Properties of the Flows Generated by Stochastic Equations with Reflection

Abstract

We consider properties of a random set $\varphi_t(\mathbb{R}_+^d)$, where $\varphi_t(x)$ is a solution of a stochastic differential equation in $\mathbb{R}_+^d$ with normal reflection on the boundary starting at the point $x$. We perform the characterization of inner and boundary points of the set $\varphi_t(\mathbb{R}_+^d)$. We prove that the Hausdorff dimension of the boundary $\partial \varphi_t(\mathbb{R}_+^d)$ is not greater than $d - 1$.