May  2012, 17(3): 735-757. doi: 10.3934/dcdsb.2012.17.735

Two theorems on singularly perturbed semigroups with applications to models of applied mathematics

1. 

Department of Mathematics, Faculty of Electrical Engineering and Computer Science, Lublin University of Technology, Nadbystrzycka 38A, 20-618 Lublin, Poland

2. 

European Actuarial Services, Ernst & Young Business Advisory Sp. z o.o. i Wspólnicy sp.k., Rondo ONZ 1, 00-124, Warsaw, Poland

Received  May 2011 Revised  October 2011 Published  January 2012

We present two theorems on convergence of semigroups related to singularly perturbed abstract Cauchy problems, and apply them to some recent models of applied mathematics. The semigroups considered are related to piecewise deterministic Markov processes jumping between several copies of a rectangle in $\mathbb{R}^M$ and moving along deterministic integral curves of some ODEs between jumps. Our theorems describe limit behavior of the processes in the cases of frequent jumps and of fast motions in the direction of chosen variables. These results are motivated by Kepler--Elston's model of gene regulation and Lipniacki's model of gene expression. Application to other models, including those of mathematical economics, are also discussed.
Citation: Adam Bobrowski, Radosław Bogucki. Two theorems on singularly perturbed semigroups with applications to models of applied mathematics. Discrete and Continuous Dynamical Systems - B, 2012, 17 (3) : 735-757. doi: 10.3934/dcdsb.2012.17.735
References:
[1]

J. Banasiak and A. Bobrowski, Interplay between degenerate convergence of semigroups and asymptotic analysis: A study of a singularly perturbed abstract telegraph system, J. Evol. Equ., 9 (2009), 293-314. doi: 10.1007/s00028-009-0009-7.

[2]

A. Bobrowski, A note on convergence of semigroups, Ann. Polon. Math., 69 (1998), 107-127.

[3]

A. Bobrowski, "Functional Analysis for Probability and Stochastic Processes. An Introduction," Cambridge University Press, Cambridge, 2005. doi: 10.1017/CBO9780511614583.

[4]

A. Bobrowski, Degenerate convergence of semigroups related to a model of stochastic gene expression, Semigroup Forum, 73 (2006), 345-366, with correction in, 77 (2008), 520-521.

[5]

A. Bobrowski, On limitations and insufficiency of the Trotter-Kato theorem, Semigroup Forum, 75 (2007), 317-336. doi: 10.1007/s00233-006-0676-4.

[6]

A. Bobrowski and R. Bogucki, Semigroups generated by convex combinations of several Feller generators in models of mathematical biology, Studia Math., 189 (2008), 287-300. doi: 10.4064/sm189-3-6.

[7]

A. Bobrowski, T. Lipniacki, K. Pichór and R. Rudnicki, Asymptotic behaviour of distributions of mRNA and protein levels in a model of stochastic gene expression, J. Math. Anal. Appl., 333 (2007), 753-769. doi: 10.1016/j.jmaa.2006.11.043.

[8]

B. M. Brown, D. Elliot and D. F. Paget, Lipschitz constants for the Bernstein polynomials of a Lipschitz continuous function, J. Approx. Theory, 49 (1987), 196-199. doi: 10.1016/0021-9045(87)90087-6.

[9]

M. H. A. Davis, Piecewise-deterministic Markov processes: A general class of nondiffusion stochastic models. With discussion, J. R. Stat. Soc. Ser. B, 46 (1984), 353-388.

[10]

M. H. A. Davis, "Lectures on Stochastic Control and Nonlinear Filtering," Tata Institute of Fundamental Research Lectures on Mathematics and Physics, 75, Published for the Tata Institute of Fundamental Research, Bombay, by Springer-Verlag, Berlin, 1984.

[11]

M. H. A. Davis, "Markov Processes and Optimization," Chapman and Hall, 1993.

[12]

R. M. Dudley, "Real Analysis and Probability," Revised reprind of the 1989 original, Cambridge Studies in Advanced Mathematics, 74, Cambridge University Press, Cambridge, 2002.

[13]

S. N. Ethier and T. G. Kurtz, "Markov Processes. Characterization and Convergence," Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1986.

