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On the Geometry of Diffusion Operators and Stochastic Flows

2007-01-05 Mathematics
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Release : 2007-01-05
Genre : Mathematics
Kind : eBook
Book Rating : 220/5 ( reviews)

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Book Synopsis On the Geometry of Diffusion Operators and Stochastic Flows by : K.D. Elworthy

Download or read book On the Geometry of Diffusion Operators and Stochastic Flows written by K.D. Elworthy. This book was released on 2007-01-05. Available in PDF, EPUB and Kindle. Book excerpt: 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.

On the Geometry of Diffusion Operators and Stochastic Flows

2014-09-01
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Release : 2014-09-01
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Kind : eBook
Book Rating : 460/5 ( reviews)

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Book Synopsis On the Geometry of Diffusion Operators and Stochastic Flows by : K. D. Elworthy

Download or read book On the Geometry of Diffusion Operators and Stochastic Flows written by K. D. Elworthy. This book was released on 2014-09-01. Available in PDF, EPUB and Kindle. Book excerpt:

The Geometry of Filtering

2010-11-27 Mathematics
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Release : 2010-11-27
Genre : Mathematics
Kind : eBook
Book Rating : 76X/5 ( reviews)

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Book Synopsis The Geometry of Filtering by : K. David Elworthy

Download or read book The Geometry of Filtering written by K. David Elworthy. This book was released on 2010-11-27. Available in PDF, EPUB and Kindle. Book excerpt: Filtering is the science of nding the law of a process given a partial observation of it. The main objects we study here are di usion processes. These are naturally associated with second-order linear di erential operators which are semi-elliptic and so introduce a possibly degenerate Riemannian structure on the state space. In fact, much of what we discuss is simply about two such operators intertwined by a smooth map, the \projection from the state space to the observations space", and does not involve any stochastic analysis. From the point of view of stochastic processes, our purpose is to present and to study the underlying geometric structure which allows us to perform the ltering in a Markovian framework with the resulting conditional law being that of a Markov process which is time inhomogeneous in general. This geometry is determined by the symbol of the operator on the state space which projects to a symbol on the observation space. The projectible symbol induces a (possibly non-linear and partially de ned) connection which lifts the observation process to the state space and gives a decomposition of the operator on the state space and of the noise. As is standard we can recover the classical ltering theory in which the observations are not usually Markovian by application of the Girsanov- Maruyama-Cameron-Martin Theorem. This structure we have is examined in relation to a number of geometrical topics.

Stochastic Processes, Physics and Geometry: New Interplays. I

2000 Mathematics
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Release : 2000
Genre : Mathematics
Kind : eBook
Book Rating : 593/5 ( reviews)

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Book Synopsis Stochastic Processes, Physics and Geometry: New Interplays. I by : Sergio Albeverio

Download or read book Stochastic Processes, Physics and Geometry: New Interplays. I written by Sergio Albeverio. This book was released on 2000. Available in PDF, EPUB and Kindle. Book excerpt: This volume and "IStochastic Processes, Physics and Geometry: New Interplays II" present state-of-the-art research currently unfolding at the interface between mathematics and physics. Included are select articles from the international conference held in Leipzig (Germany) in honor of Sergio Albeverio's sixtieth birthday. The theme of the conference, "Infinite Dimensional (Stochastic) Analysis and Quantum Physics", was chosen to reflect Albeverio's wide-ranging scientific interests. The articles in these books reflect that broad range of interests and provide a detailed overview highlighting the deep interplay among stochastic processes, mathematical physics, and geometry. The contributions are written by internationally recognized experts in the fields of stochastic analysis, linear and nonlinear (deterministic and stochastic) PDEs, infinite dimensional analysis, functional analysis, commutative and noncommutative probability theory, integrable systems, quantum and statistical mechanics, geometric quantization, and neural networks. Also included are applications in biology and other areas. Most of the contributions are high-level research papers. However, there are also some overviews on topics of general interest. The articles selected for publication in these volumes were specifically chosen to introduce readers to advanced topics, to emphasize interdisciplinary connections, and to stress future research directions. Volume I contains contributions from invited speakers; Volume II contains additional contributed papers. Members of the Canadian Mathematical Society may order at the AMS member price.

An Introduction to the Geometry of Stochastic Flows

2004 Mathematics
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Release : 2004
Genre : Mathematics
Kind : eBook
Book Rating : 817/5 ( reviews)

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Book Synopsis An Introduction to the Geometry of Stochastic Flows by : Fabrice Baudoin

Download or read book An Introduction to the Geometry of Stochastic Flows written by Fabrice Baudoin. This book was released on 2004. Available in PDF, EPUB and Kindle. Book excerpt: This book aims to provide a self-contained introduction to the local geometry of the stochastic flows associated with stochastic differential equations. It stresses the view that the local geometry of any stochastic flow is determined very precisely and explicitly by a universal formula referred to as the Chen-Strichartz formula. The natural geometry associated with the Chen-Strichartz formula is the sub-Riemannian geometry whose main tools are introduced throughout the text. By using the connection between stochastic flows and partial differential equations, we apply this point of view of the study of hypoelliptic operators written in Hormander's form.

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