A limit theorem for boundary local time of a symmetric reflected diffusion

Let $X$ be a symmetric diffusion reflecting in a $\mathcal{C}^{3}$-bounded domain $D\subset\mathbb{R}^{d}$, $d\geq 1$, with a $\mathcal{C}^{2}$-bounded and non-degenerate matrix $a$. For $t>0$ and $n,k\in \mathbb{N}$ let $N(n,t)$ be the number of dyadic intervals $I_{n,k}$ of length $2^{-n}$, $k\geq 0$, that contain a time $s\leq t$ s.t. $X(s)\in\partial D$. For a suitable normalizing factor $H(t)$ we prove, extending the one dimensional case, that a.s. for all $t>0$ the entropy functional $N(n,t)/H(2^{-n})$ converges to the boundary local time $L(t)$ as $n\rightarrow\infty$. Applications include boundary value problems in PDE theory, efficient Monte Carlo simulations and Finance.