Surjective Lévy-Prokhorov Isometries

Gehér, György Pál and Titkos, Tamás (2017) Surjective Lévy-Prokhorov Isometries. ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA-CLASSE DI SCIENZE. pp. 1-19. ISSN 0391-173X

Preview

Text 1701.04267v1.pdf Download (235kB) | Preview

Abstract

According to the fundamental work of Yu.V. Prokhorov, the general theory of stochastic processes can be regarded as the theory of probability measures in complete separable metric spaces. Since stochastic processes depending upon a continuous parameter are basically probability measures on certain subspaces of the space of all functions of a real variable, a particularly important case of this theory is when the underlying metric space has a linear structure. Prokhorov also provided a concrete metrisation of the topology of weak convergence today known as the L\'evy-Prokhorov distance. Motivated by these facts and some recent works related to the characterisation of onto isometries of spaces of Borel probability measures, here we give the complete description of the structure of surjective L\'evy-Prokhorov isometries on the space of all Borel probability measures on an arbitrary separable real Banach space. Our result can be considered as a generalisation of L. Moln\'ar's earlier result which characterises surjective L\'evy isometries of the space of all probability distribution functions on the real line. However, the present more general setting requires the development of an essentially new technique.

Actions (login required)

Edit Item