An introduction to stochastic dynamics.

*(English)*Zbl 1359.60003
Cambridge Texts in Applied Mathematics. Cambridge: Cambridge University Press (ISBN 978-1-107-42820-1/pbk; 978-1-107-07539-9/hbk). xvii, 291 p. (2015).

According to the author’s words in the preface, this book is written primarily for applied mathematicians, and its goal is to provide an introduction to basic techniques for understanding solutions of stochastic differential equations.

In order to remain at an accessible level, the author has chosen a rather expository and summary style, with few proofs. Thus, the basic concepts and objects of the theory are progressively presented, by means of precise definitions, and basic theorems are stated, with references given for the proofs.

The material is illustrated, along the whole book, by a lot of simple examples, consisting mainly in explicitly solved ordinary and stochastic differential equations or systems. In particular, the Ornstein-Uhlenbeck process and the geometrical Brownian motion (given by their S.D.E.) constitute a leitmotif along the book.

Simulations are also considered, with indications on how to implant them on Matlab. A lot of figures are obtained in this way, and depicted along the text.

Lists of problems are provided, with indications for the solutions.

After an introduction containing most classical examples (e.g., the pendulum equation), the author provides a summary of basic knowledge on elementary analysis and probability theory, and then on Brownian motion and on stochastic differential equations. Then, examples of deterministic quantities arising from stochastic dynamics are discussed, in particular mean exist times and harmonic measures (called “escape probabilities”). Simple examples are studied in detailed.

Ordinary and stochastic differential systems are then discussed to some extent. In particular, the comparison between the dynamics about a singularity and the linearized corresponding system is considered, and also is the decomposition in stable-unstable-neutral submanifolds, associated with Lyapunov exponents. A way of reducing a S.D.E. into an O.D.E. with random coefficients (“R.D.E.”) is also given.

The random dynamics in this book are driven either by Brownian motion or by Lévy processes, actually by \(\alpha\)-stable ones. For the latter, \(\alpha\)-stable Lévy processes and their Fourier transform are discussed along the last chapter of the book, together with generators and Dirichlet problems.

In order to remain at an accessible level, the author has chosen a rather expository and summary style, with few proofs. Thus, the basic concepts and objects of the theory are progressively presented, by means of precise definitions, and basic theorems are stated, with references given for the proofs.

The material is illustrated, along the whole book, by a lot of simple examples, consisting mainly in explicitly solved ordinary and stochastic differential equations or systems. In particular, the Ornstein-Uhlenbeck process and the geometrical Brownian motion (given by their S.D.E.) constitute a leitmotif along the book.

Simulations are also considered, with indications on how to implant them on Matlab. A lot of figures are obtained in this way, and depicted along the text.

Lists of problems are provided, with indications for the solutions.

After an introduction containing most classical examples (e.g., the pendulum equation), the author provides a summary of basic knowledge on elementary analysis and probability theory, and then on Brownian motion and on stochastic differential equations. Then, examples of deterministic quantities arising from stochastic dynamics are discussed, in particular mean exist times and harmonic measures (called “escape probabilities”). Simple examples are studied in detailed.

Ordinary and stochastic differential systems are then discussed to some extent. In particular, the comparison between the dynamics about a singularity and the linearized corresponding system is considered, and also is the decomposition in stable-unstable-neutral submanifolds, associated with Lyapunov exponents. A way of reducing a S.D.E. into an O.D.E. with random coefficients (“R.D.E.”) is also given.

The random dynamics in this book are driven either by Brownian motion or by Lévy processes, actually by \(\alpha\)-stable ones. For the latter, \(\alpha\)-stable Lévy processes and their Fourier transform are discussed along the last chapter of the book, together with generators and Dirichlet problems.

Reviewer: Jacques Franchi (Strasbourg)

##### MSC:

60-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to probability theory |

60H30 | Applications of stochastic analysis (to PDEs, etc.) |

60H40 | White noise theory |

37H99 | Random dynamical systems |

60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |

37H10 | Generation, random and stochastic difference and differential equations |

60J65 | Brownian motion |

60G51 | Processes with independent increments; Lévy processes |