Abstract
This paper is a brief survey of some properties of the “iterated squared gradient” associated to some diffusion semigroup. In the first part, we use this notion to give an “intrinsic” definition of the Ricci curvature and of the dimension of the semigroup: in the case of the heat semigroup on a Riemannian manifold, we recover the usual notions. In the second part, we describe some properties of diffusions with Ricci curvature bounded from below. In the third part, we show how to improve these properties in the case of diffusions with finite dimension. The fourth part is devoted to examples, and was worked out with the help of M.Emery.
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© 1990 Kluwer Academic Publishers
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Bakry, D. (1990). Ricci Curvature and Dimension for Diffusion Semigroups. In: Albeverio, S., Streit, L., Blanchard, P. (eds) Stochastic Processes and their Applications. Mathematics and Its Applications (Soviet Series), vol 61. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-2117-7_2
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DOI: https://doi.org/10.1007/978-94-009-2117-7_2
Publisher Name: Springer, Dordrecht
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