2}(0,1)|f\geq-\alpha\}$, $\alpha\geq0$.