Diffusion Semigroups Research Articles

Phi-entropy inequalities for diffusion semigroups

We obtain and study new Φ-entropy inequalities for diffusion semigroups, with Poincaré or logarithmic Sobolev inequalities as particular cases. From this study we derive the asymptotic behaviour of a large class of linear Fokker–Planck type equations under simple conditions, widely extending previous results. Nonlinear diffusion equations are also studied by means of these inequalities. The Γ 2 criterion of D. Bakry and M. Emery appears as a main tool in the analysis, in local or integral forms.

Intrinsic ultracontractivity on Riemannian manifolds with infinite volume measures

By establishing the intrinsic super-Poincare inequality, some explicit conditions are presented for diffusion semigroups on a non-compact complete Riemannian manifold to be intrinsically ultracontractive. These conditions, as well as the resulting uniform upper bounds on the intrinsic heat kernels, are sharp for some concrete examples.

A direct Lebeau-Robbiano strategy for the observability of heat-like semigroups

This paper generalizes and simplifies abstract results of Miller and Seidmanon the cost of fast control/observation. It deduces final-observability of an evolution semigroupfrom a spectral inequality, i.e. some stationary observability propertyon some spaces associated to the generator,e.g. spectral subspaces when the semigrouphas an integral representation via spectral measures. wordsContrary to the original Lebeau-Robbiano strategy,it does not have recourse to null-controllabilityand it yields the optimal bound of the cost when applied to the heat equation,i.e. $c_0\exp(c/T)$, or to the heat diffusion in potential wells observed from cones,i.e. $c_0\exp(c/T^\beta)$ with optimal $\beta$.It also yields simple upper bounds for the cost rate $c$in terms of the spectral rate.    This paper also gives geometric lower bounds on the spectral and cost ratesfor heat, diffusion and Ginzburg-Landau semigroups,including on non-compact Riemannian manifolds,based on $L^2$ Gaussian estimates.

The differentiation of hypoelliptic diffusion semigroups

Basic derivative formulas are presented for hypoelliptic heat semi- groups and harmonic functions extending earlier work in the elliptic case. Fol- lowing the approach of (22), emphasis is placed on developing integration by parts formulas at the level of local martingales. Combined with the optional sampling theorem, this turns out to be an efficient way of dealing with bound- ary conditions, as well as with finite lifetime of the underlying diffusion. Our formulas require hypoellipticity of the diffusion in the sense of Malliavin cal- culus (integrability of the inverse Malliavin covariance) and are formulated in terms of the derivative flow, the Malliavin covariance and its inverse. Finally some extensions to the nonlinear setting of harmonic mappings are discussed.

Ornstein–Uhlenbeck semi-groups on stratified groups

We consider, in the setting of stratified groups G, two analogues of the Ornstein–Uhlenbeck semi-group, namely Markovian diffusion semi-groups acting on L q ( p d γ ) , whose invariant density p is a heat kernel at time 1 on G. Both generators have the same “carré du champ”. The first semi-group is symmetric on L 2 ( p d γ ) , with generator ∑ i = 1 n X i * X i , where ( X i ) i = 1 n is a basis of the first layer of the Lie algebra of G. The second one is compact on L q ( p d γ ) , 1 < q < ∞ , non-symmetric on L 2 ( p d γ ) and the formal real part of its generator N is ∑ i = 1 n X i * X i . The spectrum of N is the set of non-negative integers if polynomials are dense in L 2 ( p d γ ) , i.e. if G has at most 4 layers; we determine in this case the eigenspaces. When G is step two, we give another description of these eigenspaces, similar to the classical definition of Hermite polynomials by their generating function.

Gradient estimates and Harnack inequalities on non-compact Riemannian manifolds

A gradient-entropy inequality is established for elliptic diffusion semigroups on arbitrary complete Riemannian manifolds. As applications, a global Harnack inequality with power and a heat kernel estimate are derived.

Log-Sobolev inequalities: Different roles of Ric and Hess

Let Pt be the diffusion semigroup generated by L:=Δ+∇V on a complete connected Riemannian manifold with Ric≥−(σ2ρo2+c) for some constants σ, c>0 and ρo the Riemannian distance to a fixed point. It is shown that Pt is hypercontractive, or the log-Sobolev inequality holds for the associated Dirichlet form, provided −HessV≥δ holds outside of a compact set for some constant $\delta >(1+\sqrt{2})\sigma \sqrt{d-1}$. This indicates, at least in finite dimensions, that Ric and −HessV play quite different roles for the log-Sobolev inequality to hold. The supercontractivity and the ultracontractivity are also studied.

A necessary and sufficient condition for the global-in-time existence of solutions to stochastic differential and parabolic equations on manifolds

A necessary and sufficient condition of one-sided type for the completeness of a stochastic flow and the corresponding diffusion semigroup on a manifold M is found under the assumption that the space C 0(M) of functions is invariant.

Intrinsic ultracontractivity for non-symmetric Lévy processes

Recently in [Kim and Song, Intrinsic ultracontractivity of non-symmetric diffusion semigroups in bounded domains: 2006, Kim and Song, Intrinsic ultracontractivity of non-symmetric diffusion with measure-valued drifts and potentials: 2006], we extended the concept of intrinsic ultracontractivity to non-symmetric semigroups and proved that for a large class of non-symmetric diffusions Z with measure-valued drift and potential, the semigroup of ZD (the process obtained by killing Z upon exiting D) in a bounded domain is intrinsic ultracontractive under very mild assumptions.

