Consider a Brownian motion $W$ reflected at the boundary of a domain $D$ in Euclidean space. I want to look at the process obtained by "restricting" it to the boundary.

I was thinking of taking the local time $L(t)$ at the boundary, and then defining my new process as $W(L^{-1}(t))$.

Is this the appropriate way to do it? Does it define a Markov process on the boundary? If so, what is known about this process for, say, a $d$-dimensional round ball (eg on its regularity/ size and frequency of jumps)?