MULTIPLICATIVE CHAOS AND DIMENSION OF A MEASURE

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Let $ X= {X_k(ω,t)}_{t \in T} $, $ k \in 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 $ \sum _k[X_k( \omega ,s) X_k( \omega ,t)] = u \times \log^{+} {1/d(s,t)}+O(1) $, $ \s,t \in 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.
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