Elements of stochastic dynamics (Q2820473)

This book is explicitly destined for researchers and students in various areas of science, such as engineering, applied physics or chemistry, aeronautics, meteorology, economics, biology. As it is not intended for mathematicians, it is not written in a rigorous mathematical style, and proofs are mostly omitted. The emphasis is rather on applications to physical modeling and on numerous examples, with a series of simulations.NEWLINENEWLINETo keep their book relatively self-contained, the authors present basic concepts in three introducing chapters, in a rather informal way, and partly from the view point of physicists on dynamics and stochastic analysis. In this way random variables, stochastic processes, and particularly stationary, Markov, Gaussian, Wiener and one-dimensional diffusion processes are introduced, with some indications on stochastic calculus; then linear systems are presented.NEWLINENEWLINEThe dynamics considered in this book are defined and described by some differential equation, possibly in several dimensions, and possibly under a Hamiltonian presentation. Both scholastic and more practically justified (arising from some concrete frameworks) types of equation are considered.NEWLINENEWLINEThe equations under consideration are mostly of the following type: NEWLINE\[NEWLINE\ddot X_t+f(\dot X_t,X_t)=a\dot W_t+b\dot B_tX_t+c\dot\beta_t\dot X_t\quad (X_t\in\mathbb R^d),NEWLINE\]NEWLINE with a given \(f\) and given Brownian excitations \(W_t\in\mathbb R^d\), \(B_t\in\mathbb R\), \(\beta_t\in\mathbb R\).NEWLINENEWLINEExact stationary solutions are computed, in some cases where they exist, mostly one-dimensional ones.NEWLINENEWLINEA central chapter is devoted to getting approximate solutions. Some methods are described, which should reasonably give approximate solutions to some non-exactly solvable equations. The main such method is by averaging the possibly random coefficients of the system. An alternative method consists in simplifying some coefficients of the equation, in a way which respects some constraints. Both methods are intended to result in a much more tractable equation, mostly linear or one-dimensional. The control of the quality of these approximations is generally not addressed, but by some simulations, for some examples. Another way of getting a simplified equation is to focus on the equation governing a one-dimensional projection (the total energy) of the initial solution. This type of simplification can be followed by the previous ones. Getting an exactly solvable approaching equation and thus an approximate stationary density is here the main aim.NEWLINENEWLINENext, the authors discuss some specific problems. For each type of problem addressed, they generically proceed by first giving some definitions and basic properties, without proof, and explaining how the deterministic case is handled. Then, they introduce randomness in the equation, either by means of some (mostly Gaussian) excitation in the right hand side, or/and by allowing some coefficients of the equation to depend on the randomness, mostly linearly as in the above displayed equation. They discuss the (rare, mostly in the only linear framework) eventual existence of exact analytical solutions or invariant laws for the randomized equation. After this, they apply the methods introduced before, in order to get a reasonable approximate solution, together with its stationary density. Finally, in the major part, the authors handle various examples where the preceding can be implemented, in a rather detailed way. Moreover, for each subject under study, a list of alternative examples is given, as a series of exercises left to the reader.NEWLINENEWLINEIn this way, the following items are successively considered and discussed. {\parindent=0.7cm \begin{itemize}\item[--] Stability and asymptotic stability: Lyapunov exponents are computed in the linear case; in the non-linear case the equation is linearized near an equilibrium point. \item[--] Bifurcation, i.e., phase transition, as in the equation some deterministic parameter varies: the study focusses on the cases where a sensible change occurs either in the sign of the first Lyapunov exponent or in the shape or integrability of an approximate stationary density. \item[--] First passage problems: the purpose is here to estimating the moments of some hitting time (typically, of some total energy level) of the solution of the equation. NEWLINENEWLINE\end{itemize}}