Distributions and Analytical Measures
on Infinite-Dimensional Spaces
A. A. Belyaeva,* and O. G. Smolyanovb,c,**
Translated by I. Ruzanova
a Financial University under the Government
of the Russian Federation, Moscow, 125993 Russia
b Faculty of Mechanics and Mathematics,
Moscow State University, Moscow, 119991 Russia
c Moscow Institute of Physics and Technology (State University), Dolgoprudnyi, Moscow oblast, 141700 Russia
Correspondence to: *e-mail: AABelyaev@fa.ru
Correspondence to: **e-mail: smolyanov@yandex.ru
Received 30 May, 2018
Abstract—Spaces of test functions and spaces of distributions (generalized measures) on infinite-dimensional spaces are constructed, which, in the finite-dimensional case, coincide with classical spaces $\mathcal{D}$ and $\mathcal{D}{\text{'}}$. These distribution spaces contain generalized Feynman measures (but do not contain a generalized Lebesgue measure, which is not considered in this paper). For broad classes of infinite-dimensional differential equations in distribution spaces, the Cauchy problem has fundamental solutions. These results are much more definitive than those of A.Yu. Khrennikov’s and A.V. Uglanov’s pioneering works.
DOI: 10.1134/S1064562418070013