General Markov Semigroups Research Articles

Modified log-Sobolev inequalities, Beckner inequalities and moment estimates

We prove that in the context of general Markov semigroups Beckner inequalities with constants separated from zero as p→1+ are equivalent to the modified log Sobolev inequality (previously only one implication was known to hold in this generality). Further, by adapting an argument by Boucheron et al. we derive Sobolev type moment estimates which hold under these functional inequalities.We illustrate our results with applications to concentration of measure estimates (also of higher order, beyond the case of Lipschitz functions) for various stochastic models, including random permutations, zero-range processes, strong Rayleigh measures, exponential random graphs, and geometric functionals on the Poisson path space.

A supersolutions perspective on hypercontractivity

The purpose of this article is to expose an algebraic closure property of supersolutions to certain diffusion equations. This closure property quickly gives rise to a monotone quantity which generates a hypercontractivity inequality. Our abstract argument applies to a general Markov semigroup whose generator is a diffusion and satisfies a curvature condition.

The Li–Yau inequality and applications under a curvature-dimension condition

We prove a global Li-Yau inequality for a general Markov semigroup under a curvature-dimension condition. This inequality is stronger than all classical Li-Yau type inequalities known to us. On a Riemannian manifold, it is equivalent to a new parabolic Harnack inequality, both in negative and positive curvature, giving new subsequents bounds on the heat kernel of the semigroup. Under positive curvature we moreover reach ultracontractive bounds by a direct and robust method.

Recent advances in various fields of numerical probability

The goal of this paper is to present a series of recent contributions on some various problems of numerical probability. Beginning with the Richardson-Romberg Multilevel Monte-Carlo method which, among other fields of applications, is a very efficient method for the approximation of diffusion processes, we focus on some adaptive multilevel splitting algorithms for rare event simulation. Then, the third part is devoted to the simulation of McKean-Vlasov forward and decoupled forward-backward stochastic differential equations by some cubature algorithms. Finally, we tackle the problem of the weak error estimation in total variation norm for a general Markov semi-group.

Dimensional contraction via Markov transportation distance

It is now well known that curvature conditions à la Bakry–Émery are equivalent to contraction properties of the heat semigroup with respect to the classical quadratic Wasserstein distance. However, this curvature condition may include a dimensional correction which up to now had not induced any strengthening of this contraction. We first consider the simplest example of the Euclidean heat semigroup, and prove that indeed it is so. To consider the case of a general Markov semigroup, we introduce a new distance between probability measures, based on the semigroup, and adapted to it. We prove, in the setting of a compact Riemannian manifold, that this Markov transportation distance satisfies the same properties as the Wasserstein distance does in the specific case of the Euclidean heat semigroup, namely dimensional contraction properties and evolution variational inequalities.

Lp-Uniqueness of Schrödinger Operators and the Capacitary Positive Improving Property

We prove several Lp-uniqueness results for Schrödinger operators −L+V by means of the Feynman–Kac formula. Using the (m, p)-capacity theory for general Markov semigroups, we show that the associated Feynman–Kac semigroup is positive improving in the sense of (m, p)-capacity, improving the well known one in the sense of measure. Using that capacitary positive improving property and two new inequalities for generalized Ornstein–Uhlenbeck generators, we show the essential self-adjointness of the ground state diffusion generator Lφ=L+2Γ(φ, ·)/φ associated with two dimensional Euclidean quantum fields.

Exponential convergence rate in Boltzmann-Shannon entropy

This paper deals with the optimal exponential convergence rate β to the equilibrium state in Boltzmann-Shannon entropy for general Markov semigroups. We prove a variational formula of β, and then discuss the relation among β, spectral gap λ and logarithmic Sobolev constant α, which is read as λ≥β≥α.

On the Convergence of the Discrete Time Dynamic Programming Equation for General Semigroups

We consider several classes of control problems for Markov processes (continuous control, optimal stopping, impulse control). The formulation we use is valid for general Markov semigroups. We study the discrete time approximation of the dynamic programming equation, using mainly an analytical approach. Probabilistic interpretation is given for some of the results.