The intrinsic local time sheet of Brownian motion

Abstract

McGill showed that the intrinsic local time process $$\tilde L$$ (t, x), t ≧ 0, x ∈ ℝ, of one-dimensional Brownian motion is, for fixedt>0, a supermartingale in the space variable, and derived an expression for its Doob-Meyer decomposition. This expression referred to the derivative of some process which was not obviously differentiable. In this paper, we provide an independent proof of the result, by analysing the local time of Brownian motion on a family of decreasing curves. The ideas involved are best understood in terms of stochastic area integrals with respect to the Brownian local time sheet, and we develop this approach in a companion paper. However, the result mentioned above admits a direct proof, which we give here; one is inevitably drawn to look at the local time process of a Dirichlet process which is not a semimartingale.