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

TL;DR

This paper extends the concept of BV functions within a Gelfand triple using Dirichlet form theory and studies the stochastic reflection problem in infinite-dimensional convex sets, proving existence and uniqueness of solutions.

Contribution

It introduces a new BV function definition in Gelfand triples and establishes existence and uniqueness results for the stochastic reflection problem in infinite-dimensional convex sets.

Findings

Proved existence and uniqueness of solutions for regular convex sets.

Extended results to non-symmetric cases.

Applied the theory to specific convex sets like K_α.

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 [2] 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 non-symmetric case. Finally, we extend our results to the case when $\Gamma=K_\alpha$, where $K_\alpha={f\in L^2 (0,1)|f\geq -\alpha},\alpha\geq0$.