Construction and Decomposition of Diffusion Processes

研究了定义在区域D（* R^d）上的扩散过程的构造和分解，其中D的边界 *D是具有H_lder尖点的Lipschitz区域. *D必须局部地表示为H_lder函数的图，并且H_lder指数的下确界gamma必须是正的。设（G，H^1（D））是对应于二阶椭圆型偏微分算子的Dirichiet形式，{G_<lambda>}是L^2（D）上与G相关联的强连续预解式.则（i）G_<lambda>（L2（D）*L^p（D））* C（D），p &gt; 1 +（d- 1）/gamma，（ii）G_<lambda>（C_*（D））是C_*（D）的稠密子空间.因此，在D上存在一个与C相关联的扩散过程M。这个过程是D上的一个反射扩散过程。因此M有一个跃迁概率密度p（t，x，y）。我们可以得到p（t，x，y）的上界，它依赖于尖点的形状。我们的扩散过程M定义在D上，应用M.福岛的分解定理，M的样本路径允许唯一的分解：局部零能量的鞅可加泛函和连续可加泛函.如果C^1（D）类的可加泛函和H_lder尖点的系数都大于1/2，则局部能量为零的连续可加泛函是有界变差的，并具有Skorohod表示.这是一个与维度d无关的有趣性质。M也是一个下鞅问题的解。证明了当d = 2且下鞅的系数为C^&lt;1，alpha&gt; -类时，下鞅问题有唯一解.

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.