Theory of random evolutions with applications to partial differential equations
Authors:
Richard Griego and Reuben Hersh
Journal:
Trans. Amer. Math. Soc. 156 (1971), 405418
MSC:
Primary 60.40; Secondary 35.00
MathSciNet review:
0275507
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: The selection from a finite number of strongly continuous semigroups by means of a finitestate 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 onedimensional 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 ndimensional 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.
 [1]
Garrett
Birkhoff and Robert
E. Lynch, Numerical solution of the telegraph and related
equations, Numerical Solution of Partial Differential Equations (Proc.
Sympos. Univ. Maryland, 1965) Academic Press, New York, 1966,
pp. 289–315. MR 0203953
(34 #3800)
 [2]
Kai
Lai Chung, Markov chains with stationary transition
probabilities, Die Grundlehren der mathematischen Wissenschaften, Bd.
104, SpringerVerlag, BerlinGöttingenHeidelberg, 1960. MR 0116388
(22 #7176)
 [3]
S.
Goldstein, On diffusion by discontinuous movements, and on the
telegraph equation, Quart. J. Mech. Appl. Math. 4
(1951), 129–156. MR 0047963
(13,960b)
 [4]
R.
J. Griego and R.
Hersh, Random evolutions, Markov chains, and systems of partial
differential equations, Proc. Nat. Acad. Sci. U.S.A.
62 (1969), 305–308. MR 0270207
(42 #5099)
 [5]
Reuben
Hersh, Explicit solution of a class of higherorder abstract Cauchy
problems., J. Differential Equations 8 (1970),
570–579. MR 0270210
(42 #5102)
 [6]
Einar
Hille and Ralph
S. Phillips, Functional analysis and semigroups, American
Mathematical Society Colloquium Publications, vol. 31, American
Mathematical Society, Providence, R. I., 1957. rev. ed. MR 0089373
(19,664d)
 [7]
M. Kac, Some stochastic problems in physics and mathematics, Magnolia Petroleum Co., Lectures in Pure and Applied Science, no. 2, 1956.
 [8]
Mark
Pinsky, Differential equations with a small parameter and the
central limit theorem for functions defined on a finite Markov chain,
Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 9 (1968),
101–111. MR 0228067
(37 #3651)
 [9]
N.
P. Romanoff, On oneparameter groups of linear transformations.
I, Ann. of Math. (2) 48 (1947), 216–233. MR 0020218
(8,520c)
 [10]
Frank
S. Scalora, Abstract martingale convergence theorems, Pacific
J. Math. 11 (1961), 347–374. MR 0123356
(23 #A684)
 [11]
Andrew
Y. Schoene, Semigroups and a class of singular perturbation
problems, Indiana Univ. Math. J. 20 (1970/1971),
247–263. MR 0283622
(44 #852)
 [12]
Joel
A. Smoller, Singular perturbations of Cauchy’s problem,
Comm. Pure Appl. Math. 18 (1965), 665–677. MR 0185240
(32 #2709)
 [13]
Stanley
Kaplan, Differential equations in which the
Poisson process plays a role, Bull. Amer. Math.
Soc. 70 (1964),
264–268. MR 0158183
(28 #1409), http://dx.doi.org/10.1090/S000299041964111125
 [14]
L.
Bobisud and R.
Hersh, Perturbation and approximation theory for higherorder
abstract Cauchy problems, Rocky Mountain J. Math. 2
(1972), no. 1, 57–73. MR 0294913
(45 #3981)
 [15]
Kôsaku
Yosida, Functional analysis, Second edition. Die Grundlehren
der mathematischen Wissenschaften, Band 123, SpringerVerlag New York Inc.,
New York, 1968. MR 0239384
(39 #741)
 [1]
 Garrett Birkhoff and Robert E. Lynch, Numerical solution of the telegraph and related equations, Proc. Sympos. Numerical Solution of Partial Differential Equations (Univ. of Maryland, 1965), Academic Press, New York, 1966, pp. 289315. MR 34 #3800. MR 0203953 (34:3800)
 [2]
 K. L. Chung, Markov chains with stationary transition probabilities, Die Grundlehren der math. Wissenschaften, Band 104, SpringerVerlag, Berlin, 1960. MR 22 #7176. MR 0116388 (22:7176)
 [3]
 S. Goldstein, On diffusion by discontinuous movements, and on the telegraph equation, Quart. J. Mech. Appl. Math. 4 (1951), 129156. MR 13, 960. MR 0047963 (13:960b)
 [4]
 R. J. Griego and R. Hersh, Random evolutions, Markov chains, and systems of partial differential equations, Proc. Nat. Acad. Sci. U.S.A. 62 (1969), 305308. MR 0270207 (42:5099)
 [5]
 R. Hersh, Explicit solution of a class of higherorder abstract Cauchy problems, J. Differential Equations 8 (1970), 570579. MR 0270210 (42:5102)
 [6]
 E. Hille and R. S. Phillips, Functional analysis and semigroups, rev. ed., Amer. Math. Soc. Colloq. Publ., vol. 31, Amer. Math. Soc., Providence, R. I., 1957. MR 19, 664. MR 0089373 (19:664d)
 [7]
 M. Kac, Some stochastic problems in physics and mathematics, Magnolia Petroleum Co., Lectures in Pure and Applied Science, no. 2, 1956.
 [8]
 M. Pinsky, Differential equations with a small parameter and the central limit theorem for functions defined on a finite Markov chain, Z. Wahrscheinlichkeitstheorie Verw. Gebiete 9 (1968), 101111. MR 37 #3651. MR 0228067 (37:3651)
 [9]
 N. P. Romanoff, On oneparameter groups of linear transformations. I, Ann. of Math. (2) 48 (1947), 216233. MR 8, 520. MR 0020218 (8:520c)
 [10]
 Frank S. Scalora, Abstract martingale convergence theorems, Pacific J. Math. (1961), 347374. MR 23 #A684. MR 0123356 (23:A684)
 [11]
 A. Schoene, Semigroups and a class of singular perturbation problems, Indiana U. Math. J. 20 (1970), 247263. MR 0283622 (44:852)
 [12]
 J. A. Smoller, Singular perturbations of Cauchy's problem, Comm. Pure Appl. Math. 18 (1965), 665677. MR 32 #2709. MR 0185240 (32:2709)
 [13]
 S. Kaplan, Differential equations in which the Poisson process plays a role, Bull. Amer. Math. Soc. 70 (1964), 264268. MR 28 #1409. MR 0158183 (28:1409)
 [14]
 L. Bobisud and R. Hersh, Perturbation and approximation theory for higherorder abstract Cauchy problems, Rocky Mt. J. Math. (to appear). MR 0294913 (45:3981)
 [15]
 K. Yosida, Functional analysis, SpringerVerlag, New York, 1968. MR 0239384 (39:741)
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC:
60.40,
35.00
Retrieve articles in all journals
with MSC:
60.40,
35.00
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197102755077
PII:
S 00029947(1971)02755077
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
