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

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

2010-11-17

arxiv.org/abs/1011.3996v1

Mathematics

Probability

Scientific paper

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

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