Geodesic
Reconstruction of characteristic functions of quadratic functionals on trajectories of Gaussian stochastic processes
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh international winter mathematical school "Modern methods of function theory and related problems", Voronezh, January 27 - February 1, 2023, Part 1, Tome 227 (2023), pp. 20-40

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In this paper, we examine the characteristic functions $Q_J(-i\lambda)$, $\lambda \in {\mathbb R}$, of stochastic variables determined by the values of the quadratic functionals $\mathsf{J}[\tilde{x}(t)]$ on the space ${\mathbb L}_2 [0, T]$ of trajectories of homogeneous Gaussian stochastic processes. We justify a method for calculating such characteristic functions, called reconstruction in the work, the application of which is not related to the use of the well-known Karhunen–Loeve–Pugachev method.
Keywords: Gaussian stochastic process, integral quadratic functional, correlation function, self-adjoint operator, characteristic function 
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     author = {Yu. P. Virchenko and A. S. Mazmanishvili},
     title = {Reconstruction of characteristic functions of quadratic functionals on trajectories of {Gaussian} stochastic processes},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
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     publisher = {mathdoc},
     volume = {227},
     year = {2023},
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Yu. P. Virchenko; A. S. Mazmanishvili. Reconstruction of characteristic functions of quadratic functionals on trajectories of Gaussian stochastic processes. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh international winter mathematical school "Modern methods of function theory and related problems", Voronezh, January 27 - February 1, 2023, Part 1, Tome 227 (2023), pp. 20-40. http://geodesic.mathdoc.fr/item/INTO_2023_227_a1/