MULTIPLICATIVE CHAOS AND DIMENSION OF A MEASURE
1994 年 48 巻 1 号 p. 87-100
詳細
1994 年 48 巻 1 号 p. 87-100
Let X= {Xk(ω, t)}t∈T, k∈N, be an independent sequence of Gaussian processes with mean 0 on a locally compact separable metric space (T, d), and assume that the covariance functions are non-negative and
??E[Xk(ω, s)Xk(ω, t)] = ulog+?? + O(1), s, t∈T,
for a positeve number u>0. Kahane [1985] defined a multiplicative chaos by X and gave sufficient conditions for the regularity or the singularity of a measure with respect to it in the case where (T, d) is compact and homogeneous in the sense of Coifman and Weiss.
The aim of this paper are to interprete Kahane's results from the viewpoint of the dimension of a measure, and extend them to the case where (T, d) is a locally compact metric space equipped with a majorizing measure and not necessarily homogeneous.
