Analysis and Geometry of Markov Diffusion Operators

This opening chapter introduces, somewhat informally, some of the basic ideas and concepts in the investigation of Markov semigroups, operators and processes, at the interface between analysis, partial differential equations, probability theory and geometry, loosely jumping from one area to the other. The first sections describe some of the basic elements in a general description of Markov semigroups, operators and processes, dealing with state space, Markov property, invariant measures, infinitesimal generators and carré du champ operators, the associated Chapman–Kolmogorov and Fokker-Planck equations. Further sections deal with symmetric semigroups, Dirichlet forms, spectral decompositions and ergodicity. The probabilistic intuition is further developed on the basis of stochastic differential equations and diffusion processes which illustrate the main diffusion hypothesis, central to the investigation. Diffusion semigroups and their domains are discussed together with ellipticity and hypo-ellipticity. Basic tools and operations on semigroups and their infinitesimal generators are presented in the final part, together with the concept of curvature and dimension of a Markov (diffusion) operator of crucial importance throughout the monograph.

This chapter is devoted to some basic model examples, which will serve as a guide throughout the book. They are the opportunity to illustrate some of the ideas, definitions and properties of Markov semigroups, operators and processes and will help to set up the framework of the investigation of more general Markov semigroups and generators which will be achieved in the next chapter. The chapter starts with the three geometric models of the heat semigroup on the Euclidean space, the sphere and the hyperbolic space. Further sections discuss the heat semigroup with Neumann, Dirichlet or periodic conditions on \({\mathbb{R}}\) or on an interval of \({\mathbb{R}}\). More general Sturm–Liouville operators on an interval of the real line are examined next, illustrated by the examples of the Ornstein–Uhlenbeck or Hermite, Laguerre and Jacobi 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.

This chapter investigates the first important family of functional inequalities for Markov semigroups, the Poincaré or spectral gap inequalities. These will provide the first results towards convergence to equilibrium, and illustrate, at a mild and accessible level, some of the basic ideas and techniques on Markov semigroups and functional inequalities developed in this work, at the interplay between analysis, probability theory and geometry. On the basis of the example of the Ornstein–Uhlenbeck semigroup, the formal definition of Poincaré or spectral gap inequality is introduced in the context of a Markov Triple (E,μ,Γ). Exponential decay, tensorization properties and exponential integrability belong to the first properties of Poincaré inequalities. Poincaré inequalities for measures on the real line or on an interval of the real line are investigated next, together with the Lyapunov function method. Local Poincaré inequalities for heat kernel measures under curvature conditions via the basic semigroup interpolation scheme are parts of the central results of this chapter. Global Poincaré inequalities for the invariant measure under curvature-dimension conditions is a further topic of interest. Further inequalities of Brascamp–Lieb-type are also discussed. The last part is a somewhat deeper investigation of the bottom of the spectrum and essential spectrum of Markov generators together with criterions towards discrete spectrum of general interest via the notion of Persson operator.

After Poincaré inequalities, logarithmic Sobolev inequalities are amongst the most studied functional inequalities for semigroups. They contain much more information than Poincaré inequalities, and are at the same time sufficiently general to be available in numerous cases of interest, in particular in infinite dimension (as limits of Sobolev inequalities on finite-dimensional spaces). After the basic definition of a logarithmic Sobolev inequality together with its first properties, the first sections of this chapter present the exponential decay in entropy and the fundamental equivalence between the logarithmic Sobolev inequality and smoothing properties of the semigroup in the form of hypercontractivity. Next, integrability properties of eigenvectors and of Lipschitz functions under a logarithmic Sobolev inequality are discussed together with a criterion for measures on the real line to satisfy a logarithmic Sobolev inequality (for the usual gradient). The further sections deal with curvature conditions, first for the local logarithmic Sobolev inequalities for heat kernel measures, then for the invariant measure with an additional dimensional information. Local hypercontractivity and some applications of the local logarithmic Sobolev inequalities towards heat kernel bounds are further presented. Harnack-type inequalities under the infinite-dimensional curvature conditions, linked with reverse local logarithmic Sobolev inequalities complete the chapter.