Some results on Gaussian Besov–Lipschitz spaces and Gaussian Triebel–Lizorkin spaces

In this paper we define Besov–Lipschitz and Triebel–Lizorkin spaces in the context of Gaussian harmonic analysis, the harmonic analysis of Hermite polynomial expansions. We study inclusion relations among them, some interpolation results and continuity results of some important operators (the Ornstein–Uhlenbeck and the Poisson–Hermite semigroups and the Bessel potentials) on them. We also prove that the Gaussian Sobolev spaces L α p ( γ d ) are contained in them. The proofs are general enough to allow extensions of these results to the case of Laguerre or Jacobi expansions and even further in the general framework of diffusion semigroups.

Entropy-cost inequalities for diffusion semigroups with curvature unbounded below

The weighted log-Sobolev inequality and the entropy-cost inequality are established for a class of diffusion semigroups with curvature unbounded below. Concrete examples are presented to illustrate the main results.

Pointwise convergence for semigroups in vector-valued L p spaces

Suppose that {T t : t ≥ 0} is a symmetric diffusion semigroup on L 2(X) and denote by \({\{\widetilde{T}_t:t\geq0\}}\) its tensor product extension to the Bochner space \({L^p(X,\,\mathcal{B})}\), where \({\mathcal{B}}\) belongs to a certain broad class of UMD spaces. We prove a vector-valued version of the Hopf–Dunford–Schwartz ergodic theorem and show that this extends to a maximal theorem for analytic continuations of \({\{\widetilde{T}_t\,:\,t\,\geq\,0\}}\) on \({L^p(X,\,\mathcal{B})}\). As an application, we show that such continuations exhibit pointwise convergence.

Littlewood-Paley-Stein theory for semigroups in UMD spaces

The Littlewood-Paley theory for a symmetric diffusion semigroup T^t , as developed by Stein, is here generalized to deal with the tensor extensions of these operators on the Bochner spaces L^p(\mu,X) , where X is a Banach space. The g -functions in this situation are formulated as expectations of vector-valued stochastic integrals with respect to a Brownian motion. A two-sided g -function estimate is then shown to be equivalent to the UMD property of X . As in the classical context, such estimates are used to prove the boundedness of various operators derived from the semigroup T^t , such as the imaginary powers of the generator.

Gradient flows and diffusion semigroups in metric spaces under lower curvature bounds

We present some new results concerning well-posedness of gradient flows generated by λ-convex functionals in a wide class of metric spaces, including Alexandrov spaces satisfying a lower curvature bound and the corresponding L 2 -Wasserstein spaces. Applications to the gradient flow of Entropy functionals in metric-measure spaces with Ricci curvature bounded from below and to the corresponding diffusion semigroup are also considered. These results have been announced during the workshop on “Optimal Transport: theory and applications” held in Pisa, November 2006. To cite this article: G. Savaré, C. R. Acad. Sci. Paris, Ser. I 345 (2007).

Weak logarithmic Sobolev inequalities and entropic convergence

In this paper we introduce and study a weakened form of logarithmic Sobolev inequalities in connection with various others functional inequalities (weak Poincare inequalities, general Beckner inequalities...). We also discuss the quantitative behaviour of relative entropy along a symmetric diffusion semi-group. In particular, we exhibit an example where Poincare inequality can not be used for deriving entropic convergence whence weak logarithmic Sobolev inequality ensures the result.

Weighted norm inequalities for heat-diffusion Laguerre’s semigroups

Fil: Chicco Ruiz, Anibal Leonardo. Consejo Nacional de Investigaciones Cientificas y Tecnicas. Centro Cientifico Tecnologico Conicet - Santa Fe. Instituto de Matematica Aplicada del Litoral. Universidad Nacional del Litoral. Instituto de Matematica Aplicada del Litoral; Argentina

Formulae for the derivatives of degenerate diffusion semigroups

We consider the diffusion semigroup Pt associated to a class of degenerate elliptic operators \(\mathcal{A}\; \text{on}\; \mathbb{R}^n\). This class includes the hypoelliptic Ornstein-Uhlenbeck operator but does not satisfy in general the well-known Hormander condition on commutators for sums of squares of vector fields. We establish probabilistic formulae for the spatial derivatives of Ptf up to the third order. We obtain L∞-estimates for the derivatives of Ptf and show the existence of a classical bounded solution for the parabolic Cauchy problem involving \(\mathcal{A}\) and having \(f\in C_b(\mathbb{R}^n)\) as initial datum.

A Harnack-type inequality for non-symmetric Markov semigroups

For a strong Feller and irreducible Markov semigroup on a locally compact Polish space, the Harnack-type inequality (1.1) holds if and only if the semigroup has a unique invariant probability measure and is ultracontractive. Moreover, new sufficient conditions for this inequality to hold, as well as upper bound estimates of the underlying constant, are presented for diffusion semigroups on Riemannian manifolds.

Bismut Type Formulae for Diffusion Semigroups on Riemannian Manifolds

Applying the stochastic calculus of variations on frame bundles along tangent processes, we derive Bismut type formulae for the derivatives of diffusion semigroups on Riemannian manifold in both variables. We also obtain the Bismut formulae expressed in terms of the Ricci and torsion tensors for the connection with torsion.

A SIMPLE PROOF OF LOG-SOBOLEV INEQUALITIES ON A PATH SPACE WITH GIBBS MEASURES

In this paper, we give a simple proof of log-Sobolev inequalities on an infinite volume path space C (ℝ, ℝd) with Gibbs measures. We introduce a parabolic stochastic partial differential equation which is reversible with respect to the Gibbs measures. In the proof, the gradient estimate for the diffusion semigroup which is derived from the stochastic flow plays a central role.