Geodesic
Transformation Semigroups of the Space of Functions That Are Square Integrable with respect to a Translation-Invariant Measure on a Banach Space
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Quantum probability, Tome 151 (2018), pp. 73-90.

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We examine measures on a Banach space $E$ that are invariant under shifts by arbitrary vectors of the space and are additive extensions of a set function defined on the family of bars with converging products of edge lengths that do not satisfy the $\sigma$-finiteness condition and, perhaps, the countable additivity condition. We introduce the Hilbert space $\mathcal{H}$ of complex-valued functions of the space $E$ of functions that are square integrable with respect to a shift-invariant measure. We analyze properties of semigroups of shift operators in the space $\mathcal{H}$ and the corresponding generators and resolvents. We obtain a criterion of the strong continuity of such semigroups. We introduce and examine mathematical expectations of operators of shifts along random vectors by a one-parameter family of Gaussian measures that form a semigroup with respect to the convolution. We prove that the family of mathematical expectations is a one-parameter semigroup of linear self-adjoint contraction mappings of the space $\mathcal{H}$, find invariant subspaces of operators of this semigroup, and obtain conditions of its strong continuity.
Keywords: finitely additive measure, invariant measure on a group, random walk, continuous one-parameter semigroup, generator, resolvent. 
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     author = {V. Zh. Sakbaev},
     title = {Transformation {Semigroups} of the {Space} of {Functions} {That} {Are} {Square} {Integrable} with respect to a {Translation-Invariant} {Measure} on a {Banach} {Space}},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {73--90},
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     volume = {151},
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     url = {http://geodesic.mathdoc.fr/item/INTO_2018_151_a7/}
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V. Zh. Sakbaev. Transformation Semigroups of the Space of Functions That Are Square Integrable with respect to a Translation-Invariant Measure on a Banach Space. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Quantum probability, Tome 151 (2018), pp. 73-90. http://geodesic.mathdoc.fr/item/INTO_2018_151_a7/

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