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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

**[1]**R. M. Blumenthal and R. K. Getoor,*Markov processes and potential theory*, Pure and Applied Mathematics, Vol. 29, Academic Press, New York-London, 1968. MR**0264757****[2]**E. Çinlar and M. Pinsky,*A stochastic integral in storage theory*, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete**17**(1971), 227–240. MR**0292194**, https://doi.org/10.1007/BF00536759**[3]**R. Cogburn and R. Hersh,*Two limit theorems for random differential equations*, Indiana Univ. Math. J.**22**(1972/73), 1067–1089. MR**0319267**, https://doi.org/10.1512/iumj.1973.22.22090**[4]**J. L. Doob,*Stochastic processes*, John Wiley & Sons, Inc., New York; Chapman & Hall, Limited, London, 1953. MR**0058896****[5]**E. B. Dynkin,*\cyr Markovskie protsessy*, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1963 (Russian). MR**0193670****[6]**Richard Griego and Reuben Hersh,*Theory of random evolutions with applications to partial differential equations*, Trans. Amer. Math. Soc.**156**(1971), 405–418. MR**0275507**, https://doi.org/10.1090/S0002-9947-1971-0275507-7**[7]**R. Hersh and G. Papanicolaou,*Non-commuting random evolutions, and an operator-valued Feynman-Kac formula*, Comm. Pure Appl. Math.**25**(1972), 337–367. MR**0310940**, https://doi.org/10.1002/cpa.3160250307**[8]**Einar Hille and Ralph S. Phillips,*Functional analysis and semi-groups*, American Mathematical Society Colloquium Publications, vol. 31, American Mathematical Society, Providence, R. I., 1957. rev. ed. MR**0089373****[9]**A. M. Il′in and R. Z. Has′minskiĭ,*On the equations of Brownian motion*, Teor. Verojatnost. i Primenen.**9**(1964), 466–491 (Russian, with English summary). MR**0168018****[10]**R. P. Kertz,*Limit theorems for discontinuous random evolutions*, Ph. D. Dissertation, Northwestern University, Evanston, Ill., 1972.**[11]**Robert P. Kertz,*Limit theorems for discontinuous random evolutions with applications to initial value problems and to Markov processes on 𝑁 lines*, Ann. Probability**2**(1974), 1046–1064. MR**0368180****[12]**Thomas G. Kurtz,*Extensions of Trotter’s operator semigroup approximation theorems*, J. Functional Analysis**3**(1969), 354–375. MR**0242016****[13]**Thomas G. Kurtz,*A general theorem on the convergence of operator semigroups*, Trans. Amer. Math. Soc.**148**(1970), 23–32. MR**0256210**, https://doi.org/10.1090/S0002-9947-1970-0256210-5**[14]**Thomas G. Kurtz,*A limit theorem for perturbed operator semigroups with applications to random evolutions*, J. Functional Analysis**12**(1973), 55–67. MR**0365224****[15]**Mark A. Pinsky,*Multiplicative operator functionals of a Markov process*, Bull. Amer. Math. Soc.**77**(1971), 377–380. MR**0298769**, https://doi.org/10.1090/S0002-9904-1971-12703-9**[16]**Mark A. Pinsky,*Multiplicative operator functionals and their asymptotic properties*, Advances in probability and related topics, Vol. 3, Dekker, New York, 1974, pp. 1–100. MR**0368182****[17]**A. Sommerfeld,*Mechanics. Lectures on theoretical physics*. Vol. 1, Translated from the 4th German edition; Academic Press, New York, 1952. MR**14**, 419.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
60J75,
47D05

Retrieve articles in all journals with MSC: 60J75, 47D05

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