An occupation time formula for semimartingales in RN

Inspired by coarea formula in geometric measure theory, an occupation time formula for continuous semimartingales in RN is proven. The occupation measure of a semimartingale, for N≥2, is singular with respect to Lebesgue measure but it has a bounded density “transversal” to a foliation, under proper assumptions. In the particular case of the foliation given locally by the distance function from a manifold, the transversal density is related to a geometric local time of the semimartingale at the manifolds of the foliation.