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Transactions of the American Mathematical Society

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Perturbed semigroup limit theorems with applications to discontinuous random evolutions


Author: Robert P. Kertz
Journal: Trans. Amer. Math. Soc. 199 (1974), 29-53
MSC: Primary 60J75; Secondary 47D05
DOI: https://doi.org/10.1090/S0002-9947-1974-0362521-9
MathSciNet review: 0362521
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Abstract: For $ \varepsilon > 0$ small, let $ {U^\varepsilon }(t)$ and $ S(t)$ be strongly continuous semigroups of linear contractions on a Banach space $ L$ with infinitesimal operators $ A(\varepsilon )$ and $ B$ respectively, where $ A(\varepsilon ) = {A^{(1)}} + {\varepsilon A^{(2)}} + o()$ as $ \varepsilon \to 0$. Let $ \{ B(u);u \geqslant 0\} $ be a family of linear operators on $ L$ satisfying $ B(\varepsilon ) = B + {\varepsilon \Pi ^{(1)}} + {\varepsilon ^2}{\Pi ^{\varepsilon (2)}} + o({\varepsilon ^2})$ as $ \varepsilon \to 0$. Assume that $ A(\varepsilon ) + {\varepsilon ^{ - 1}}B()$ is the infinitesimal operator of a strongly continuous contraction semigroup $ {T_\varepsilon }(t)$ on $ L$ and that for each $ f \in L,{\lim _{\lambda \to 0}}\lambda \int_0^\infty {{e^{ - \lambda t}}} S(t)fdt \equiv Pf$ exists. We give conditions under which $ {T_\varepsilon }(t)$ converges as $ \to 0$ to the semigroup generated by the closure of $ P({A^{(1)}} + {\Pi ^{(1)}})$ on $ \mathcal{R}(P) \cap \mathcal{D}({A^{(1)}}) \cap \mathcal{D}({\Pi ^{(1)}})$. If $ P({A^{(1)}} + {\Pi ^{(1)}})f = 0,Bh = - ({A^{(1)}} + {\Pi ^{(1)}})f$, and we let $ \hat Vf = P({A^{(1)}} + {\Pi ^{(1)}})h$, then we show that $ {T_\varepsilon }(t/\varepsilon )f$ converges as $ \varepsilon \to 0$ to the strongly continuous contraction semigroup generated by the closure of $ {V^{(2)}} + \hat V$.

From these results we obtain new limit theorems for discontinuous random evolutions and new characterizations of the limiting infinitesimal operators of the discontinuous random evolutions. We apply these results in a model for the approximation of physical Brownian motion and in a model of the content of an infinite capacity dam.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1974-0362521-9
Keywords: Perturbation, random evolution, Markov process, operator semigroup, multiplicative operator functional, central limit theorem, stochastic initial value problem, approximation of physical Brownian motion
Article copyright: © Copyright 1974 American Mathematical Society

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