Geodesic
Random point fields with Markovian refinements and the geometry of fractally disordered media
Teoretičeskaâ i matematičeskaâ fizika, Tome 124 (2000) no. 3, pp. 490-505

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We give a general construction of the probability measure for describing stochastic fractals that model fractally disordered media. For these stochastic fractals, we introduce the notion of a metrically homogeneous fractal Hausdorff–Karathéodory measure of a nonrandom type. We select a class $\mathbf F[q]$ of random point fields with Markovian refinements for which we explicitly construct the probability distribution. We prove that under rather weak conditions, the fractal dimension $D$ for random fields of this class is a self-averaging quantity and a fractal measure of a nonrandom type (the Hausdorff $D$-measure) can be defined on these fractals with probability 1. 
@article{TMF_2000_124_3_a9,
     author = {Yu. P. Virchenko and O. L. Shpilinskaya},
     title = {Random point fields with {Markovian} refinements and the geometry of fractally disordered media},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {490--505},
     publisher = {mathdoc},
     volume = {124},
     number = {3},
     year = {2000},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2000_124_3_a9/}
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Yu. P. Virchenko; O. L. Shpilinskaya. Random point fields with Markovian refinements and the geometry of fractally disordered media. Teoretičeskaâ i matematičeskaâ fizika, Tome 124 (2000) no. 3, pp. 490-505. http://geodesic.mathdoc.fr/item/TMF_2000_124_3_a9/