Summary.In earlier works, the gauge theorem was proved for additive functionals of Brownian motion of the form f ~o q(Bs)ds, where q is a function in the Kato class.Subsequently, the theorem was extended to additive functionals with Revuz measures/t in the Kato class. We prove that the gauge theorem holds for a large class of additive functionals of zero energy which are, in general, of unbounded variation. These additive functionals may not be semi-martingales, but correspond to a collection of distributions that belong to the Kato class in a suitable sense. Our gauge theorem generalizes the earlier versions of the gauge theorem.
Mathematics Subject Classification (1991): 60J65, 60J55, 60J57Brownian motion has long been used to treat the Dirichlet problem and other differential equations involving the Laplacian in a probabilistic manner. In resolving the Dirichlet problem, one tries to solve Au = 0 on some bounded open domain D subject to the condition u = f on the boundary of D, for some specified function f Kac showed that the probabilistic approach can be extended to treat differential operators of the form A -q, or more generally, of the form A -# by considering Brownian motion "killed" by a multiplicative functional associated with q or #.