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Robustness of nonuniform mean-square exponential dichotomies

Part of: Stochastic analysis Stability theory

Published online by Cambridge University Press:  18 April 2023

Hailong Zhu Open the ORCID record for Hailong Zhu [Opens in a new window]
Hailong Zhu*
Affiliation:
Anhui University of Finance and Economics, Bengbu 233030, China (hai-long-zhu@163.com)
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Abstract

For linear stochastic differential equations with bounded coefficients, we establish the robustness of nonuniform mean-square exponential dichotomy (NMS-ED) on $[t_{0},\,+\infty )$, $(-\infty,\,t_{0}]$ and the whole ${\Bbb R}$ separately, in the sense that such an NMS-ED persists under a sufficiently small linear perturbation. The result for the nonuniform mean-square exponential contraction is also discussed. Moreover, in the process of proving the existence of NMS-ED, we use the observation that the projections of the ‘exponential growing solutions’ and the ‘exponential decaying solutions’ on $[t_{0},\,+\infty )$, $(-\infty,\,t_{0}]$ and ${\Bbb R}$ are different but related. Thus, the relations of three types of projections on $[t_{0},\,+\infty )$, $(-\infty,\,t_{0}]$ and ${\Bbb R}$ are discussed.

Keywords

robustnessnonuniform mean-square exponential contractionnonuniform mean-square exponential dichotomystochastic differential equations

MSC classification

Primary: 60H10: Stochastic ordinary differential equations
Secondary: 34D09: Dichotomy, trichotomy
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 154 , Issue 2 , April 2024 , pp. 525 - 567
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Copyright
Copyright © The Author(s), 2023. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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