Convergence of One-Parameter Operator Semigroups: In Models of Mathematical Biology and Elsewhere (New Mathematical Monographs, Series Number 30)

This book presents a detailed and contemporary account of the classical theory of convergence of semigroups and its more recent development treating the case where the limit semigroup, in contrast to the approximating semigroups, acts merely on a subspace of the original Banach space (this is the case, for example, with singular perturbations). The author demonstrates the far-reaching applications of this theory using real examples from various branches of pure and applied mathematics, with a particular emphasis on mathematical biology. The book may serve as a useful reference, containing a significant number of new results ranging from the analysis of fish populations to signaling pathways in living cells. It comprises many short chapters, which allows readers to pick and choose those topics most relevant to them, and it contains 160 end-of-chapter exercises so that readers can test their understanding of the material as they go along. Contents......Page 8 Preface......Page 12 1 Semigroups of Operators and Cosine Operator Functions......Page 16 PART I REGULAR CONVERGENCE......Page 26 2 The First Convergence Theorem......Page 28 3 Continuous Dependence on Boundary Conditions......Page 32 4 Semipermeable Membrane......Page 39 5 Convergence of Forms......Page 47 6 Uniform Approximation of Semigroups......Page 55 7 Convergence of Resolvents......Page 60 8 (Regular) Convergence of Semigroups......Page 66 9 A Queue in Heavy Traffic......Page 71 10 Elastic Brownian Motions......Page 75 11 Back to the Membrane......Page 80 12 Telegraph with Small Parameter......Page 85 13 Minimal Markov Chains......Page 89 14 Outside of the Regularity Space: A Bird’s-Eye View......Page 96 15 Hasegawa’s Condition......Page 100 16 Blackwell’s Example......Page 105 17 Wright’s Diffusion......Page 111 18 Discrete-Time Approximation......Page 115 19 Discrete-Time Approximation: Examples......Page 120 20 Back to Wright’s Diffusion......Page 127 21 Kingman’s n-Coalescent......Page 131 22 The Feynman–Kac Formula......Page 136 23 The Two-Dimensional Dirac Equation......Page 143 24 Approximating Spaces......Page 147 25 Boundedness, Stabilization......Page 151 PART II IRREGULAR CONVERGENCE......Page 158 26 First Examples......Page 160 27 Extremely Strong Genetic Drift......Page 167 28 The Nature of Irregular Convergence......Page 172 29 Irregular Convergence Is Preserved Under Bounded Perturbations......Page 178 30 Stein’s Model......Page 181 31 Uniformly Holomorphic Semigroups......Page 186 32 Asymptotic Behavior of Semigroups......Page 192 33 Fast Neurotransmitters......Page 204 34 Fast Neurotransmitters II......Page 212 35 From Diffusions on Graphs to Markov Chains and Back Again......Page 218 36 Semilinear Equations, Early Cancer Modeling......Page 225 37 Coagulation-Fragmentation Equation......Page 234 38 Homogenization Theorem......Page 243 39 Shadow Systems......Page 251 40 Kinases......Page 256 41 Uniformly Differentiable Semigroups......Page 265 42 Kurtz’s Singular Perturbation Theorem......Page 268 43 A Singularly Perturbed Markov Chain......Page 273 44 A Tikhonov-Type Theorem......Page 278 45 Fast Motion and Frequent Jumps Theorems for Piecewise Deterministic Processes......Page 286 46 Models of Gene Regulation and Gene Expression......Page 296 47 Oligopolies, Manufacturing Systems, and Climate Changes......Page 302 48 Convex Combinations of Feller Generators......Page 307 49 The Dorroh Theorem and the Volkonskii Formula......Page 314 50 Convex Combinations in Biological Models......Page 318 51 Recombination......Page 326 52 Recombination (Continued)......Page 333 53 Averaging Principle of Freidlin and Wentzell: Khasminskii’s Example......Page 342 54 Comparing Semigroups......Page 350 55 Relations to Asymptotic Analysis......Page 356 56 Greiner’s Theorem......Page 360 57 Fish Population Dynamics and Convex Combination of Boundary Conditions......Page 367 58 Averaging Principle of Freidlin and Wentzell: Emergence of Transmission Conditions......Page 376 59 Averaging Principle Continued: L1-Setting......Page 385 PART III CONVERGENCE OF COSINE FAMILIES......Page 396 60 Regular Convergence of Cosine Families......Page 398 61 Cosines Converge in a Regular Way......Page 405 PART IV APPENDIXES......Page 410 62 Appendix A: Representation Theorem for the Laplace Transform......Page 412 63 Appendix B: Measurable Cosine Functions Are Continuous......Page 420 References......Page 429 Index......Page 450 The study of model spaces, the closed invariant subspaces of the backward shift operator, is a vast area of research with connections to complex analysis, operator theory and functional analysis. This self-contained text is the ideal introduction for newcomers to the field. It sets out the basic ideas and quickly takes the reader through the history of the subject before ending up at the frontier of mathematical analysis. Open questions point to potential areas of future research, offering plenty of inspiration to graduate students wishing to advance further Written by a leading expert in the field, this book presents the classical theory of convergence of semigroups and then uses real examples to show how it can be applied to models of mathematical biology as well as other branches of mathematics.