## An Introduction to Stochastic DynamicsThe mathematical theory of stochastic dynamics has become an important tool in the modeling of uncertainty in many complex biological, physical, and chemical systems and in engineering applications - for example, gene regulation systems, neuronal networks, geophysical flows, climate dynamics, chemical reaction systems, nanocomposites, and communication systems. It is now understood that these systems are often subject to random influences, which can significantly impact their evolution. This book serves as a concise introductory text on stochastic dynamics for applied mathematicians and scientists. Starting from the knowledge base typical for beginning graduate students in applied mathematics, it introduces the basic tools from probability and analysis and then develops for stochastic systems the properties traditionally calculated for deterministic systems. The book's final chapter opens the door to modeling in non-Gaussian situations, typical of many real-world applications. Rich with examples, illustrations, and exercises with solutions, this book is also ideal for self-study. |

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### Contents

Introduction | 1 |

Background in Analysis and Probability | 12 |

Noise | 59 |

A Crash Course in Stochastic Differential Equations | 61 |

2 | 65 |

1 | 77 |

22 | 85 |

Deterministic Quantities for Stochastic Dynamics | 99 |

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### Common terms and phrases

a-stable a-stable Lévy motion Brownian motion Bt called Cauchy principal value Chapter cocycle Consider convergence defined definition denoted distribution domain eigenspace eigenvalues equilibrium point escape probability Example Figure finite variation Fokker-Planck equation Gaussian random variable Hölder impact of noise independent inequality initial condition integrand interval invariant manifolds Itˆo Itˆo’s formula Itó Itô integral Itô's formula jump measure L´evy motion linear system Lyapunov exponents mean exit mean square non-Gaussian nonnegative parameter partial differential equations phase portrait probability density function probability space random dynamical system random invariant random numbers random variable Riemann-Stieltjes integral sample paths scalar Brownian motion scalar SDE SDE dX SDE system Section sequence solution mapping solution paths solution process stable random variable stochastic differential equations stochastic dynamics stochastic integral stochastic process stochastic system Stratonovich integral Stratonovich SDE symmetric a-stable unstable manifolds vector zero