The measure of a translated ball in uniformly convex spaces

Let (E, ¦·¦) be a uniformly convex Banach space with the modulus of uniform convexity of power type. Let be the convolution of the distribution of a random series inE with independent one-dimensional components and an arbitrary probability measure onE. Under some assumptions about the components and the smoothness of the norm we show that there exists a constant such that |{·<t}–{·+r<t}|r ^q , whereq depends on the properties of the norm. We specify it in the case ofL spaces, >1.     

Statistical mechanics of probabilistic cellular automata

We investigate the behavior of discrete-time probabilistic cellular automata (PCA), which are Markov processes on spin configurations on ad-dimensional lattice, from a rigorous statistical mechanics point of view. In particular, we exploit, whenever possible, the correspondence between stationary measures on the space-time histories of PCAs on ^d and translation-invariant Gibbs states for a related Hamiltonian on ( ^d+1). This leads to a simple large-deviation formula for the space-time histories of the PCA and a proof that in a high-temperature regime the stationary states of the PCA are Gibbsian. We also obtain results about entropy, fluctuations, and correlation inequalities, and demonstrate uniqueness of the invariant state and exponential decay of correlations in a high-noise regime. We discuss phase transitions in the low-noise (or low-temperature) regime and review Toom's proof of nonergodicity of a certain class of PCAs.     

Duality in parameter space and approximation of measures for mixing repellers

For one-dimensional expanding mapsT with an invariant measure we consider, in a parameter space, the envelope _n of the real lines associated to any couple of points of the orbit, connected byn iterations ofT. If the map hass inverses and is piecewise linear, then the sets _n are just the union ofs ⁿ points and converge to the invariant Cantor set ofT. A correspondence between all the sets and their measures is established and allows one to associate the atomic measure on ₁ to the completly continuous measure on the Cantor set. If the map is nonlinear, hyperbolic, and hass inverses, the sets _n are homeomorphic to the Cantor set; they converge to the Cantor set ofT and their measures converge to the measure of the Cantor set whenn. The correspondence between the sets _n allows one to define converging approximation schemes for the map an its measure: one replaces each of thes ⁿ disjoint sets with a point in a convenient neighborhood and a probability equal to its measure and transforms it back to the original set ₁. All the approximations with linear Cantor systems previously proposed are recovered, the converging proprties being straightforward in the present scheme. Moreover, extensions to higher dimensionality and to nondisconnected repellers arte possible and are briefly examined.     

On the invariant measures for the two-dimensional euler flow

It is proven that the canonical Gibbs measure associated with a gas of vortices of intensity ± converges, in the limitN, 0,Nconst, to a Gaussian measure, which is invariant for the two-dimensional Euler equation.On leave from Dipartimento di Matematica Università di Roma Tor Vergata Roma, Italy.On leave from Dipartimento di Matematica Università di Roma La Sapienza, Roma, Italy.     

Extremality and the global Markov property II: The global markov property for non-FKG maximal Gibbs measures

We give a condition on a Gibbs measure for an attractive Markov specification, which assures extremality and the global Markov property. As an example of application we consider the class of attractive Markov specifications defined on a compact configuration space over a two-dimensional lattice by the interaction Hamiltonians (assumed to have a finite set of periodic ground configurations) satisfying Peierl's condition. We prove that each extremal Gibbs measure for such a specification, at sufficiently low temperature, has the global Markov property.On leave of absence from the Institute of Theoretical Physics, University of Wrocaw, Poland.     

Distributionally robust modeling of optimal control

The aim of this paper is to formulate several questions related to distributionally robust Stochastic Optimal Control modeling. As an example the distributionally robust counterpart of the classical inventory model is discussed in details. Finite and infinite horizon stationary settings are considered.     

Cramér theorem on symmetric spaces of noncompact type

We prove the Cramér theorem forK-invariant Gaussian measures on irreducible symmetric spacesX=G/K withG semisimple noncompact. To do this we use a kind of Abel transform ofK-invariant measures onX.This research is supported by KBN Grant.     

Characterization of measures by potentials

Let μ be a measure in a Banach spaceE, f be an even function onR. We consider the potentialg(a)=f _E f(‖x?a‖)dμ(x). The question is as follows: For whichf does the potentialg determine μ uniquely? In this article we give answers in the cases whereE=l _∞ ⁿ and wheref(t)=|t| ^p andE is a finite dimensional Banach space with symmetric analytic norm. Calculating the Fourier transform of the functionf(‖x‖ _∞) we give a new proof of the J. Misiewicz's result that the functionf(‖x‖ _∞) is positive definite only iff is a constant function.     

The projection method for computing multidimensional absolutely continuous invariant measures

We present an algorithm for numerically computing an absolutely continuous invariant measure associated with a piecewiseC ² expanding mappingS: on a bounded region R ^N. The method is based on the Galerkin projection principle for solving an operator equation in a Banach space. With the help of the modern notion of functions of bounded variation in multidimension, we prove the convergence of the algorithm.     

Some analytical semigroups occurring in probability theory

One-parameter semigroups occurring in operator-limit distributions are investigated. The topological-algebraic background of the relevant monoids is discussed and Lie semigroup theory is applied to the Urbanik Decomposability Semigroup.