The gauge theorem for a class of additive functionals of zero energy

In earlier works, the gauge theorem was proved for additive functionals of Brownian motion of the form ∫₀^tq(B_s)ds, where q is a function in the Kato class. Subsequently, the theorem was extended to additive functionals with Revuz measures μ 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.