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Robustness of nonuniform mean-square exponential dichotomies
- Part of
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 154 / Issue 2 / April 2024
- Published online by Cambridge University Press:
- 18 April 2023, pp. 525-567
- Print publication:
- April 2024
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- Article
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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.