[14]

W. H. Fleming and Q. Zhang, Risk-sensitive production planning of a stochastic manufacturing system, SIAM J. Control Optim., 36 (1998), 1147-1170. doi: 10.1137/S036301299631034X.

[15]

B. Hat, P. Paszek, M. Kimmel, K. Piechór and T. Lipniacki, How the number of alleles influences gene expression, J. Statist. Phys., 128 (2007), 511-533. doi: 10.1007/s10955-006-9218-4.

[16]

A. Haurie and F. Moresino, Singularly perturbed piecewise deterministic games, SIAM J. Control Optim., 47 (2008), 73-91. doi: 10.1137/050627599.

[17]

A. Haurie, A two-timescale stochastic game framework for climate change policy assessment, in "Dynamic Games: Theory and Applications" (eds. A. Haurie and G. Zaccour), GERAD 25th Anniv. Ser., 10, Springer, New York, (2005), 193-211.

[18]

E. Hille and R. S. Phillips, "Functional Analysis and Semi-Groups," rev. ed., American Mathematical Society Colloquium Publications, vol. 31, Amer. Math. Soc., Providence, R.I., 1957.

[19]

F. Hoppensteadt, Stability in systems with parameter, J. Math. Anal. Appl., 18 (1967), 129-134. doi: 10.1016/0022-247X(67)90187-4.

[20]

T. B. Kepler and T. C. Elston, Stochasticity in transcriptional regulation: Origins, consequences, and mathematical representations, Biophys. J., 81 (2001), 3116-3136. doi: 10.1016/S0006-3495(01)75949-8.

[21]

V. S. Korolyuk and A. F. Turbin, "Mathematical Foundations of the State Lumping of Large Systems," Translated from the 1978 Russian original by V. V. Zayats and Y. A. Atanov and revised by the authors, Mathematics and its Applications, 264, Kluwer Academic Publishers Group, Dordrecht, 1993.

[22]

T. G. Kurtz, Extensions of Trotter’s operator semigroup approximation theorems, J. Funct. Anal., 3 (1969), 354-375. doi: 10.1016/0022-1236(69)90031-7.

[23]

T. G. Kurtz, A limit theorem for perturbed operator semigroups with applications to random evolutions, J. Funct. Anal., 12 (1973), 55-67. doi: 10.1016/0022-1236(73)90089-X.

[24]

T. G. Kurtz, Applications of an abstract perturbation theorem to ordinary differential equations, Houston J. Math., 3 (1977), 67-82.

[25]

P. D. Lax, "Linear Algebra and Its Applications," 2 edition, Pure and Applied Mathematics (Hoboken), Wiley-Interscience [John Wiley & Sons], Hoboken, NJ, 2007.

[26]

T. Lipniacki, P. Paszek, A. Marciniak-Czochra, A. R. Brasier and M. Kimmel, Transcriptional stochasticity in gene expression, J. Theor. Biol., 238 (2006), 348-367. doi: 10.1016/j.jtbi.2005.05.032.

[27]

T. Lipniacki, P. Paszek, A. R. Brasier, B. Luxon and M. Kimmel, Stochastic regulation in early immune response, Biophys. J., 90 (2006), 725-742. doi: 10.1529/biophysj.104.056754.

[28]

G. G. Lorentz, "Bernstein Polynomials," 2nd edition, Chelsea Publishing Co., New York, 1986.

[29]

J. L. Massera, On Liapounoff's conditions of stability, Ann. Math. (2), 50 (1949), 705-721. doi: 10.2307/1969558.

[30]

R. E. Megginson, "An Introduction to Banach Space Theory," Graduate Texts in Mathematics, 183, Springer-Verlag, New York, 1998.

[31]

C. Meyer, "Matrix Analysis and Applied Linear Algebra," SIAM, Philadelphia, PA, 2000. doi: 10.1137/1.9780898719512.

[32]

J. R. Mika and J. Banasiak, "Singularly Perturbed Evolution Equations with Applications to Kinetic Theory," Series on Advances in Mathematics for Applied Sciences, 34, World Scientific Publishing Co., Inc., River Edge, NJ, 1995.

