Construction and Decomposition of Diffusion Processes

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We investigated construction and decomposition of diffusion processes defined on a domain D (* R^d) whose boundary *D is of Lipschitz with H_lder cusps. *D must be locally expressed as a graph of a H_lder function and the infimum gamma of H_lder exponents must be positive. Let (epsilon, H^1 (D)) be a Dirichiet form corresponding to second order partial differential operator of elliptic type, {G_<lambda> } the strongly continu- ous resolvent on L^2 (D) assosiated with epsilon. Then (i) G_<lambda> (L2(D)*L^p (D)) * C(D), p > 1 + (d- 1)/gamma, and (ii) G_<lambda> (C_* (D)) is a dense subspace of C_* (D). Hence there exists a diffusion process M on D associated with C.This process is a reflecting diffusion process on D.The resolvent as above has a density which is continuous on D chi D off diagonal. Thus M has a transition probability density p(t, x, y) . We can obtain upper bounds of p(t, x, y), which depend on the shape of cusps. Our diffusion pro- cess M is defined on D.Applying a decompositin theorem due to M.Fukushima, sample paths of M admit a unique decomposition : martingale additive functionals and continuous additive functionals locally of zero energy. If coefficients of epsilon is of C^1 (D)-class and H_lder cusps are greater than 1/2 in addition, then continuous additive functionals locally of zero energy are of bounded variation and have Skoro- hod representations. This is an interesting property independ of dimension d. M is also a solution of a submartingale problem. We can prove that the submartingale problem has a unique solution if d = 2 and the coefficients of epsilon are of C^<1, alpha> -class.