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Transformation semigroups and their applications

Katarzyna Pichór, Ryszard Rudnicki

Abstract

In this chapter we present transformation semigroups and their applications. We begin with Klein's approach to geometry based on invariants of transformation groups. Then we present symmetry groups in chemistry and in classical mechanics. Next we introduce one-parameter semigroups of transformations and their applications in ergodic theory. Our main subject are one-parameter semigroups of operators, in particular stochastic semigroups. We present general results on their existence and long-time behaviour. We also give examples of one-parameter semigroups related to Markov chains, diffusion and processes with jumps. We focus on the applications of semigroups of operators in biology. Among other things, we study models of: DNA evolution; growth of erythrocyte population; gene expression; cell cycle; the movement of bacteria and insects. We also consider models with stochastic noise and different population models.

Transformation semigroups and their applications

Abstract

In this chapter we present transformation semigroups and their applications. We begin with Klein's approach to geometry based on invariants of transformation groups. Then we present symmetry groups in chemistry and in classical mechanics. Next we introduce one-parameter semigroups of transformations and their applications in ergodic theory. Our main subject are one-parameter semigroups of operators, in particular stochastic semigroups. We present general results on their existence and long-time behaviour. We also give examples of one-parameter semigroups related to Markov chains, diffusion and processes with jumps. We focus on the applications of semigroups of operators in biology. Among other things, we study models of: DNA evolution; growth of erythrocyte population; gene expression; cell cycle; the movement of bacteria and insects. We also consider models with stochastic noise and different population models.
Paper Structure (31 sections, 27 theorems, 160 equations, 5 figures, 2 tables)

This paper contains 31 sections, 27 theorems, 160 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Transformation semigroups everywhere
  3. Basic definitions and examples
  4. Transformation groups in geometry
  5. Symmetry groups in chemistry
  6. Applications to Newtonian mechanics
  7. Measure-preserving dynamical systems
  8. Stochastic semigroups
  9. Semigroups of operators
  10. Stochastic and substochastic semigroups
  11. Semigroups related to Markov chains
  12. Diffusion semigroup
  13. Processes with jumps
  14. Diffusion process with random switching
  15. Processes with non-random moments of jumps
  16. ...and 16 more sections

Key Result

theorem 1

Let $E$ be a Banach space, and let $A$ be a linear operator on $E$ with domain $\mathfrak D(A)$. The operator $A$ is a generator of a strongly continuous semigroup of contractions if and only if the following conditions are satisfied:

Figures (5)

  • Figure 1: The map $Q$ transfers the line $l$ to the point $B$.
  • Figure 2: Rotation $S_l$ around the axis $l$ by an angle of $120^{\circ}$ does not change an ammonia molecule. The mirror plane $\sigma_2$ is marked on the right picture.
  • Figure 3: Example of a trajectory of a diffusion process with random switches. In this case $X=\mathbb R^2\times\{0,1\}$ and $t_0,t_1,t_2,t_3$ are the successive switching times.
  • Figure 4: Left: classical billiards -- angles of incidence are equal to the angles of reflection. Right: stochastic billiards. Regular and diffuse reflection: $\mathcal{V}$ -- the specular reflection, thin vectors -- diffuse reflection.
  • Figure 5: Gene expression scheme with jumps on the boundary.

Theorems & Definitions (31)

  • theorem 1: Hille--Yosida
  • remark 1
  • remark 2
  • theorem 2
  • theorem 3
  • theorem 4
  • remark 3
  • theorem 5
  • theorem 6
  • lemma 1
  • ...and 21 more