[33]

M. Rausand and A. Høyland, "System Reliability Theory: Models, Statistical Methods, and Applications," Second edition, Wiley-Interscience [John Wiley & Sons], Hoboken, NJ, 2004.

[34]

L. Saloff-Coste, Lectures on Finite Markov Chains, in "Lectures on Probability Theory and Statistics" (Saint-Flour, 1996), Lecture Notes in Mathematics, 1665, Springer, Berlin, 1997.

[35]

V. V. Sarafyan and A. V. Skorokhod, On fast switching dynamical systems, Theory Probab. Appl., 32 (1987), 595-607. doi: 10.1137/1132092.

[36]

V. V. Sarafyan, Weak convergence of measures and asymptotic behavior of solutions of the Cauchy problem for systems of differential equations with a small parameter, Theory Probab. Appl., 34 (1989), 351-357. doi: 10.1137/1134035.

[37]

S. Sethi, H. Zhang and Q. Zhang, "Average-Cost Control of Stochastic Manufacturing Systems," Stochastic Modelling and Applied Probability, 54, Springer-Verlag, New York, 2005.

[38]

M. Sova, Convergence d'opérations linéaires non bornées, (French) Rev. Roumaine Math. Pures Appl., 12 (1967), 373-389.

[39]

A. N. Tikhonov, Systems of differential equations containing small parameters in the derivatives, (Russian) Mat. Sb. N. S., 31(73) (1952), 575-586.

[40]

M. E. Taylor, "Partial Differential Equations. Basic Theory," Texts in Applied Mathematics, 23, Springer-Verlag, New York, 1996.

[41]

A. B. Vasil'eva and B. F. Butuzov, "Asymptotic Expansions of the Solutions of Singularly Perturbed Equations," (Russian) Izdat. "Nauka," Moscow, 1973.

[42]

W. Walter, "Differential and Integral Inequalities," Translated from German by Lisa Rosenblatt and Lawrence Shampine, Ergebnisse d. Mathematik u. ihrer Granzgebiete, Vol. 55, Springer-Verlag, New York-Berlin, 1970.

[43]

W. Walter, Differential inequalities and maximum principles: Theory, new methods and applications, in "Preceedings of the Second World Congress of Nonlinear Analysts, Part 8" (Athens, 1996), Nonlinear Anal., 30 (1997), 4695-4711. doi: 10.1016/S0362-546X(96)00259-3.

[44]

W. Walter, "Ordinary Differential Equations," Translated from the sixth German (1996) edition by Russell Thompson, Graduate Texts in Mathematics, 182, Readings in Mathematics, Springer-Verlag, New York, 1998.

[45]

G. G. Yin and Q. Zhang, "Continuous-Time Markov Chains and Applications. A Singular Perturbation Approach," Applications of Mathematics (New York), 37, Springer-Verlag, New York, 1998.

show all references

References:
[1]

J. Banasiak and A. Bobrowski, Interplay between degenerate convergence of semigroups and asymptotic analysis: A study of a singularly perturbed abstract telegraph system, J. Evol. Equ., 9 (2009), 293-314. doi: 10.1007/s00028-009-0009-7.

[2]

A. Bobrowski, A note on convergence of semigroups, Ann. Polon. Math., 69 (1998), 107-127.

[3]

A. Bobrowski, "Functional Analysis for Probability and Stochastic Processes. An Introduction," Cambridge University Press, Cambridge, 2005. doi: 10.1017/CBO9780511614583.

[4]

A. Bobrowski, Degenerate convergence of semigroups related to a model of stochastic gene expression, Semigroup Forum, 73 (2006), 345-366, with correction in, 77 (2008), 520-521.

[5]

A. Bobrowski, On limitations and insufficiency of the Trotter-Kato theorem, Semigroup Forum, 75 (2007), 317-336. doi: 10.1007/s00233-006-0676-4.

[6]

A. Bobrowski and R. Bogucki, Semigroups generated by convex combinations of several Feller generators in models of mathematical biology, Studia Math., 189 (2008), 287-300. doi: 10.4064/sm189-3-6.

