Formulation and properties of a divergence used to compare probability measures without absolute continuity

This paper develops a new divergence that generalizes relative entropy and can be used to compare probability measures without a requirement of absolute continuity. We establish properties of the divergence, and in particular derive and exploit a representation as an infimum convolution of optimal transport cost and relative entropy. Also included are examples of computation and approximation of the divergence, and the demonstration of properties that are useful when one quantifies model uncertainty.

Statistical solution and Liouville type theorem for coupled Schrödinger-Boussinesq equations on infinite lattices

<p style='text-indent:20px;'>In this article, we are concerned with statistical solutions for the nonautonomous coupled Schrödinger-Boussinesq equations on infinite lattices. Firstly, we verify the existence of a pullback-<inline-formula><tex-math id="M1">\begin{document}$ {\mathcal{D}} $\end{document}</tex-math></inline-formula> attractor and establish the existence of a unique family of invariant Borel probability measures carried by the pullback-<inline-formula><tex-math id="M2">\begin{document}$ {\mathcal{D}} $\end{document}</tex-math></inline-formula> attractor for this lattice system. Then, it will be shown that the family of invariant Borel probability measures is a statistical solution and satisfies a Liouville type theorem. Finally, we illustrate that the invariant property of the statistical solution is indeed a particular case of the Liouville type theorem.</p>

Coercive Inequalities and U-Bounds on Step-Two Carnot Groups

We prove Poincaré and Logβ-Sobolev inequalities for a class of probability measures on step-two Carnot groups.

Multifractal dimensions for projections of measures

In this paper, we calculate the multifractal Hausdorff and packing dimensions of Borel probability measures and study their behaviors under orthogonal projections. In particular, we try through these results to improve the main result of M. Dai in \cite{D} about the multifractal analysis of a measure of multifractal exact dimension.

Probabilities

This chapter offers a BST theory of propensities (i.e., of objective single-case probabilities), which builds on the account of indeterministic causation developed in Chapter 6. Propensities are shown to deliver classical (Kolmogorovian) probability spaces. The chapter draws a distinction between propensities and probability measures. The former are assigned to sets of BST transitions, in particular to sets of causae causantes of transitions, and are interpreted as degrees of possibility of these transitions. The latter are defined in terms of propensities and are measures of Komogorovian probability spaces. Features of propensities are derived from a logico-causal analysis. Finally, the chapter discusses how the theory developed here handles well-known objections to propensities due to Humphreys and to Salmon, especially Humphreys’s paradox.

Dimensions of ‘self-affine sponges’ invariant under the action of multiplicative integers

Let $m_1 \geq m_2 \geq 2$ be integers. We consider subsets of the product symbolic sequence space $(\{0,\ldots ,m_1-1\} \times \{0,\ldots ,m_2-1\})^{\mathbb {N}^*}$ that are invariant under the action of the semigroup of multiplicative integers. These sets are defined following Kenyon, Peres, and Solomyak and using a fixed integer $q \geq 2$ . We compute the Hausdorff and Minkowski dimensions of the projection of these sets onto an affine grid of the unit square. The proof of our Hausdorff dimension formula proceeds via a variational principle over some class of Borel probability measures on the studied sets. This extends well-known results on self-affine Sierpiński carpets. However, the combinatoric arguments we use in our proofs are more elaborate than in the self-similar case and involve a new parameter, namely $j = \lfloor \log _q ( {\log (m_1)}/{\log (m_2)} ) \rfloor $ . We then generalize our results to the same subsets defined in dimension $d \geq 2$ . There, the situation is even more delicate and our formulas involve a collection of $2d-3$ parameters.

Expectations

The expectation is defined, applying integration concepts to probability measures. Leading examples are given and the different characterizations of the expectation of a function are compared. The Markov and Jensen inequalities are given and consideration of multivariate distributions then leads to the treatment of the Cauchy–Schwarz, Hölder, Liapunov, Minkowski, and Loève inequalities. The final section treats the calculus of random functions of a real variable.

Weak Convergence

This chapter reviews the theory of weak convergence in metric spaces. Topics include Skorokhod’s representation theorem, the metrization of spaces of measures, and the concept of tightness of probability measures. The key relation is shown between weak convergence and uniform tightness. Considering the space C of continuous functions in particular, the functional central limit theorem is proved for martingales, together with extensions to the multivariate case.

Probability Spaces

This chapter defines probability measures and probability spaces in a general context, as a case of the concepts introduced in Chapter 3. The axioms of probability are explained, and the important concepts of conditional probability and independence are introduced and linked to the role of product spaces and product measures.