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Presents recent developments in the areas of differential
equations, dynamical systems, and control of finke and infinite
dimensional systems. Focuses on current trends in differential
equations and dynamical system research-from Darameterdependence of
solutions to robui control laws for inflnite dimensional systems.
Stochastic differential equations, and Hoermander form
representations of diffusion operators, can determine a linear
connection associated to the underlying (sub)-Riemannian structure.
This is systematically described, together with its invariants, and
then exploited to discuss qualitative properties of stochastic
flows, and analysis on path spaces of compact manifolds with
diffusion measures. This should be useful to stochastic analysts,
especially those with interests in stochastic flows, infinite
dimensional analysis, or geometric analysis, and also to
researchers in sub-Riemannian geometry. A basic background in
differential geometry is assumed, but the construction of the
connections is very direct and itself gives an intuitive and
concrete introduction. Knowledge of stochastic analysis is also
assumed for later chapters.
The aims of this book, originally published in 1982, are to give an
understanding of the basic ideas concerning stochastic differential
equations on manifolds and their solution flows, to examine the
properties of Brownian motion on Riemannian manifolds when it is
constructed using the stochiastic development and to indicate some
of the uses of the theory. The author has included two appendices
which summarise the manifold theory and differential geometry
needed to follow the development; coordinate-free notation is used
throughout. Moreover, the stochiastic integrals used are those
which can be obtained from limits of the Riemann sums, thereby
avoiding much of the technicalities of the general theory of
processes and allowing the reader to get a quick grasp of the
fundamental ideas of stochastic integration as they are needed for
a variety of applications.
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