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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Theory of random evolutions with applications to partial differential equations
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by Richard Griego and Reuben Hersh PDF
Trans. Amer. Math. Soc. 156 (1971), 405-418 Request permission


The selection from a finite number of strongly continuous semigroups by means of a finite-state Markov chain leads to the new notion of a random evolution. Random evolutions are used to obtain probabilistic solutions to abstract systems of differential equations. Applications include one-dimensional first order hyperbolic systems. An important special case leads to consideration of abstract telegraph equations and a generalization of a result of Kac on the classical n-dimensional telegraph equation is obtained and put in a more natural setting. In this connection a singular perturbation theorem for an abstract telegraph equation is proved by means of a novel application of the classical central limit theorem and a representation of the solution for the limiting equation is found in terms of a transformation formula involving the Gaussian distribution.
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Additional Information
  • © Copyright 1971 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 156 (1971), 405-418
  • MSC: Primary 60.40; Secondary 35.00
  • DOI:
  • MathSciNet review: 0275507