Extremes of space–time Gaussian processes

Let Z={_Zt(h);h∈R^d,t∈R}$Z=\{Z_t\left(h\right);h\in {\mathbb{R}}^d,t\in \mathbb{R}\}$ be a space–time Gaussian process which is stationary in the time variable t$t$. We study _Mn(h)=_{supt∈[0,n]Zt}(_snh)$M_n\left(h\right)={sup}_{t\in [0,n]}{Z}_t\left(s_{n}h\right)$, the supremum of Z$Z$ taken over t∈[0,n]$t\in [0,n]$ and rescaled by a properly chosen sequence _sn→0$s_n\to 0$. Under appropriate conditions on Z$Z$, we show that for some normalizing sequence _bn→∞$b_n\to \infty$, the process _bn(_Mn−_bn)$b_n(M_n-b_n)$ converges as n→∞$n\to \infty$ to a stationary max-stable process of Brown–Resnick type. Using strong approximation, we derive an analogous result for the empirical process.     

Weitzenböck derivations of free metabelian Lie algebras

A nonzero locally nilpotent linear derivation δ   of the polynomial algebra K[_Xd]=K[_x1,…,_xd]$K[X_d]=K[x_1\text{,}\dots \text{,}{x}_d]$ in several variables over a field K   of characteristic 0 is called a Weitzenböck derivation. The classical theorem of Weitzenböck states that the algebra of constants K[_Xd]^δ$K{[X_d]}^{\delta }$ (which coincides with the algebra of invariants of a single unipotent transformation) is finitely generated. Similarly one may consider the algebra of constants of a locally nilpotent linear derivation δ of a finitely generated (not necessarily commutative or associative) algebra which is relatively free in a variety of algebras over K  . Now the algebra of constants is usually not finitely generated. Except for some trivial cases this holds for the algebra of constants (_Ld/_Ld^″)^δ${(L_d/{L_d}^{″})}^{\delta }$ of the free metabelian Lie algebra _Ld/_Ld″$L_d/L_d″$ with d   generators. We show that the vector space of the constants (_Ld/_Ld^″)^δ${(L_d/{L_d}^{″})}^{\delta }$ in the commutator ideal _Ld′/_Ld″$L_d\prime /L_d″$ is a finitely generated K[_Xd]^δ$K{[X_d]}^{\delta }$-module. For small d  , we calculate the Hilbert series of (_Ld/_Ld^″)^δ${(L_d/{L_d}^{″})}^{\delta }$ and find the generators of the K[_Xd]^δ$K{[X_d]}^{\delta }$-module (_Ld/_Ld^″)^δ${(L_d/{L_d}^{″})}^{\delta }$. This gives also an (infinite) set of generators of the algebra (_Ld/_Ld^″)^δ${(L_d/{L_d}^{″})}^{\delta }$.     

Canonical variation of a Lorentzian metric

Given a Lorentzian manifold (M,_gL)$(M,g_L)$ and a timelike unitary vector field E  , we can construct the Riemannian metric _gR=_gL+2ω⊗ω$g_R=g_L+2\omega \otimes \omega$, ω being the metrically equivalent one form to E. We relate the curvature of both metrics, especially in the case of E   being Killing or closed, and we use the relations obtained to give some results about (M,_gL)$(M,g_L)$.     

On the number of empty boxes in the Bernoulli sieve II

The Bernoulli sieve is the infinite “balls-in-boxes” occupancy scheme with random frequencies _Pk=_W1?_Wk−1(1−_Wk)$P_k=W_1?W_{k-1}(1-W_k)$, where _(Wk)k∈N${\left(W_k\right)}_{k\in \mathbb{N}}$ are independent copies of a random variable W$W$ taking values in (0,1)$(0,1)$. Assuming that the number of balls equals n$n$, let _Ln$L_n$ denote the number of empty boxes within the occupancy range. In this paper, we investigate convergence in distribution of _Ln$L_n$ in the two cases which remained open after the previous studies. In particular, provided that E|logW|=E|log(1−W)|=∞$\mathbb{E}|logW|=\mathbb{E}|log(1-W)|=\infty$ and that the law of W$W$ assigns comparable masses to the neighborhoods of 0 and 1, it is shown that _Ln$L_n$ weakly converges to a geometric law. This result is derived as a corollary to a more general assertion concerning the number of zero decrements of nonincreasing Markov chains. In the case that E|logW|<∞$\mathbb{E}|logW|<\infty$ and E|log(1−W)|=∞$\mathbb{E}|log(1-W)|=\infty$, we derive several further possible modes of convergence in distribution of _Ln$L_n$. It turns out that the class of possible limiting laws for _Ln$L_n$, properly normalized and centered, includes normal laws and spectrally negative stable laws with finite mean. While investigating the second problem, we develop some general results concerning the weak convergence of renewal shot-noise processes. This allows us to answer a question asked by Mikosch and Resnick (2006) [18].     