[7]

A. Bobrowski, T. Lipniacki, K. Pichór and R. Rudnicki, Asymptotic behaviour of distributions of mRNA and protein levels in a model of stochastic gene expression, J. Math. Anal. Appl., 333 (2007), 753-769. doi: 10.1016/j.jmaa.2006.11.043.

[8]

B. M. Brown, D. Elliot and D. F. Paget, Lipschitz constants for the Bernstein polynomials of a Lipschitz continuous function, J. Approx. Theory, 49 (1987), 196-199. doi: 10.1016/0021-9045(87)90087-6.

[9]

M. H. A. Davis, Piecewise-deterministic Markov processes: A general class of nondiffusion stochastic models. With discussion, J. R. Stat. Soc. Ser. B, 46 (1984), 353-388.

[10]

M. H. A. Davis, "Lectures on Stochastic Control and Nonlinear Filtering," Tata Institute of Fundamental Research Lectures on Mathematics and Physics, 75, Published for the Tata Institute of Fundamental Research, Bombay, by Springer-Verlag, Berlin, 1984.

[11]

M. H. A. Davis, "Markov Processes and Optimization," Chapman and Hall, 1993.

[12]

R. M. Dudley, "Real Analysis and Probability," Revised reprind of the 1989 original, Cambridge Studies in Advanced Mathematics, 74, Cambridge University Press, Cambridge, 2002.

[13]

S. N. Ethier and T. G. Kurtz, "Markov Processes. Characterization and Convergence," Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1986.

[14]

W. H. Fleming and Q. Zhang, Risk-sensitive production planning of a stochastic manufacturing system, SIAM J. Control Optim., 36 (1998), 1147-1170. doi: 10.1137/S036301299631034X.

[15]

B. Hat, P. Paszek, M. Kimmel, K. Piechór and T. Lipniacki, How the number of alleles influences gene expression, J. Statist. Phys., 128 (2007), 511-533. doi: 10.1007/s10955-006-9218-4.

[16]

A. Haurie and F. Moresino, Singularly perturbed piecewise deterministic games, SIAM J. Control Optim., 47 (2008), 73-91. doi: 10.1137/050627599.

[17]

A. Haurie, A two-timescale stochastic game framework for climate change policy assessment, in "Dynamic Games: Theory and Applications" (eds. A. Haurie and G. Zaccour), GERAD 25th Anniv. Ser., 10, Springer, New York, (2005), 193-211.

[18]

E. Hille and R. S. Phillips, "Functional Analysis and Semi-Groups," rev. ed., American Mathematical Society Colloquium Publications, vol. 31, Amer. Math. Soc., Providence, R.I., 1957.

[19]

F. Hoppensteadt, Stability in systems with parameter, J. Math. Anal. Appl., 18 (1967), 129-134. doi: 10.1016/0022-247X(67)90187-4.

[20]

T. B. Kepler and T. C. Elston, Stochasticity in transcriptional regulation: Origins, consequences, and mathematical representations, Biophys. J., 81 (2001), 3116-3136. doi: 10.1016/S0006-3495(01)75949-8.

[21]

V. S. Korolyuk and A. F. Turbin, "Mathematical Foundations of the State Lumping of Large Systems," Translated from the 1978 Russian original by V. V. Zayats and Y. A. Atanov and revised by the authors, Mathematics and its Applications, 264, Kluwer Academic Publishers Group, Dordrecht, 1993.

[22]

T. G. Kurtz, Extensions of Trotter’s operator semigroup approximation theorems, J. Funct. Anal., 3 (1969), 354-375. doi: 10.1016/0022-1236(69)90031-7.

[23]

T. G. Kurtz, A limit theorem for perturbed operator semigroups with applications to random evolutions, J. Funct. Anal., 12 (1973), 55-67. doi: 10.1016/0022-1236(73)90089-X.

[24]

T. G. Kurtz, Applications of an abstract perturbation theorem to ordinary differential equations, Houston J. Math., 3 (1977), 67-82.

[25]

P. D. Lax, "Linear Algebra and Its Applications," 2 edition, Pure and Applied Mathematics (Hoboken), Wiley-Interscience [John Wiley & Sons], Hoboken, NJ, 2007.

