Jan Maas

Ricci Curvature of Finite Markov Chains via Convexity of the Entropy

We study a new notion of Ricci curvature that applies to Markov chains on discrete spaces. This notion relies on geodesic convexity of the entropy and is analogous to the one introduced by Lott, Sturm, and Villani for geodesic measure spaces. In order to apply to the discrete setting, the role of the Wasserstein metric is taken over by a different metric, having the property that continuous time Markov chains are gradient flows of the entropy. Using this notion of Ricci curvature we prove discrete analogues of fundamental results by Bakry–Émery and Otto–Villani. Further, we show that Ricci curvature bounds are preserved under tensorisation. As a special case we obtain the sharp Ricci curvature lower bound for the discrete hypercube.

APPROXIMATING ROUGH STOCHASTIC PDES

We study approximations to a class of vector-valued equations of Burgers type driven by a multiplicative space-time white noise. A solution theory for this class of equations has been developed recently in Probability Theory Related Fields by Hairer and Weber. The key idea was to use the theory of controlled rough paths to give definitions of weak/mild solutions and to set up a Picard iteration argument. In this article the limiting behavior of a rather large class of (spatial) approximations to these equations is studied. These approximations are shown to converge and convergence rates are given, but the limit may depend on the particular choice of approximation. This effect is a spatial analogue to the Ito-Stratonovich correction in the theory of stochastic ordinary differential equations, where it is well known that different approximation schemes may converge to different solutions.

Entropic Ricci curvature bounds for discrete interacting systems

We develop a new and systematic method for proving entropic Ricci curvature lower bounds for Markov chains on discrete sets. Using different methods, such bounds have recently been obtained in several examples (e.g., 1-dimensional birth and death chains, product chains, Bernoulli-Laplace models, and random transposition models). However, a general method to obtain discrete Ricci bounds had been lacking. Our method covers all of the examples above. In addition, we obtain new Ricci curvature bounds for zero-range processes on the complete graph. The method is inspired by recent work of Caputo, Dai Pra and Posta on discrete functional inequalities.

Whitney coverings and the tent spaces T1, q(γ) for the Gaussian measure

We introduce a technique for handling Whitney decompositions in Gaussian harmonic analysis and apply it to the study of Gaussian analogues of the classical tent spaces T1,q of Coifman–Meyer–Stein.

A spatial version of the Itô–Stratonovich correction

We consider a class of stochastic PDEs of Burgers type in spatial dimension 1, driven by space–time white noise. Even though it is well known that these equations are well posed, it turns out that if one performs a spatial discretization of the nonlinearity in the “wrong” way, then the sequence of approximate equations does converge to a limit, but this limit exhibits an additional correction term. This correction term is proportional to the local quadratic cross-variation (in space) of the gradient of the conserved quantity with the solution itself. This can be understood as a consequence of the fact that for any fixed time, the law of the solution is locally equivalent to Wiener measure, where space plays the role of time. In this sense, the correction term is similar to the usual Ito–Stratonovich correction term that arises when one considers different temporal discretizations of stochastic ODEs.

Conical square functions and non-tangential maximal functions with respect to the gaussian measure

We study, in L 1 (R n ;) with respect to the gaussian measure, nontangential maximal functions and conical square functions associated with the Ornstein-Uhlenbeck operator by developing a set of techniques which allow us, to some extent, to compensate for the non-doubling character of the gaussian measure. The main result asserts that conical square functions can be controlled in L 1 -norm by non-tangential maximal functions. Along the way we prove a change of aperture result for the latter. This complements recent results on gaussian Hardy spaces due to Mauceri and Meda.

Gradient Estimates and Domain Identification for Analytic Ornstein-Uhlenbeck Operators

Let P be the Ornstein-Uhlenbeck semigroup associated with the stochastic Cauchy problem

Gradient flows of the entropy for finite Markov chains

dU(t) = AU(t)\,dt + dW_{H}(t),