2017/01/16 by György Pál Gehér, Gehér, György Pál, Tamás Titkos +1
Economics, Econometrics and Finance · Mathematics · #46E27 #47B49 #54E40 #60A10 #60B05 #60B10 #Advanced Banach Space Theory #Advanced Topology and Set Theory #FOS: Mathematics #Functional Analysis (math.FA) #Primary: 46B04 #Secondary: 28A33 #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.1701.04267
openalex publication_date 2017/01/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
According to the fundamental work of Yu.V. Prokhorov, the general theory of\nstochastic processes can be regarded as the theory of probability measures in\ncomplete separable metric spaces. Since stochastic processes depending upon a\ncontinuous parameter are basically probability measures on certain subspaces of\nthe space of all functions of a real variable, a particularly important case of\nthis theory is when the underlying metric space has a linear structure.\nProkhorov also provided a concrete metrisation of the topology of weak\nconvergence today known as the L 'evy-Prokhorov distance. Motivated by these\nfacts, the famous Banach-Stone theorem, and some recent works related to\ncharacterisations of onto isometries of spaces of Borel probability measures,\nhere we give a complete description of surjective isometries with respect to\nthe L 'evy-Prokhorov metric in case when the underlying metric space is a\nseparable Banach space. Our result can be considered as a generalisation of L.\nMoln 'ar's earlier Banach-Stone-type result which characterises onto isometries\nof the space of all probability distribution functions on the real line wit\nrespect to the L 'evy distance. However, the present more general setting\nrequires the development of an essentially new technique.\n