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Stochastic analysis on manifolds. (English) Zbl 0994.58019
Graduate Studies in Mathematics. 38. Providence, RI: American Mathematical Society (AMS). xiv, 281 p. (2002).
The purpose of this fine book is to explore connections between Brownian motion and analysis in the area of differential geometry, from a probabilist’s point of view.
The background of differential geometry which is needed is progressively explained along the text, in a synthetic and comprehensive way.
The author has succeeded in developing in a clear way, together the base of Brownian motion theory on manifolds, and recent achievements of this theory.
Moreover the overlap of this new excellent book with the previous books on stochastic analysis is not very significant, illustrating the richness of this active domain of research.
Starting with Euclidean stochastic differential equations and basic stochastic differential geometry, the author deals then mainly with: horizontal lifts of semimartingales, radial processes, heat kernels, Laplacians, short time asymptotics, Brownian bridge, angular convergence, spectral gap, probabilistic proofs of the Gauss-Bonnet-Chern and Atiyah-Singer index theorems, and ends with a basic course on the analysis of the path space over a compact manifold, including integration by parts and logarithmic Sobolev inequality.

58J65 Diffusion processes and stochastic analysis on manifolds
60J60 Diffusion processes
60J65 Brownian motion
58-02 Research exposition (monographs, survey articles) pertaining to global analysis
60H07 Stochastic calculus of variations and the Malliavin calculus
60-02 Research exposition (monographs, survey articles) pertaining to probability theory