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 small, let and be strongly continuous semigroups of linear contractions on a Banach space with infinitesimal operators and respectively, where as . Let be a family of linear operators on satisfying as . Assume that is the infinitesimal operator of a strongly continuous contraction semigroup on and that for each exists. We give conditions under which converges as to the semigroup generated by the closure of on . If , and we let , then we show that converges as to the strongly continuous contraction semigroup generated by the closure of .

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