Joint continuity and representations of additive functionals of d-dimensional Brownian motion

AbstractConditions are given on a family of measures {μa, 0⩽a⩽1} so that the corresponding family {Aat, 0⩽a⩽1} of additive functionals of d-dimensional Brownian motion will be jointly continuous in a and t, a.s. This is then used to give a d-dimensional analogue to the representation At = ∝ Lytμ(dy) that is valid for one-dimensional Brownian motion, where Lyt is local time at y. In place of local times at points, local times at hyperplanes are used