Following the seminal paper by Bourgain, Brezis, and Mironescu, we focus on the asymptotic behaviour of some nonlocal functionals that, for each $u\in L^2(\mathbb {R}^N)$, are defined as the double integrals of weighted, squared difference quotients of $u$. Given a family of weights $\{\rho _{\varepsilon} \}$, $\varepsilon \in (0,\,1)$, we devise sufficient and necessary conditions on $\{\rho _{\varepsilon} \}$ for the associated nonlocal functionals to converge as $\varepsilon \to 0$ to a variant of the Dirichlet integral. Finally, some comparison between our result and the existing literature is provided.

We prove a joint partial equidistribution result for common perpendiculars with given density on equidistributing equidistant hypersurfaces, towards a measure supported on truncated stable leaves. We recover a result of Marklof on the joint partial equidistribution of Farey fractions at a given density, and give several analogous arithmetic applications, including in Bruhat–Tits trees.

For given Boolean algebras $\mathbb {A}$ and $\mathbb {B}$ we endow the space $\mathcal {H}(\mathbb {A},\mathbb {B})$ of all Boolean homomorphisms from $\mathbb {A}$ to $\mathbb {B}$ with various topologies and study convergence properties of sequences in $\mathcal {H}(\mathbb {A},\mathbb {B})$ . We are in particular interested in the situation when $\mathbb {B}$ is a measure algebra as in this case we obtain a natural tool for studying topological convergence properties of sequences of ultrafilters on $\mathbb {A}$ in random extensions of the set-theoretical universe. This appears to have strong connections with Dow and Fremlin’s result stating that there are Efimov spaces in the random model. We also investigate relations between topologies on $\mathcal {H}(\mathbb {A},\mathbb {B})$ for a Boolean algebra $\mathbb {B}$ carrying a strictly positive measure and convergence properties of sequences of measures on $\mathbb {A}$ .

Let $(X,T)$ be a topological dynamical system consisting of a compact metric space X and a continuous surjective map $T : X \to X$ . By using local entropy theory, we prove that $(X,T)$ has uniformly positive entropy if and only if so does the induced system $({\mathcal {M}}(X),\widetilde {T})$ on the space of Borel probability measures endowed with the weak* topology. This result can be seen as a version for the notion of uniformly positive entropy of the corresponding result for topological entropy due to Glasner and Weiss.

We study a class of optimal transport planning problems where the reference cost involves a non-linear function G(x, p) representing the transport cost between the Dirac measure δx and a target probability p. This allows to consider interesting models which favour multi-valued transport maps in contrast with the classical linear case ($G(x,p)=\int c(x,y)dp$) where finding single-valued optimal transport is a key issue. We present an existence result and a general duality principle which apply to many examples. Moreover, under a suitable subadditivity condition, we derive a Kantorovich–Rubinstein version of the dual problem allowing to show existence in some regular cases. We also consider the well studied case of Martingale transport and present some new perspectives for the existence of dual solutions in connection with Γ-convergence theory.

In the realm of Delone sets in locally compact, second countable Hausdorff groups, we develop a dynamical systems approach in order to study the continuity behavior of measured quantities arising from point sets. A special focus is both on the autocorrelation, as well as on the density of states for random bounded operators. It is shown that for uniquely ergodic limit systems, the latter measures behave continuously with respect to the Chabauty–Fell convergence of hulls. In the special situation of Euclidean spaces, our results complement recent developments in describing spectra as topological limits: we show that the measured quantities under consideration can be approximated via periodic analogs.

We consider a meromorphic family of endomorphisms of degree at least 2 of a complex projective space that is parameterized by the unit disk. We prove that the measure of maximal entropy of these endomorphisms converges to the equilibrium measure of the associated non-Archimedean dynamical system when the system degenerates. The convergence holds in the hybrid space constructed by Berkovich and further studied by Boucksom and Jonsson. We also infer from our analysis an estimate for the blow-up of the Lyapunov exponent near a pole in one-dimensional families of endomorphisms.

We look at joint regular variation properties of MA(∞) processes of the form X = (Xk, k ∈ Z), where Xk = ∑j=0∞ψjZk-j and the sequence of random variables (Zi, i ∈ Z) are independent and identically distributed with regularly varying tails. We use the setup of MO-convergence and obtain hidden regular variation properties for X under summability conditions on the constant coefficients (ψj: j ≥ 0). Our approach emphasizes continuity properties of mappings and produces regular variation in sequence space.

Given a non-atomic, finite and complete measure space $(\Omega,\Sigma,\mu)$ and a Banach space $X$, the modulus of continuity for a vector measure $F$ is defined as the function $\omega_F(t) = \sup_{\mu(E)\le t} |F|(E)$ and the space ${V}^{p,q}(X)$ of vector measures such that $t^{-1/p^\prime}\wf (t)\in L^q((0,\mu(\Omega)],dt/t)$ is introduced. It is shown that $V^{p,q}(X)$ contains isometrically $\lpq(X)$ and that $L^{p,q}(X)=V^{p,q}(X)$ if and only if $X$ has the Radon–Nikodym property. It is also proved that ${V}^{p,q}(X)$ coincides with the space of cone absolutely summing operators from $L^{p^\prime,q^\prime}$ into $X$ and the duality ${V}^{p,q}(X^*)= (L^{p^\prime,q^\prime}(X))^*$ where $1/p+1/p^\prime= 1/q+1/q^\prime=1$. Finally, $V^{p,q}(X)$ is identified with the interpolation space obtained by the real method $(V^{1}(X)$, $V^{\infty}(X))_{1/p',q}$. Spaces where the variation of $F$ is replaced by the semivariation are also considered.

A new construction of the Gaussian distribution is introduced and proven. The procedure consists of using fractal interpolating functions, with graphs having increasing fractal dimensions, to transform arbitrary continuous probability measures defined over a closed interval. Specifically, let X be any probability measure on the closed interval I with a continuous cumulative distribution. And let fΘ,D:I → R be a deterministic continuous fractal interpolating function, as introduced by Barnsley (1986), with parameters Θ and fractal dimension for its graph D. Then, the derived measure Y = fΘ,D(X) tends to a Gaussian for all parameters Θ such that D → 2, for all X. This result illustrates that plane-filling fractal interpolating functions are ‘intrinsically Gaussian'. It explains that close approximations to the Gaussian may be obtained transforming any continuous probability measure via a single nearly-plane filling fractal interpolator.

In this paper we consider improvements in the rate of approximation for the distribution of sums of independent Bernoulli random variables via convolutions of Poisson measures with signed measures of specific type. As a special case, the distribution of the number of records in an i.i.d. sequence of length n is investigated. For this particular example, it is shown that the usual rate of Poisson approximation of O(1/log n) can be lowered to O(1/n2). The general case is discussed in terms of operator semigroups.

A geometric version of the Vitali integral convergence theorem is established. Parameterized versions of results on uniform absolute continuity in spaces of measures suggested by the convergence theorem are studied.