On decomposition of additive functionals of reflecting Brownian motions

Abstract

Let X be a locally compact separable Hausdorff metric space and m be a positive Radon measure on X with full support. For an m-symmetric Hunt process M = (X_t, P_x) on X with the associated Dirichlet form (ε,.F) being regular on L ² (X; m), the following decomposition of additive functionals (AF’s in abbreviaton) is known ([11]):

$$u({{X}_{t}})-u({{X}_{0}})={{M}_{t}}^{[u]}+{{N}_{t}}^{[u]}$$

P _x — almost surely, which holds for quasi every (q.e. in abbreviation) x ∈ X Here u is a quasi- continuous function in the space F, Ma_t ^[u] is a martingale AF with quadratic variation being associated with the energy measure of u, N_t ^[u] is a continuous AF of zero energy and `for q.e. x ∈ X’ means `for every x ∈ X outside a set of zero capacity’. (1.1) is beyond a semimartingale decomposition in that N _t ^[u] is of zero quadratic variation P_m-a.s. but not necessarily of bounded variation P_x-a.s. on each finite time interval. Nevertheless both M _t ^[u] and N _t ^[u] are well computable from u through the Dirichlet form ε and accordingly the decomposition (1.1) has proved to be a useful substitute of Ito’s formula for symmetric Markov processes.