Construction and decomposition of reflecting diffusions on Lipschitz domains with Holder cusps

We consider a d-dimensional Euclidean domain D whose boundary is Lipschitz continuous but admits locally finite number of outward or inward Holder cusp points. Using a method of Stampacchia and Moser for PDE, we first construct a conservative diffusion process on the Euclidean closure of D possessing a strong Feller resolvent and associated with a second order uniformly elliptic differential operator of divergence form with measurable coefficients a(ij). The sample path of the constructed diffusion can be uniquely decomposed as a sum of a martingale additive functional and an additive functional locally of zero energy. The second additive functional will be proved to be of bounded variation with a Skorohod type expression whenever ail is weakly differentiable and the Holder exponent at each outward cusp boundary point is greater than 1/2 regardless the dimension d.