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Spatial or geometric quantiles are among the most celebrated concepts of multivariate quantiles. The spatial quantile \(\mu _{\alpha ,u}(P)\) of a probability measure P over \(\mathbb {R}^d\) is a point in \(\mathbb R^d\) indexed by an order \(\alpha \in [0,1)\) and a direction u in the unit sphere \(\mathcal {S}^{d-1}\) of \(\mathbb R^d\)—or equivalently by a vector \(\alpha u\) in the open unit ball of \(\mathbb R^d\). Recently, Girard and Stupfler (2017) proved that (i) the extreme quantiles \(\mu _{\alpha ,u}(P)\) obtained as \(\alpha \rightarrow 1\) exit all compact sets of \(\mathbb R^d\) and that (ii) they do so in a direction converging to u. These results help understanding the nature of these quantiles: the first result is particularly striking as it holds even if P has a bounded support, whereas the second one clarifies the delicate dependence of spatial quantiles on u. However, they were established under assumptions imposing that P is non-atomic, so that it is unclear whether they hold for empirical probability measures. We improve on this by proving these results under much milder conditions, allowing for the sample case. This prevents using gradient condition arguments, which makes the proofs very challenging. We also weaken the well-known sufficient condition for the uniqueness of finite-dimensional spatial quantiles.

The paper studies compactness properties of the affine Sobolev inequality of Lutwak et al. (J Differ Geom 62:17–38, 2002) and Zhang (J Differ Geom 53:183–202, 1999) in the case $$p=2$$ , and existence and regularity of related minimizers, in particular, solutions to the nonlocal Dirichlet problems $$\begin{aligned} -\sum _{i,j=1}^{N}(A^{-1}[u])_{ij}\frac{\partial ^2u}{\partial x_i\partial x_j}=f \text{ in } \Omega \subset {\mathbb {R}}^N, \end{aligned}$$ and $$\begin{aligned} -\sum _{i,j=1}^{N}(A^{-1}[u])_{ij}\frac{\partial ^2u}{\partial x_i\partial x_j}=u^{q-1},\quad u>0, \text{ in } \Omega \subset {\mathbb {R}}^N, \end{aligned}$$ where $$A_{ij}[u]=\int _\Omega \frac{\partial u}{\partial x_i}\frac{\partial u}{\partial x_j}{\mathrm {d}}{x}$$ and $$q\in (2,\frac{2N}{N-2})$$ .

Let $G$ be a real connected algebraic semi-simple Lie group, and $H$ an algebraic subgroup of $G$. Let $\mu$ be a probability measure on $G$, with finite exponential moment, whose support spans a Zariski-dense subsemigroup of $G$. Let $X=G/H$ be the quotient of $G$ by $H$. We study the Markov chain on $X$ with transition probability $P_x=\mu *\delta_x$ for $x\in X$. We prove that either for every $x\in X$, almost every trajectory starting from $x$ is transient or for every $x\in X$, almost every trajectory starting from $x$ is recurrent. In fact, this recurrence is uniform over all $X$, i.e. there exists a compact set $C\subset X$ such that for each point $x\in X$, every trajectory starting in $x$ almost surely returns to $C$ infinitely often. Furthermore, we give a criterion for recurrence depending on $G$, $H$, and $\mu$., Comment: in French

Soit T une transformation mesurable d’un espace probabilisé $(E,\mathcal {E},\pi)$ préservant la mesure π et soit $B\in \mathcal {E}$. Nous donnons une condition suffisante pour que l’orbite sous T de π-presque tout point visite B : il suffit que B soit accessible depuis π-presque tout point pour une chaîne de Markov de noyau K, où K(⋅, ⋅) est une version régulière de la loi conditionnelle de X sachant T(X) lorsque X est une variable aléatoire de loi π. ¶ Nous appliquons ensuite ce résultat général à la transformation de Lévy, qui à un mouvement brownien W associe le mouvement brownien |W| − L où L est le temps local en 0 de W. Cela nous permet de donner une nouvelle démonstration du théorème de Malric qui affirme que l’orbite sous la transformation de Lévy de presque toute trajectoire visite tout ouvert non vide de l’espace de Wiener pour la topologie de la convergence uniforme sur les compacts.

We prove existence (uniqueness is easy) of a weak solution to a boundary value problem for an equation like $(v-1)^+_t = v_{xx} + F(v)_x$ where the function $F: \Bbb R\rightarrow\Bbb R$ is only supposed to be locally lipschitz continuous. In order to replace the lack of compactness in t on v , we use nonlinear semigroup theory.

The special setQ ofWightman's functionals is considered. All the pure functionals that belong toQ have the strong spectral property and the single vacuum. All other functionals fromQ may be represented by integrals over (weak) compact sets of pure functionals fromQ. The Borchers structure theorem is proved for this set. The conditions of the unitary equivalence of the representations and their similarity are given. These results are applied for the analysis of the symmetry problem. In particular we formulate the necessary and sufficient condition for the given group of automorphisms to be unitarily implemented.