Motivated by free boundary problems under uncertainties, we consider the oriented distance function as a way to define the expectation for a random compact or open set. In order to provide a law of large numbers and a central limit theorem for this notion of expectation, we also address the question of the convergence of the level sets of f$$ to the level sets of f when (f$$) is a sequence of functions uniformly converging to f. We provide error estimates in term of Hausdorff convergence. We illustrate our results on a free boundary problem.
Keywords: Random sets, continuity of level sets, oriented distance functions, law of large numbers, central limit theorem, free boundary problem
@article{COCV_2020__26_1_A84_0,
author = {Dambrine, M. and Puig, B.},
title = {Oriented distance point of view on random sets},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2020},
publisher = {EDP Sciences},
volume = {26},
doi = {10.1051/cocv/2020007},
mrnumber = {4167084},
zbl = {1459.60020},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2020007/}
}
TY - JOUR AU - Dambrine, M. AU - Puig, B. TI - Oriented distance point of view on random sets JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2020 VL - 26 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2020007/ DO - 10.1051/cocv/2020007 LA - en ID - COCV_2020__26_1_A84_0 ER -
Dambrine, M.; Puig, B. Oriented distance point of view on random sets. ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 84. doi: 10.1051/cocv/2020007
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