Titlebook: Global and Stochastic Analysis with Applications to Mathematical Physics; Yuri E. Gliklikh Book 2011 Springer-Verlag London Limited 2011 G

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Connections . on a manifold . with local coordinates (..,…,..) and a trivialization . of the bundle over that chart. Let ..,…,.. be the standard basis in ℝ.. Since ., this basis generates a basis in every fiber Θ., .. We obtain a smooth field of bases that will also be denoted by ..,…,... Thus every cross-sect

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Essentials from Stochastic Analysis in Linear Spacesly included in standard university courses on these subjects. This is done mainly for convenience of reference, but if necessary this material can be used as an introduction to the subject. Nevertheless the reader is assumed to be familiar with the main notions of probability theory including the no

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Stochastic Analysis on Manifoldson stochastic differential equations. A monographic presentation of various alternative aspects of and approaches to stochastic analysis on manifolds can be found in (Belopolskaya and Dalecky, .), (Elworthy, .), (Emery, .), (Hsu, .), Meyer (Lecture Notes in Mathematics 850, .; Lecture Notes in Mathe

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Mean Derivatives in Linear Spaceselson, ., .). This notion was first introduced by E. Nelson (., ., .) for the needs of so-called stochastic mechanics (see Chapter 15) but it turns out to be useful in some other problems of mathematical physics, economics, and elsewhere.

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Mean Derivatives on Manifoldsd and mean backward derivatives of .(.), if they exist, in any chart. However, from formula (7.19) it follows that for solutions of (7.18) we would obtain the mean derivatives depending on the local connector of the connection . in the chart and even on ., while for physical reasons the derivatives

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Stochastic Analysis on Groups of Diffeomorphismsn those groups of diffeomorphisms that arise in the applications to viscous hydrodynamics described in Section 16.4 below (see, e.g., Gliklikh (., ., .)). This class of equations is characterized by the fact that they involve finite-dimensional Wiener processes. It should be pointed out that a theor