(r, p)-Capacity on the Wiener space and properties of Brownian motion
(r, p)-Capacity on the Wiener space and properties of Brownian motion
复制标题
(r, p)-维纳空间的容量和布朗运动的性质
DOI：
10.1007/bf00531775
复制
发表时间：
1984
期刊：
影响因子：
--
通讯作者：
M. Takeda
中科院分区：
文献类型：
--
作者：
M. Takeda
Let W d be a set of all continuous functions on [0, oo) taking values in the ddimensional Euclidean space R a and pw be the Wiener measure on W a. The basic properties of Brownian motion such as nowhere differentiability, L6vy's HNder continuity, the law of the iterated logarithm and so forth have been formulated to hold almost everywhere with respect to the measure pw. On the other hand, we can define the capacity C on W d by means of the Dirichlet space associated with the Ornstein-Uhlenbeck process on W d ([3]). By the definition" C-capacity 0" implies" PW-measure 0" but the converse does not hold generally. Therefore it is interesting to ask what property holds not only almost everywhere but also quasi-everywhere, namely except on a set of C-capacity zero. Fukushima [3] has shown that some properties including the above classical ones hold quasi-everywhere. Recently, P. Malliavin [8] has introduced a family of capacities Cr, p (r= l, 2...., l< p< oo) on W a as an infinite dimensional analogue of the Bessel capacities in the non linear potential theory. Now C1, a is equal to C-capacity. The main objective of the present paper is to show that Fukushima's statements can also be refined into Cr, pq. e. ones.
