Abstract
This chapter provides a general framework for the investigation of symmetric Markov diffusion semigroups and operators. Based on the early observations of the first chapter and on the investigation of the model examples, it develops all the fundamental properties which will justify the computations in the following chapters in the most convenient framework. The main setting consisting of a Markov Triple (E,μ,Γ) describes a convenient framework to develop the formalism of Markov semigroups and the Γ-calculus towards functional inequalities and convergence to equilibrium. The analysis of the concrete example of second order differential operators on smooth complete connected manifolds is the guide for the description, in this framework, of some main features such as connexity, completeness, weak hypo-ellipticity etc. With the main tool of essential self-adjointness at the center of the construction, the chapter emphasizes the relevant hypotheses and properties of Full Markov Triples, in force throughout the monograph, covering the main examples of illustrations.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Bibliography
D. Bakry, Transformations de Riesz pour les semi-groupes symétriques. II. Étude sous la condition Γ2≥0, in Séminaire de Probabilités, XIX, 1983/84. Lecture Notes in Math., vol. 1123 (Springer, Berlin, 1985), pp. 145–174
D. Bakry, Étude des transformations de Riesz dans les variétés Riemanniennes à courbure de Ricci minorée, in Séminaire de Probabilités, XXI. Lecture Notes in Math., vol. 1247 (Springer, Berlin, 1987), pp. 137–172
D. Bakry, The Riesz transforms associated with second order differential operators, in Seminar on Stochastic Processes, Gainesville, FL, 1988. Progr. Probab., vol. 17 (Birkhäuser, Boston, 1989), pp. 1–43
D. Bakry, L’hypercontractivité et son utilisation en théorie des semigroupes, in Lectures on Probability Theory, Saint-Flour, 1992. Lecture Notes in Math., vol. 1581 (Springer, Berlin, 1994), pp. 1–114
D. Bakry, On Sobolev and logarithmic Sobolev inequalities for Markov semigroups, in New Trends in Stochastic Analysis, Charingworth, 1994 (World Sci., River Edge, 1997), pp. 43–75
D. Bakry, Functional inequalities for Markov semigroups, in Probability Measures on Groups: Recent Directions and Trends (Tata Inst. Fund. Res, Mumbai, 2006), pp. 91–147
D. Bakry, F. Baudoin, M. Bonnefont, D. Chafaï, On gradient bounds for the heat kernel on the Heisenberg group. J. Funct. Anal. 255(8), 1905–1938 (2008)
D. Bakry, M. Émery, Diffusions hypercontractives, in Séminaire de Probabilités, XIX, 1983/1984. Lecture Notes in Math., vol. 1123 (Springer, Berlin, 1985), pp. 177–206
D. Bakry, M. Ledoux, Sobolev inequalities and Myers’s diameter theorem for an abstract Markov generator. Duke Math. J. 85(1), 253–270 (1996)
F. Baudoin, N. Garofalo, Curvature-dimension inequalities and Ricci lower bounds for sub-Riemannian manifolds with transverse symmetries. Preprint, 2012
J.-M. Bismut, Large Deviations and the Malliavin Calculus. Progress in Mathematics, vol. 45 (Birkhäuser, Boston, 1984)
N. Bouleau, F. Hirsch, Dirichlet Forms and Analysis on Wiener Space. de Gruyter Studies in Mathematics, vol. 14 (Walter de Gruyter, Berlin, 1991)
K.D. Elworthy, Stochastic Differential Equations on Manifolds. London Mathematical Society Lecture Note Series, vol. 70 (Cambridge University Press, Cambridge, 1982)
K.D. Elworthy, Geometric aspects of diffusions on manifolds, in École d’Été de Probabilités de Saint-Flour XV–XVII, 1988–87. Lecture Notes in Math., vol. 1362 (Springer, Berlin, 1985), pp. 277–425
M. Fukushima, Dirichlet Forms and Markov Processes. North-Holland Mathematical Library, vol. 23 (North-Holland, Amsterdam, 1980)
M. Fukushima, Y. Oshima, M. Takeda, Dirichlet Forms and Symmetric Markov Processes. de Gruyter Studies in Mathematics, vol. 19 (Walter de Gruyter, Berlin, 2011), extended edn.
