Perturbed semigroup limit theorems with applications to discontinuous random evolutions
Author:
Robert P. Kertz
Journal:
Trans. Amer. Math. Soc. 199 (1974), 2953
MSC:
Primary 60J75; Secondary 47D05
MathSciNet review:
0362521
Fulltext 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 YorkLondon, 1968. MR 0264757
(41 #9348)
 [2]
E.
Çinlar and M.
Pinsky, A stochastic integral in storage theory, Z.
Wahrscheinlichkeitstheorie und Verw. Gebiete 17 (1971),
227–240. MR 0292194
(45 #1281)
 [3]
R.
Cogburn and R.
Hersh, Two limit theorems for random differential equations,
Indiana Univ. Math. J. 22 (1972/73), 1067–1089. MR 0319267
(47 #7811)
 [4]
J.
L. Doob, Stochastic processes, John Wiley & Sons, Inc.,
New York; Chapman & Hall, Limited, London, 1953. MR 0058896
(15,445b)
 [5]
E.
B. Dynkin, Markovskie protsessy, Gosudarstv. Izdat. Fiz.Mat.
Lit., Moscow, 1963 (Russian). MR 0193670
(33 #1886)
 [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
(43 #1261), http://dx.doi.org/10.1090/S00029947197102755077
 [7]
R.
Hersh and G.
Papanicolaou, Noncommuting random evolutions, and an
operatorvalued FeynmanKac formula, Comm. Pure Appl. Math.
25 (1972), 337–367. MR 0310940
(46 #10038)
 [8]
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)
 [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
(29 #5283)
 [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
(51 #4421)
 [12]
Thomas
G. Kurtz, Extensions of Trotter’s operator semigroup
approximation theorems, J. Functional Analysis 3
(1969), 354–375. MR 0242016
(39 #3351)
 [13]
Thomas
G. Kurtz, A general theorem on the convergence
of operator semigroups, Trans. Amer. Math.
Soc. 148 (1970),
23–32. MR
0256210 (41 #867), http://dx.doi.org/10.1090/S00029947197002562105
 [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
(51 #1477)
 [15]
Mark
A. Pinsky, Multiplicative operator functionals of
a Markov process, Bull. Amer. Math. Soc. 77 (1971), 377–380.
MR
0298769 (45 #7818), http://dx.doi.org/10.1090/S000299041971127039
 [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
(51 #4423)
 [17]
A. Sommerfeld, Mechanics. Lectures on theoretical physics. Vol. 1, Translated from the 4th German edition; Academic Press, New York, 1952. MR 14, 419.
 [1]
 R. M. Blumenthal and R. K. Getoor, Markov processes and potential theory, Pure and Appl. Math., vol. 29, Academic Press, New York, 1968. MR 41 #9348. MR 0264757 (41:9348)
 [2]
 E. Çinlar and M. Pinsky, A stochastic integral in storage theory, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 17 (1971), 227240. MR 45 #1281. MR 0292194 (45:1281)
 [3]
 R. Cogburn and R. Hersh, Two limit theorems for random differential equations, Indiana Univ. Math. J. 22 (1973), 10671089. MR 0319267 (47:7811)
 [4]
 J. L. Doob, Stochastic processes, Wiley, New York; Chapman & Hall, London, 1953. MR 15, 445. MR 0058896 (15:445b)
 [5]
 E. B. Dynkin, Markov processes, Fizmatgiz, Moscow, 1963; English transl., Vol. I, Die Grundlehren der math. Wissenschaften, Bände 121, Academic Press, New York; SpringerVerlag, Berlin, 1965. MR 33 #1886; #1887. MR 0193670 (33:1886)
 [6]
 R. Griego and R. Hersh, Theory of random evolutions with applications to partial differential equations, Trans. Amer. Math. Soc. 156 (1971), 405418. MR 43 #1261. MR 0275507 (43:1261)
 [7]
 R. Hersh and G. C. Papanicolaou, Noncommuting random evolutions, and an operatorvalued FeynmanKac formula, Comm. Pure Appl. Math. 25 (1972), 337367. MR 0310940 (46:10038)
 [8]
 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)
 [9]
 A. M. Il'in and R. Z. Has'minskii, On equations of Brownian motion, Teor. Verojatnost. i Primenen. 9 (1964), 466491 = Theor. Probability Appl. 9 (1964), 421444. MR 29 #5283. MR 0168018 (29:5283)
 [10]
 R. P. Kertz, Limit theorems for discontinuous random evolutions, Ph. D. Dissertation, Northwestern University, Evanston, Ill., 1972.
 [11]
 , Limit theorems for discontinuous random evolutions with applications to initial value problems and to Markov processes on lines, Ann. Probability (to appear). MR 0368180 (51:4421)
 [12]
 T. G. Kurtz, Extensions of Trotter's operator semigroup approximation theorems, J. Functional Analysis 3 (1969), 354375. MR 39 #3351. MR 0242016 (39:3351)
 [13]
 , A general theorem on the convergence of operator semigroups, Trans. Amer. Math. Soc. 148 (1970), 2332. MR 41 #867. MR 0256210 (41:867)
 [14]
 , A limit theorem for perturbed operator semigroups with applications to random evolutions, J. Functional Analysis 12 (1973), 5567. MR 0365224 (51:1477)
 [15]
 M. A. Pinsky, Multiplicative operator functionals of a Markov process, Bull. Amer. Math. Soc. 77 (1971), 377380. MR 45 #7818. MR 0298769 (45:7818)
 [16]
 , Multiplicative operator functional and their asymptotic properties, Advances in Probability, Vol. III, Marcel Dekker, New York (to appear). MR 0368182 (51:4423)
 [17]
 A. Sommerfeld, Mechanics. Lectures on theoretical physics. Vol. 1, Translated from the 4th German edition; Academic Press, New York, 1952. MR 14, 419.
Similar Articles
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:
http://dx.doi.org/10.1090/S00029947197403625219
PII:
S 00029947(1974)03625219
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
