On stochastic distributions and currents

摘要

In many applications, it is of great importance to handle random closed sets of different (even though integer) Hausdorff dimensions, including local information about initial conditions and growth parameters. Following a standard approach in geometric measure theory, such sets may be described in terms of suitable measures. For a random closed set of lower dimension with respect to the environment space, the relevant measures induced by its realizations are singular with respect to the Lebesgue measure, and so their usual Radon–Nikodym derivatives are zero almost everywhere. In this paper, how to cope with these difficulties has been suggested by introducing random generalized densities (distributions) á la Dirac–Schwarz, for both the deterministic case and the stochastic case. For the last one, mean generalized densities are analyzed, and they have been related to densities of the expected values of the relevant measures. Actually, distributions are a subclass of the larger class of currents; in the usual Euclidean space of dimension d, currents of any order k ∈{0,1,…,d} or k-currents may be introduced. In this paper, the cases of 0-currents (distributions), 1-currents, and their stochastic counterparts are analyzed. Of particular interest in applications is the case in which a 1-current is associated with a path (curve). The existence of mean values has been discussed for currents too. In the case of 1-currents associated with random paths, two cases are of interest: when the path is differentiable, and also when it is the path of a Brownian motion or (more generally) of a diffusion. Differences between the two cases have been discussed, and nontrivial problems are mentioned which arise in the case of diffusions. Two significant applications to real problems have been presented too: tumor driven angiogenesis, and turbulence.