Symmetric Markov Diffusion Operators

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.

[1]  D. Bakry.  The Riesz transforms associated with second order differential operators , 1989 .

[2]  L. Hörmander,et al.  The Analysis of Linear Partial Differential Operators II: Differential Operators with Constant Coefficients , 1983 .

[3]  N. H. Bingham,et al.  Seminar on Stochastic Processes , 1993 .

[4]  Djalil CHAFAÏ,et al.  On gradient bounds for the heat kernel on the Heisenberg group , 2007, 0710.3139.

[5]  Elton P. Hsu.  Stochastic analysis on manifolds , 2002 .

[6]  Hong-Quan Li.  Estimation optimale du gradient du semi-groupe de la chaleur sur le groupe de Heisenberg , 2006 .

[7]  D. Elworthy.  Geometric aspects of diffusions on manifolds , 1988 .

[8]  Jiang-Hua Lu,et al.  Progress in Mathematics , 2013 .

[9]  D. Stroock.  An Introduction to the Analysis of Paths on a Riemannian Manifold , 2005 .

[10]  Vladimir Maz’ya,et al.  Sobolev Spaces: with Applications to Elliptic Partial Differential Equations , 2011 .

[11]  Hong-Quan Li.  Estimations asymptotiques du noyau de la chaleur sur les groupes de Heisenberg , 2007 .

[12]  L. Hörmander,et al.  Differential operators with constant coefficients , 1983 .

[13]  M. Fukushima,et al.  Dirichlet forms and symmetric Markov processes , 1994 .

[14]  M. P. Gaffney.  The conservation property of the heat equation on riemannian manifolds , 1959 .

[15]  Karl-Theodor Sturm,et al.  Analysis on local Dirichlet spaces. I. Recurrence, conservativeness and Lp-Liouville properties. , 1994 .

[16]  M. Fukushima.  Dirichlet forms and Markov processes , 1980 .

[17]  Lizhen Ji,et al.  Geometry and analysis , 2011 .

[18]  Nicolas Bouleau,et al.  Dirichlet Forms and Analysis on Wiener Space , 1991 .

[19]  D. Bakry.  Transformations de Riesz pour les semi-groupes symetriques Seconde patrie: Etude sous la condition Γ2≧0 , 1985 .

[20]  Fabrice Baudoin,et al.  Curvature-dimension inequalities and Ricci lower bounds for sub-Riemannian manifolds with transverse symmetries , 2011, 1101.3590.

[21]  Alexander Grigor'yan,et al.  Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds , 1999 .

[22]  R. Strichartz.  Analysis of the Laplacian on the Complete Riemannian Manifold , 1983 .

[23]  tballest.  Séminaire de probabilités , 2013 .

[24]  J. Bismut.  Large Deviations and the Malliavin Calculus , 1984 .

[25]  A. Grigor’yan.  Heat Kernel and Analysis on Manifolds , 2012 .

[26]  Séminaire de Probabilités XXI , 1977 .

[27]  M. Ledoux,et al.  Sobolev inequalities and Myers’s diameter theorem for an abstract Markov generator , 1996 .

[28]  L. Hörmander,et al.  The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis , 1983 .

[29]  L. Hörmander.  Distribution theory and Fourier analysis , 1990 .

[30]  Nicolas Victoir,et al.  Analysis on local Dirichlet spaces , 2010 .

[31]  F. Hirsch.  Intrinsic metrics and Lipschitz functions , 2003 .

[32]  Pierre Bernard,et al.  Lectures on probability theory , 1994 .

[33]  Karl-Theodor Sturm,et al.  Analysis on local Dirichlet spaces. II. Upper Gaussian estimates for the fundamental solutions of parabolic equations , 1995 .

[34]  K. Elworthy.  Stochastic Differential Equations on Manifolds , 1982 .

[35]  B. Zegarliński.  Analysis on Extended Heisenberg Group , 2011 .

[36]  Dominique Bakry,et al.  Functional Inequalities for Markov semigroups , 2009 .

[37]  L. Hörmander.  The analysis of linear partial differential operators , 1990 .

[38]  D. Bakry.  L'hypercontractivité et son utilisation en théorie des semigroupes , 1994 .

[39]  D. Bakry.  Etude des transformations de Riesz dans les variétés riemanniennes à courbure de Ricci minorée , 1987 .