After the Poincaré and logarithmic Sobolev inequalities, this chapter is devoted to the investigation of Sobolev inequalities. Sobolev inequalities play a central role in analysis, providing in particular compact embeddings and tight connections with heat kernels bounds. They are also deeply linked with the geometric structure of the underlying state space through conformal invariance. The study here focuses on the aspects of Sobolev inequalities in the context of Markov diffusion operators and semigroups. The chapter starts with a brief exposition of the classical Sobolev inequalities on the model spaces, namely the Euclidean, spherical and hyperbolic spaces. Next, various definitions of Sobolev-type inequalities in the Markov Triple context are emphasized, in particular (logarithmic) entropy-energy and Nash-type inequalities. The basic equivalence between Sobolev inequalities and (uniform) heat kernel bounds (ultracontractivity) is studied in the further sections. Local inequalities under the semigroup are investigated next, providing in particular a heat flow approach to the celebrated Li–Yau parabolic inequality. The sharp Sobolev inequality under curvature-dimension condition, covering the example of the standard sphere, is established in the framework of this monograph. Further sections describe the conformal invariance properties of Sobolev inequalities, and, as a consequence, the sharp Sobolev inequalities in the Euclidean and hyperbolic spaces on the basis of the one on the sphere, Gagliardo–Nirenberg inequalities and non-linear porous medium and fast diffusion equations and their geometric counterparts, with in particular a fast diffusion approach to several Sobolev inequalities of interest.

The Poincaré, logarithmic Sobolev and Sobolev inequalities capture different features of the associated semigroup or the invariant measure, in terms of convergence to equilibrium, estimates on the heat kernels or tail behaviors of the invariant measure. This chapter investigates intermediate or more general families of functional inequalities which are suited to a wide variety of regimes as well as to more precise, or different, features. The study is concerned with three main examples, entropy-energy, generalized Nash and weak Poincaré inequalities. The first part of the chapter describes the family of functional entropy–energy inequalities well-suited to heat kernel bounds by means of the method developed for hypercontractivity under logarithmic Sobolev inequalities. Off-diagonal heat kernel estimates may be achieved in the same way. Further sections in this chapter investigate generalized Nash inequalities on the basis of the example of the classical Nash inequality in Euclidean space, as well as weighted Nash inequalities, with a focus on non-uniform heat kernel bounds and tail inequalities. Weak Poincaré inequalities are studied as the main minimal tool to tighten families of standard functional inequalities. Weak Poincaré inequalities may be further studied in their own from the viewpoint of heat kernel bounds and tail estimates. Further related families of functional inequalities of interest and illustrative examples complete the exposition of this chapter.

This chapter focuses on inequalities comparing measure and capacity uniformly over a given class of sets as equivalent forms of functional inequalities. The chapter concentrates on the so-called 2-capacities which capture the relevant Dirichlet form information on sets and on the 1-capacities which are related to boundary or surface measures and isoperimetric-type inequalities. The first part of the chapter introduces the basic notions on capacities and co-area formulas to transfer (and back) functional inequalities into measure-capacity inequalities. Sobolev-type, Poincaré and logarithmic Sobolev inequalities are analyzed in this respect. The second part is concerned with measure-capacity inequalities. Together with the heat kernel tools under curvature conditions, measure-capacity inequalities of isoperimetric-type are investigated, leading in particular to the Gaussian isoperimetric inequality as well as to comparison results under curvature conditions. Combination of the heat kernel isoperimetric inequality together with its reverse form further yields an isoperimetric-type Harnack inequality. The last section addresses the relationships between Poincaré and logarithmic Sobolev inequalities and their associated concentration properties.

This chapter is a brief investigation of the links between optimal transportation methods and functional inequalities in the Markov operator framework of this monograph. After a brief introduction to the basic material on optimal transportation, the main topic of transportation cost inequalities and first examples for Gaussian measures are presented. Interpolation along the geodesics of optimal transport is used towards logarithmic Sobolev inequalities and transportation cost inequalities comparing relative entropy and Wasserstein distances between probability measures. An alternate approach to sharp Sobolev or Gagliardo–Nirenberg inequalities in Euclidean space is provided next along these lines. Non-linear Hamilton–Jacobi equations and hypercontractivity properties of their solutions, analogous to the ones for linear heat equations, are investigated in the further sections towards the relationships between (quadratic) transportation cost inequalities and logarithmic Sobolev inequalities. Contraction properties in Wasserstein space along with the heat semigroup are investigated in the Markov operator setting. The last section is a very brief overview of recent developments towards a notion of Ricci curvature lower bounds based on optimal transportation and the connection with the Γ-calculus developed in this work.