Theory of random evolutions with applications to partial differential equations

Authors:
Richard Griego and Reuben Hersh

Journal:
Trans. Amer. Math. Soc. **156** (1971), 405-418

MSC:
Primary 60.40; Secondary 35.00

DOI:
https://doi.org/10.1090/S0002-9947-1971-0275507-7

MathSciNet review:
0275507

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Abstract: 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

DOI:
https://doi.org/10.1090/S0002-9947-1971-0275507-7

Keywords:
Singular perturbations,
integral representation of solutions,
linear systems,
hyperbolic systems,
telegraph equation,
wave equation,
semigroups of operators,
groups of operators,
probabilistic solution of differential equations,
Markov chains,
central limit theorem

Article copyright:
© Copyright 1971
American Mathematical Society