Theory of random evolutions with applications to partial differential equations

Griego, Richard; Hersh, Reuben

doi:10.1090/S0002-9947-1971-0275507-7

Theory of random evolutions with applications to partial differential equations
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by Richard Griego and Reuben Hersh

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

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

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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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