M.P. Gaffney, The conservation property of the heat equation on Riemannian manifolds. Commun. Pure Appl. Math. 12, 1–11 (1959)
A. Grigor’yan, Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds. Bull. Am. Math. Soc. (N.S.) 36(2), 135–249 (1999)
A. Grigor’yan, Heat Kernel and Analysis on Manifolds. AMS/IP Studies in Advanced Mathematics, vol. 47 (American Mathematical Society, Providence, 2009)
A. Grigor’yan, Yau’s work on heat kernels, in Geometry and Analysis. No. 1. Adv. Lect. Math. (ALM), vol. 17 (Int. Press, Somerville, 2011), pp. 113–117
F. Hirsch, Intrinsic metrics and Lipschitz functions. J. Evol. Equ. 3(1), 11–25 (2003). Dedicated to Philippe Bénilan
L. Hörmander, Differential operators with constant coefficients, in The Analysis of Linear Partial Differential Operators. II. Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 257 (Springer, Berlin, 1983)
L. Hörmander, Distribution theory and Fourier analysis, in The Analysis of Linear Partial Differential Operators. I, 2nd edn. (Springer, Berlin, 1990). Springer Study Edition
E.P. Hsu, Stochastic Analysis on Manifolds. Graduate Studies in Mathematics, vol. 38 (American Mathematical Society, Providence, 2002)
H.-Q. Li, Estimation optimale du gradient du semi-groupe de la chaleur sur le groupe de Heisenberg. J. Funct. Anal. 236(2), 369–394 (2006)
H.-Q. Li, Estimations asymptotiques du noyau de la chaleur sur les groupes de Heisenberg. C. R. Math. Acad. Sci. Paris 344(8), 497–502 (2007)
Z.M. Ma, M. Röckner, Introduction to the Theory of (Nonsymmetric) Dirichlet Forms. Universitext (Springer, Berlin, 1992)
V.G. Maz’ya, Sobolev Spaces with Applications to Elliptic Partial Differential Equations. Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 342 (Springer, Heidelberg, 2011). augmented ed.
R.S. Strichartz, Analysis of the Laplacian on the complete Riemannian manifold. J. Funct. Anal. 52(1), 48–79 (1983)
D.W. Stroock, An Introduction to the Analysis of Paths on a Riemannian Manifold. Mathematical Surveys and Monographs, vol. 74 (American Mathematical Society, Providence, 2000)
K.-T. Sturm, Analysis on local Dirichlet spaces. I. Recurrence, conservativeness and L p-Liouville properties. J. Reine Angew. Math. 456, 173–196 (1994)
K.-T. Sturm, Analysis on local Dirichlet spaces. II. Upper Gaussian estimates for the fundamental solutions of parabolic equations. Osaka J. Math. 32(2), 275–312 (1995)
C. Villani, Hypocoercivity. Mem. Am. Math. Soc. 202, 950 (2009), pp. iv+141
S.-T. Yau, On the heat kernel of a complete Riemannian manifold. J. Math. Pures Appl. 57(2), 191–201 (1978)
B. Zegarliński, Analysis on extended Heisenberg group. Ann. Fac. Sci. Toulouse 20(2), 379–405 (2011)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Bakry, D., Gentil, I., Ledoux, M. (2014). Symmetric Markov Diffusion Operators. In: Analysis and Geometry of Markov Diffusion Operators. Grundlehren der mathematischen Wissenschaften, vol 348. Springer, Cham. https://doi.org/10.1007/978-3-319-00227-9_3
Download citation
DOI: https://doi.org/10.1007/978-3-319-00227-9_3
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-00226-2
Online ISBN: 978-3-319-00227-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)