| Co-Investigator(Kenkyū-buntansha) |
MASUMOTO Makoto Faculty of Science, Yamaguchi University Associate Professor, 理学部, 助教授 (50173761)
YANAGIHARA Hiroshi Faculty of Engineering, Yamaguchi University Associate Professor, 工学部, 助教授 (30200538)
OKADA Mari Faculty of Engineering, Yamaguchi University Associate Professor, 工学部, 助教授 (40201389)
YANAGI Kenjiro Faculty of Engineering, Yamaguchi University Professor, 工学部, 教授 (90108267)
KAWAZU Kiyoshi Faculty of Education, Yamaguchi University Professor, 教育学部, 教授 (70037258)
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| Research Abstract |
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.
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