BV functions in a Gelfand triple and the stochastic reflection problem on a convex set of a Hilbert space

Abstract

In this Note we introduce BV functions in a Gelfand triple, which is an extension of BV functions in Ambrosio et al., preprint [1], 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. 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 non-symmetric case. Finally, we extend our results to the case when Γ = K α , where K α = { f ∈ L 2 ( 0 , 1 ) | f ⩾ − α } , α ⩾ 0 .