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Tr. Mat. Inst. Steklova, 2019, Volume 306, Pages 112–130 (Mi tm3999)  

Diffusion on a Hilbert Space Equipped with a Shift- and Rotation-Invariant Measure

D. V. Zavadskya, V. Zh. Sakbaevab

a Moscow Institute of Physics and Technology (State University), Institutskii per. 9, Dolgoprudnyi, Moscow oblast, 141701 Russia
b Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 Russia

Abstract: We study measures on a real separable Hilbert space $E$ that are invariant with respect to both shifts by arbitrary vectors of the space and orthogonal transformations. In particular, our first concern is a finitely additive analog of the Lebesgue measure. We present such an analog; namely, we construct a nonnegative finitely additive measure that is invariant with respect to shifts and rotations and is defined on the minimal ring of subsets of $E$ that contains all infinite-dimensional rectangles such that the products of their side lengths converge absolutely. We also define a Hilbert space $\mathcal H$ of complex-valued functions on $E$ that are square integrable with respect to a shift- and rotation-invariant measure. For random vectors whose distributions are given by families of Gaussian measures on $E$ that form semigroups with respect to convolution, we define expectations of the corresponding shift operators. We establish that such expectations form a semigroup of self-adjoint contractions in $\mathcal H$ that is not strongly continuous, and find invariant subspaces of strong continuity for this semigroup. We examine the structure of an arbitrary semigroup of self-adjoint contractions of the Hilbert space, which may not be strongly continuous. Finally, we show that the method of Feynman averaging of strongly continuous semigroups based on the notion of Chernoff equivalence of operator-valued functions is also applicable to discontinuous semigroups.

Funding Agency Grant Number
Ministry of Education and Science of the Russian Federation 5-100
This work was performed within the joint project with the Laboratory of Infinite-Dimensional Analysis and Mathematical Physics at the Faculty of Mechanics and Mathematics, Moscow State University, and was supported by the Ministry of Science and Higher Education of the Russian Federation within the Russian Academic Excellence Project “5-100.”


DOI: https://doi.org/10.4213/tm3999

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English version:
Proceedings of the Steklov Institute of Mathematics, 2019, 306, 102–119

Bibliographic databases:

UDC: 517.982+517.983
Received: May 10, 2019
Revised: May 28, 2019
Accepted: June 23, 2019

Citation: D. V. Zavadsky, V. Zh. Sakbaev, “Diffusion on a Hilbert Space Equipped with a Shift- and Rotation-Invariant Measure”, Mathematical physics and applications, Collected papers. In commemoration of the 95th anniversary of Academician Vasilii Sergeevich Vladimirov, Tr. Mat. Inst. Steklova, 306, Steklov Math. Inst. RAS, Moscow, 2019, 112–130; Proc. Steklov Inst. Math., 306 (2019), 102–119

Citation in format AMSBIB
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\by D.~V.~Zavadsky, V.~Zh.~Sakbaev
\paper Diffusion on a Hilbert Space Equipped with a Shift- and Rotation-Invariant Measure
\inbook Mathematical physics and applications
\bookinfo Collected papers. In commemoration of the 95th anniversary of Academician Vasilii Sergeevich Vladimirov
\serial Tr. Mat. Inst. Steklova
\yr 2019
\vol 306
\pages 112--130
\publ Steklov Math. Inst. RAS
\publaddr Moscow
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\crossref{https://doi.org/10.4213/tm3999}
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\pages 102--119
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