It is known (see e.g. Revuz/Yor 1991, Chapter VI Corollary 1.6) that for a one-dim. Brownian motion $B$ and a nonegative function $\phi$ it holds
$$\int_0^t\phi(B_s)\text{d}s=\int \phi(a)L_t^a(B)\text{d}a$$
where $L_t^a$ is the local time of $B$ at the point $a$ up to time $t$. I am wondering, whether there is an analogous result in the multi-dimensional case. So $B=(B^1,B^2, \dots , B^n)$ and $\phi: \mathbb{R}^n \to\mathbb{R}$. I know that the concept of local times must be "reinvented" for higher dimensions but I was hoping that one can break the problem down to $n$ independent one-dimensional Brownian motions, so that maybe the right hand side of the formula becomes a multidimensional integral?! I couldn't come up with a result by myself and didn't find anything in the literature.
More general, I am asking myself whether there exists some concept of local times which measure the time $d$-dim. Brownian motion has spent in a set $A \subset \mathbb{R}^d$. Does anybody know?