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Differential equations --- Operational research. Game theory --- differentiaalvergelijkingen --- Laplacetransformatie --- stochastische analyse --- kansrekening

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The purpose of these notes is to explore some simple relations between Markovian path and loop measures, the Poissonian ensembles of loops they determine, their occupation fields, uniform spanning trees, determinants, and Gaussian Markov fields such as the free field. These relations are first studied in complete generality for the finite discrete setting, then partly generalized to specific examples in infinite and continuous spaces.

Differential equations --- Operational research. Game theory --- differentiaalvergelijkingen --- Laplacetransformatie --- stochastische analyse --- kansrekening

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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.

Filters (Mathematics). --- Gaussian processes. --- Markov processes. --- Stochastic processes --- Filters (Mathematics) --- Gaussian processes --- Markov processes --- Mathematics --- Engineering & Applied Sciences --- Physical Sciences & Mathematics --- Applied Mathematics --- Geometry --- Mathematical Statistics --- Diffusion processes. --- Analysis, Markov --- Chains, Markov --- Markoff processes --- Markov analysis --- Markov chains --- Markov models --- Models, Markov --- Processes, Markov --- Mathematics. --- Global analysis (Mathematics). --- Manifolds (Mathematics). --- Numerical analysis. --- Differential geometry. --- Probabilities. --- Numerical Analysis. --- Probability Theory and Stochastic Processes. --- Differential Geometry. --- Global Analysis and Analysis on Manifolds. --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk --- Differential geometry --- Mathematical analysis --- Geometry, Differential --- Topology --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Math --- Science --- Distribution (Probability theory. --- Global differential geometry. --- Global analysis. --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities

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This volume presents original research articles and extended surveys related to the mathematical interest and work of Jean-Michel Bismut. His outstanding contributions to probability theory and global analysis on manifolds have had a profound impact on several branches of mathematics in the areas of control theory, mathematical physics and arithmetic geometry. Contributions by: K. Behrend N. Bergeron S. K. Donaldson J. Dubédat B. Duplantier G. Faltings E. Getzler G. Kings R. Mazzeo J. Millson C. Moeglin W. Müller R. Rhodes D. Rössler S. Sheffield A. Teleman G. Tian K-I. Yoshikawa H. Weiss W. Werner The collection is a valuable resource for graduate students and researchers in these fields.

Mathematics. --- Algebraic geometry. --- Global analysis (Mathematics). --- Manifolds (Mathematics). --- Number theory. --- Probabilities. --- Global Analysis and Analysis on Manifolds. --- Algebraic Geometry. --- Probability Theory and Stochastic Processes. --- Number Theory. --- Geometry --- Mathematics --- Probabilities --- Control theory --- Math --- Science --- Global analysis. --- Geometry, algebraic. --- Distribution (Probability theory. --- Number study --- Numbers, Theory of --- Algebra --- Distribution functions --- Frequency distribution --- Characteristic functions --- Algebraic geometry --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk --- Geometry, Differential --- Topology --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic

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This volume presents original research articles and extended surveys related to the mathematical interest and work of Jean-Michel Bismut. His outstanding contributions to probability theory and global analysis on manifolds have had a profound impact on several branches of mathematics in the areas of control theory, mathematical physics and arithmetic geometry. Contributions by: K. Behrend N. Bergeron S. K. Donaldson J. Dubédat B. Duplantier G. Faltings E. Getzler G. Kings R. Mazzeo J. Millson C. Moeglin W. Müller R. Rhodes D. Rössler S. Sheffield A. Teleman G. Tian K-I. Yoshikawa H. Weiss W. Werner The collection is a valuable resource for graduate students and researchers in these fields.

Number theory --- Algebraic geometry --- Differential geometry. Global analysis --- Geometry --- Operational research. Game theory --- Probability theory --- Mathematics --- landmeetkunde --- topologie (wiskunde) --- waarschijnlijkheidstheorie --- stochastische analyse --- statistiek --- wiskunde --- kansrekening --- getallenleer --- geometrie