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Mathematics 2018, 6(4), 55; https://doi.org/10.3390/math6040055

On Small Deviation Asymptotics In L2 of Some Mixed Gaussian Processes

1
St. Petersburg Department of the Steklov Mathematical Institute, Fontanka 27, 191023 St. Petersburg, Russia
2
Saint-Petersburg State University, Universitetskaya nab. 7/9, 199034 St. Petersburg, Russia
3
National Research University, Higher School of Economics, Souza Pechatnikov 16, 190008 St. Petersburg, Russia
*
Author to whom correspondence should be addressed.
Received: 23 March 2018 / Revised: 2 April 2018 / Accepted: 3 April 2018 / Published: 5 April 2018
(This article belongs to the Special Issue Stochastic Processes with Applications)
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Abstract

We study the exact small deviation asymptotics with respect to the Hilbert norm for some mixed Gaussian processes. The simplest example here is the linear combination of the Wiener process and the Brownian bridge. We get the precise final result in this case and in some examples of more complicated processes of similar structure. The proof is based on Karhunen–Loève expansion together with spectral asymptotics of differential operators and complex analysis methods. View Full-Text
Keywords: mixed Gaussian process; small deviations; exact asymptotics mixed Gaussian process; small deviations; exact asymptotics
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Nazarov, A.I.; Nikitin, Y.Y. On Small Deviation Asymptotics In L2 of Some Mixed Gaussian Processes. Mathematics 2018, 6, 55.

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