[26]

T. Lipniacki, P. Paszek, A. Marciniak-Czochra, A. R. Brasier and M. Kimmel, Transcriptional stochasticity in gene expression, J. Theor. Biol., 238 (2006), 348-367. doi: 10.1016/j.jtbi.2005.05.032.

[27]

T. Lipniacki, P. Paszek, A. R. Brasier, B. Luxon and M. Kimmel, Stochastic regulation in early immune response, Biophys. J., 90 (2006), 725-742. doi: 10.1529/biophysj.104.056754.

[28]

G. G. Lorentz, "Bernstein Polynomials," 2nd edition, Chelsea Publishing Co., New York, 1986.

[29]

J. L. Massera, On Liapounoff's conditions of stability, Ann. Math. (2), 50 (1949), 705-721. doi: 10.2307/1969558.

[30]

R. E. Megginson, "An Introduction to Banach Space Theory," Graduate Texts in Mathematics, 183, Springer-Verlag, New York, 1998.

[31]

C. Meyer, "Matrix Analysis and Applied Linear Algebra," SIAM, Philadelphia, PA, 2000. doi: 10.1137/1.9780898719512.

[32]

J. R. Mika and J. Banasiak, "Singularly Perturbed Evolution Equations with Applications to Kinetic Theory," Series on Advances in Mathematics for Applied Sciences, 34, World Scientific Publishing Co., Inc., River Edge, NJ, 1995.

[33]

M. Rausand and A. Høyland, "System Reliability Theory: Models, Statistical Methods, and Applications," Second edition, Wiley-Interscience [John Wiley & Sons], Hoboken, NJ, 2004.

[34]

L. Saloff-Coste, Lectures on Finite Markov Chains, in "Lectures on Probability Theory and Statistics" (Saint-Flour, 1996), Lecture Notes in Mathematics, 1665, Springer, Berlin, 1997.

[35]

V. V. Sarafyan and A. V. Skorokhod, On fast switching dynamical systems, Theory Probab. Appl., 32 (1987), 595-607. doi: 10.1137/1132092.

[36]

V. V. Sarafyan, Weak convergence of measures and asymptotic behavior of solutions of the Cauchy problem for systems of differential equations with a small parameter, Theory Probab. Appl., 34 (1989), 351-357. doi: 10.1137/1134035.

[37]

S. Sethi, H. Zhang and Q. Zhang, "Average-Cost Control of Stochastic Manufacturing Systems," Stochastic Modelling and Applied Probability, 54, Springer-Verlag, New York, 2005.

[38]

M. Sova, Convergence d'opérations linéaires non bornées, (French) Rev. Roumaine Math. Pures Appl., 12 (1967), 373-389.

[39]

A. N. Tikhonov, Systems of differential equations containing small parameters in the derivatives, (Russian) Mat. Sb. N. S., 31(73) (1952), 575-586.

[40]

M. E. Taylor, "Partial Differential Equations. Basic Theory," Texts in Applied Mathematics, 23, Springer-Verlag, New York, 1996.

[41]

A. B. Vasil'eva and B. F. Butuzov, "Asymptotic Expansions of the Solutions of Singularly Perturbed Equations," (Russian) Izdat. "Nauka," Moscow, 1973.

[42]

W. Walter, "Differential and Integral Inequalities," Translated from German by Lisa Rosenblatt and Lawrence Shampine, Ergebnisse d. Mathematik u. ihrer Granzgebiete, Vol. 55, Springer-Verlag, New York-Berlin, 1970.

[43]

W. Walter, Differential inequalities and maximum principles: Theory, new methods and applications, in "Preceedings of the Second World Congress of Nonlinear Analysts, Part 8" (Athens, 1996), Nonlinear Anal., 30 (1997), 4695-4711. doi: 10.1016/S0362-546X(96)00259-3.

[44]

W. Walter, "Ordinary Differential Equations," Translated from the sixth German (1996) edition by Russell Thompson, Graduate Texts in Mathematics, 182, Readings in Mathematics, Springer-Verlag, New York, 1998.

[45]

G. G. Yin and Q. Zhang, "Continuous-Time Markov Chains and Applications. A Singular Perturbation Approach," Applications of Mathematics (New York), 37, Springer-Verlag, New York, 1998.

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