Stochastic processes that take values in the group of orthogonal transformations of a finite-dimensional Euclidean space and are noncommutative analogues of processes with independent increments are considered. Such processes are defined as limits of noncommutative analogues of random walks in the group of orthogonal transformations. These random walks are compositions of independent random orthogonal transformations of Euclidean space. In particular, noncommutative analogues of diffusion processes with values in the group of orthogonal transformations are defined in this manner. Kolmogorov backward equations are derived for these processes.
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This work was performed at the Laboratory of Infinite-Dimensional Analysis and Mathematical Physics (headed by Smolyanov) of the Faculty of Mechanics and Mathematics of Lomonosov Moscow State University.
Smolyanov acknowledges the support of the Scientific Program “Fundamental Problems in Mechanics and Mathematics” of the Faculty of Mechanics and Mathematics of Lomonosov Moscow State University.
Translated by I. Ruzanova
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Zamana, K.Y., Sakbaev, V.Z. & Smolyanov, O.G. Stochastic Processes on the Group of Orthogonal Matrices and Evolution Equations Describing Them. Comput. Math. and Math. Phys. 60, 1686–1700 (2020). https://doi.org/10.1134/S0965542520100140
- random linear operator
- random operator-valued function
- operator-valued stochastic process
- law of large numbers
- Kolmogorov equation