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.
Author(s): Adam Bobrowski
Series: New Mathematical Monographs 30
Publisher: Cambridge University Press
Year: 2016
Language: English
Pages: 454
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
