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 Γ in a Hilbert space H. We prove the existence and uniqueness of a strong solution of this problem when Γ is a regular convex set. The result is also extended to the nonsymmetric case. Finally, we extend our results to the case when Γ=Kα, where Kα={f∈L2(0,1)|f≥−α}, α≥0.