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This book is devoted to unstable solutions of stochastic
differential equations (SDEs). Despite the huge interest in the
theory of SDEs, this book is the first to present a systematic
study of the instability and asymptotic behavior of the
corresponding unstable stochastic systems. The limit theorems
contained in the book are not merely of purely mathematical value;
rather, they also have practical value. Instability or violations
of stability are noted in many phenomena, and the authors attempt
to apply mathematical and stochastic methods to deal with them. The
main goals include exploration of Brownian motion in environments
with anomalies and study of the motion of the Brownian particle in
layered media. A fairly wide class of continuous Markov processes
is obtained in the limit. It includes Markov processes with
discontinuous transition densities, processes that are not
solutions of any Ito's SDEs, and the Bessel diffusion process. The
book is self-contained, with presentation of definitions and
auxiliary results in an Appendix. It will be of value for
specialists in stochastic analysis and SDEs, as well as for
researchers in other fields who deal with unstable systems and
practitioners who apply stochastic models to describe phenomena of
instability.
This book is devoted to unstable solutions of stochastic
differential equations (SDEs). Despite the huge interest in the
theory of SDEs, this book is the first to present a systematic
study of the instability and asymptotic behavior of the
corresponding unstable stochastic systems. The limit theorems
contained in the book are not merely of purely mathematical value;
rather, they also have practical value. Instability or violations
of stability are noted in many phenomena, and the authors attempt
to apply mathematical and stochastic methods to deal with them. The
main goals include exploration of Brownian motion in environments
with anomalies and study of the motion of the Brownian particle in
layered media. A fairly wide class of continuous Markov processes
is obtained in the limit. It includes Markov processes with
discontinuous transition densities, processes that are not
solutions of any Ito's SDEs, and the Bessel diffusion process. The
book is self-contained, with presentation of definitions and
auxiliary results in an Appendix. It will be of value for
specialists in stochastic analysis and SDEs, as well as for
researchers in other fields who deal with unstable systems and
practitioners who apply stochastic models to describe phenomena of
instability.
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