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 & 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.  doi: 10.1007/s00028-009-0009-7.  Google Scholar

[2]

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

[3]

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

[4]

A. Bobrowski, Degenerate convergence of semigroups related to a model of stochastic gene expression,, Semigroup Forum, 73 (2006), 345.   Google Scholar

[5]

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

[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.  doi: 10.4064/sm189-3-6.  Google Scholar

[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.  doi: 10.1016/j.jmaa.2006.11.043.  Google Scholar

[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.  doi: 10.1016/0021-9045(87)90087-6.  Google Scholar

[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.   Google Scholar

[10]

M. H. A. Davis, "Lectures on Stochastic Control and Nonlinear Filtering,", Tata Institute of Fundamental Research Lectures on Mathematics and Physics, 75 (1984).   Google Scholar

[11]

M. H. A. Davis, "Markov Processes and Optimization,", Chapman and Hall, (1993).   Google Scholar

[12]

R. M. Dudley, "Real Analysis and Probability,", Revised reprind of the 1989 original, 74 (1989).   Google Scholar

[13]

S. N. Ethier and T. G. Kurtz, "Markov Processes. Characterization and Convergence,", Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, (1986).   Google Scholar

[14]

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

[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.  doi: 10.1007/s10955-006-9218-4.  Google Scholar

[16]

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

[17]

A. Haurie, A two-timescale stochastic game framework for climate change policy assessment,, in, 10 (2005), 193.   Google Scholar

[18]

E. Hille and R. S. Phillips, "Functional Analysis and Semi-Groups,", rev. ed., (1957).   Google Scholar

[19]

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

[20]

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

[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, 264 (1978).   Google Scholar

[22]

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

[23]

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

[24]

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

[25]

P. D. Lax, "Linear Algebra and Its Applications,", 2 edition, (2007).   Google Scholar

[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.  doi: 10.1016/j.jtbi.2005.05.032.  Google Scholar

[27]

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

[28]

G. G. Lorentz, "Bernstein Polynomials,", 2nd edition, (1986).   Google Scholar

[29]

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

[30]

R. E. Megginson, "An Introduction to Banach Space Theory,", Graduate Texts in Mathematics, 183 (1998).   Google Scholar

[31]

C. Meyer, "Matrix Analysis and Applied Linear Algebra,", SIAM, (2000).  doi: 10.1137/1.9780898719512.  Google Scholar

[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 (1995).   Google Scholar

[33]

M. Rausand and A. Høyland, "System Reliability Theory: Models, Statistical Methods, and Applications,", Second edition, (2004).   Google Scholar

[34]

L. Saloff-Coste, Lectures on Finite Markov Chains,, in, 1665 (1996).   Google Scholar

[35]

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

[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.  doi: 10.1137/1134035.  Google Scholar

[37]

S. Sethi, H. Zhang and Q. Zhang, "Average-Cost Control of Stochastic Manufacturing Systems,", Stochastic Modelling and Applied Probability, 54 (2005).   Google Scholar

[38]

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

[39]

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

[40]

M. E. Taylor, "Partial Differential Equations. Basic Theory,", Texts in Applied Mathematics, 23 (1996).   Google Scholar

[41]

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

[42]

W. Walter, "Differential and Integral Inequalities,", Translated from German by Lisa Rosenblatt and Lawrence Shampine, (1970).   Google Scholar

[43]

W. Walter, Differential inequalities and maximum principles: Theory, new methods and applications,, in, 30 (1997), 4695.  doi: 10.1016/S0362-546X(96)00259-3.  Google Scholar

[44]

W. Walter, "Ordinary Differential Equations,", Translated from the sixth German (1996) edition by Russell Thompson, 182 (1996).   Google Scholar

[45]

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

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.  doi: 10.1007/s00028-009-0009-7.  Google Scholar

[2]

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

[3]

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

[4]

A. Bobrowski, Degenerate convergence of semigroups related to a model of stochastic gene expression,, Semigroup Forum, 73 (2006), 345.   Google Scholar

[5]

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

[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.  doi: 10.4064/sm189-3-6.  Google Scholar

[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.  doi: 10.1016/j.jmaa.2006.11.043.  Google Scholar

[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.  doi: 10.1016/0021-9045(87)90087-6.  Google Scholar

[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.   Google Scholar

[10]

M. H. A. Davis, "Lectures on Stochastic Control and Nonlinear Filtering,", Tata Institute of Fundamental Research Lectures on Mathematics and Physics, 75 (1984).   Google Scholar

[11]

M. H. A. Davis, "Markov Processes and Optimization,", Chapman and Hall, (1993).   Google Scholar

[12]

R. M. Dudley, "Real Analysis and Probability,", Revised reprind of the 1989 original, 74 (1989).   Google Scholar

[13]

S. N. Ethier and T. G. Kurtz, "Markov Processes. Characterization and Convergence,", Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, (1986).   Google Scholar

[14]

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

[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.  doi: 10.1007/s10955-006-9218-4.  Google Scholar

[16]

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

[17]

A. Haurie, A two-timescale stochastic game framework for climate change policy assessment,, in, 10 (2005), 193.   Google Scholar

[18]

E. Hille and R. S. Phillips, "Functional Analysis and Semi-Groups,", rev. ed., (1957).   Google Scholar

[19]

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

[20]

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

[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, 264 (1978).   Google Scholar

[22]

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

[23]

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

[24]

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

[25]

P. D. Lax, "Linear Algebra and Its Applications,", 2 edition, (2007).   Google Scholar

[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.  doi: 10.1016/j.jtbi.2005.05.032.  Google Scholar

[27]

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

[28]

G. G. Lorentz, "Bernstein Polynomials,", 2nd edition, (1986).   Google Scholar

[29]

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

[30]

R. E. Megginson, "An Introduction to Banach Space Theory,", Graduate Texts in Mathematics, 183 (1998).   Google Scholar

[31]

C. Meyer, "Matrix Analysis and Applied Linear Algebra,", SIAM, (2000).  doi: 10.1137/1.9780898719512.  Google Scholar

[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 (1995).   Google Scholar

[33]

M. Rausand and A. Høyland, "System Reliability Theory: Models, Statistical Methods, and Applications,", Second edition, (2004).   Google Scholar

[34]

L. Saloff-Coste, Lectures on Finite Markov Chains,, in, 1665 (1996).   Google Scholar

[35]

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

[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.  doi: 10.1137/1134035.  Google Scholar

[37]

S. Sethi, H. Zhang and Q. Zhang, "Average-Cost Control of Stochastic Manufacturing Systems,", Stochastic Modelling and Applied Probability, 54 (2005).   Google Scholar

[38]

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

[39]

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

[40]

M. E. Taylor, "Partial Differential Equations. Basic Theory,", Texts in Applied Mathematics, 23 (1996).   Google Scholar

[41]

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

[42]

W. Walter, "Differential and Integral Inequalities,", Translated from German by Lisa Rosenblatt and Lawrence Shampine, (1970).   Google Scholar

[43]

W. Walter, Differential inequalities and maximum principles: Theory, new methods and applications,, in, 30 (1997), 4695.  doi: 10.1016/S0362-546X(96)00259-3.  Google Scholar

[44]

W. Walter, "Ordinary Differential Equations,", Translated from the sixth German (1996) edition by Russell Thompson, 182 (1996).   Google Scholar

[45]

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

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