Ikeda, Nobuyuki; Watanabe, Shinzo Stochastic differential equations and diffusion processes. 2nd ed. (English) Zbl 0684.60040 North-Holland Mathematical Library, 24. Amsterdam etc.: North-Holland; Tokyo: Kodansha Ltd. xvi, 555 p. $ 147.25; Dfl. 280.00 (1989). The first edition of the book has been extensively reviewed in Zbl 0495.60005. In the mean time it became a world wide known standard reference text from which also a Russian translation appeared (see Zbl 0607.60041). Besides many smaller corrections two chapters are completely new written and a lot of achievements obtained in the mean time are incorporated. The headlines of the new chapters III and V are: The space of stochastic differentials; Stochastic differential equations with respect to quasimartingales; Moment inequalities for martingales; Some applications of stochastic calculus to Brownian motions; Exponential martingales; Conformal martingales. Stochastic differential equations on manifolds; Flow of diffeomorphisms; Heat equation on a manifold and horizontal lifts; Non-degenerate diffusions on a manifold and their horizontal lifts; Stochastic parallel displacement and heat equation for tensor fields; The case with boundary conditions; Kähler diffusions; Malliavin’s stochastic calculus of variation for Wiener functionals; Pull-back of Schwartz distributions under Wiener mappings and the regularity of induced measures (probability laws); The case of stochastic differential equations: Applications to heat kernels. Reviewer: M.Breger Cited in 12 ReviewsCited in 1139 Documents MSC: 60Hxx Stochastic analysis 60J60 Diffusion processes 60-02 Research exposition (monographs, survey articles) pertaining to probability theory 58J65 Diffusion processes and stochastic analysis on manifolds Keywords:Stochastic differential equations; Moment inequalities; Exponential martingales; Conformal martingales; Flow of diffeomorphisms; Stochastic parallel displacement; Malliavin’s stochastic calculus of variation Citations:Zbl 0495.60005; Zbl 0607.60041 PDFBibTeX XMLCite \textit{N. Ikeda} and \textit{S. Watanabe}, Stochastic differential equations and diffusion processes. 2nd ed. Amsterdam etc.: North-Holland; Tokyo: Kodansha Ltd. (1989; Zbl 0684.60040)