Poincaré type inequalities for group measure spaces and related transportation cost inequalities

Let G be a countable discrete group with an orthogonal representation α on a real Hilbert space H  . We prove _Lp$L_p$ Poincaré inequalities for the group measure space _L∞(_ΩH,γ)?G$L_{\infty }({\Omega }_H,\gamma )?G$, where both the group action and the Gaussian measure space (_ΩH,γ)$({\Omega }_H,\gamma )$ are associated with the representation α  . The idea of proof comes from Pisier?s method on the boundedness of Riesz transform and Lust-Piquard?s work on spin systems. Then we deduce a transportation type inequality from the _Lp$L_p$ Poincaré inequalities in the general noncommutative setting. This inequality is sharp up to a constant (in the Gaussian setting). Several applications are given, including Wiener/Rademacher chaos estimation and new examples of Rieffel?s compact quantum metric spaces.     

On the tractability of linear tensor product problems in the worst case

It has been an open problem to derive a necessary and sufficient condition for a linear tensor product problem to be weakly tractable in the worst case. The complexity of linear tensor product problems in the worst case depends on the eigenvalues _{λi}i∈N${\left\{{\lambda }_i\right\}}_{i\in \mathbb{N}}$ of a certain operator. It is known that if _λ1=1${\lambda }_1=1$ and _λ2∈(0,1)${\lambda }_2\in (0,1)$ then _λn=o((lnn)⁻²)${\lambda }_n=o\left({(lnn)}^{-2}\right)$, as n→∞$n\to \infty$, is a necessary condition for a problem to be weakly tractable. We show that this is a sufficient condition as well.     

The Calderón–Zygmund inequality and Sobolev spaces on noncompact Riemannian manifolds

We introduce the concept of Calderón–Zygmund inequalities on Riemannian manifolds. For 1<p<∞$1<p<\infty$, these are inequalities of the form

_{‖Hess(u)‖Lp}≤_C1‖u‖Lp+_{C2‖Δu‖Lp},${\Vert \mathrm{Hess}(u)\Vert }_{{\mathsf{L}}^p}\le C_1{\Vert u\Vert }_{{\mathsf{L}}^p}+C_2{\Vert \mathrm{\Delta }u\Vert }_{{\mathsf{L}}^p},$

valid a priori for all smooth functions u   with compact support, and constants _C1≥0$C_1\ge 0$, _C2>0$C_2>0$. Such an inequality can hold or fail, depending on the underlying Riemannian geometry. After establishing some generally valid facts and consequences of the Calderón–Zygmund inequality (like new denseness results for second order L^p${\mathsf{L}}^p$-Sobolev spaces and gradient estimates), we establish sufficient geometric criteria for the validity of these inequalities on possibly noncompact Riemannian manifolds. These results in particular apply to many noncompact hypersurfaces of constant mean curvature.     

Hypergroups and invariant complemented subspaces

Let K   be a hypergroup with a Haar measure. The purpose of the present paper is to initiate a systematic approach to the study of the class of invariant complemented subspaces of _L∞(K)$L_{\infty }(K)$ and _C0(K)$C_0(K)$, the class of left translation invariant w^?$w^{?}$-subalgebras of _L∞(K)$L_{\infty }(K)$ and finally the class of non-zero left translation invariant C^?$C^{?}$-subalgebras of _C0(K)$C_0(K)$ in the hypergroup context with the goal of finding some relations between these function spaces. Among other results, we construct two correspondences: one, between closed Weil subhypergroups and certain left translation invariant w^?$w^{?}$-subalgebras of _L∞(K)$L_{\infty }(K)$, and another, between compact subhypergroups and a specific subclass of the class of left translation invariant C^?$C^{?}$-subalgebras of _C0(K)$C_0(K)$. By the help of these two characterizations, we extract some results about invariant complemented subspaces of _L∞(K)$L_{\infty }(K)$ and _C0(K)$C_0(K)$.