This chapter is the core of this short monograph. It is devoted to a possibly unified treatment of three real applications from medicine, material science, and statistical shape analysis, taken from current literature, by different authors.
The first one concerns tumor-driven angiogenesis. A schematic mathematical representation includes as main features branching of new vessels from the existing vessel network; vessel extension; anastomosis, due to the fusion of growing vessels. Randomness derives from the intrinsic randomness of the above-mentioned processes, which induce the randomness of the vessel network so produced. This is then mathematically described as a random distribution.
The second example concerns the mathematical modelling of dislocations in crystals. Evolution equations for the distributions which model random dislocations are derived, together with their mean field approximation, so to obtain an evolution equation for the total dislocation density.
Finally a problem of Statistical Shape Analysis is presented. It is shown how to build a Gaussian statistical model for a family of 2-dimensional manifolds in a three-dimensional space. Crucial ingredients of the relevant model are random currents on a suitable Reproducing Kernel Hilbert Space, and their finite dimensional approximation, for their numerical treatment.
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