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 Teor. Veroyatnost. i Primenen., 2004, Volume 49, Issue 1, Pages 145–155 (Mi tvp240)

Short Communications

On Markovian perturbations of the group of unitary operators associated with a stochastic process with stationary increments

G. G. Amosov

Moscow Institute of Physics and Technology

Abstract: We introduce “Markovian” cocycle perturbations of the group of unitary operators associated with a stochastic process with stationary increments, which are characterized by a localization of the perturbation to the algebra of past events. The definition we give is necessary because the Markovian perturbation of the group associated with a stochastic process with noncorrelated increments results in the perturbed group for which there exists a stochastic process with noncorrelated increments associated with it. On the other hand, some “deterministic” stochastic process lying in the past can also be associated with the perturbed group. The model of Markovian perturbations describing all Markovian cocycles up to a unitary equivalence of the perturbations has been constructed. Using this model, we construct Markovian cocycles transforming Gaussian measures to the equivalent Gaussian measures.

Keywords: stochastic process with stationary increments, group of unitary operators, cocycle perturbation.

DOI: https://doi.org/10.4213/tvp240

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English version:
Theory of Probability and its Applications, 2005, 49:1, 123–132

Bibliographic databases:

Citation: G. G. Amosov, “On Markovian perturbations of the group of unitary operators associated with a stochastic process with stationary increments”, Teor. Veroyatnost. i Primenen., 49:1 (2004), 145–155; Theory Probab. Appl., 49:1 (2005), 123–132

Citation in format AMSBIB
\Bibitem{Amo04} \by G.~G.~Amosov \paper On Markovian perturbations of the group of unitary operators associated with a stochastic process with stationary increments \jour Teor. Veroyatnost. i Primenen. \yr 2004 \vol 49 \issue 1 \pages 145--155 \mathnet{http://mi.mathnet.ru/tvp240} \crossref{https://doi.org/10.4213/tvp240} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2141334} \zmath{https://zbmath.org/?q=an:1096.47065} \transl \jour Theory Probab. Appl. \yr 2005 \vol 49 \issue 1 \pages 123--132 \crossref{https://doi.org/10.1137/S0040585X97980907} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000228185300008} 

• http://mi.mathnet.ru/eng/tvp240
• https://doi.org/10.4213/tvp240
• http://mi.mathnet.ru/eng/tvp/v49/i1/p145

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This publication is cited in the following articles:
1. G. G. Amosov, “On Markov perturbations of quantum random problems with stationary increments”, Theory Probab. Appl., 50:4 (2006), 650–658
2. G. G. Amosov, A. D. Baranov, “Dilations of Contraction Cocycles and Cocycle Perturbations of the Translation Group of the Line”, Math. Notes, 79:1 (2006), 3–17
3. G. G. Amosov, “Evolution Equations for Markov Cocycles Obtained by Second Quantization in the Symplectic Fock Space”, Theoret. and Math. Phys., 146:1 (2006), 152–157
4. G. G. Amosov, A. D. Baranov, V. V. Kapustin, “O primenenii modelnykh prostranstv dlya postroeniya kotsiklicheskikh vozmuschenii polugruppy sdvigov na polupryamoi”, Ufimsk. matem. zhurn., 4:1 (2012), 17–28
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