New Bounds for Edge-Cover by Random Walk

We show that the expected time for a random walk on a (multi-)graph $G$ to traverse all $m$ edges of $G$, and return to its starting point, is at most $2m^2$; if each edge must be traversed in both directions, the bound is $3m^2$. Both bounds are tight and may be applied to graphs with arbitrary edge lengths, with implications for Brownian motion on a finite or infinite network of total edge-length $m$

Recurrence and pressure for group extensions

We investigate the thermodynamic formalism for recurrent potentials on group extensions of countable Markov shifts. Our main result characterises recurrent potentials depending only on the base space, in terms of the existence of a conservative product measure and a homomorphism from the group into the multiplicative group of real numbers. We deduce that, for a recurrent potential depending only on the base space, the group is necessarily amenable. Moreover, we give equivalent conditions for the base pressure and the skew product pressure to coincide. Finally, we apply our results to analyse the Poincar\'e series of Kleinian groups and the cogrowth of group presentations

Approximations of Sobolev norms in Carnot groups

This paper deals with a notion of Sobolev space $W^{1,p}$ introduced by J.Bourgain, H.Brezis and P.Mironescu by means of a seminorm involving local averages of finite differences. This seminorm was subsequently used by A.Ponce to obtain a Poincar\'e-type inequality. The main results that we present are a generalization of these two works to a non-Euclidean setting, namely that of Carnot groups. We show that the seminorm expressd in terms of the intrinsic distance is equivalent to the $L^p$ norm of the intrinsic gradient, and provide a Poincar\'e-type inequality on Carnot groups by means of a constructive approach which relies on one-dimensional estimates. Self-improving properties are also studied for some cases of interest

Kramers-Moyall cumulant expansion for the probability distribution of parallel transporters in quantum gauge fields

A general equation for the probability distribution of parallel transporters on the gauge group manifold is derived using the cumulant expansion theorem. This equation is shown to have a general form known as the Kramers-Moyall cumulant expansion in the theory of random walks, the coefficients of the expansion being directly related to nonperturbative cumulants of the shifted curvature tensor. In the limit of a gaussian-dominated QCD vacuum the obtained equation reduces to the well-known heat kernel equation on the group manifold.Comment: 7 page

The Corona Theorem for the Drury-Arveson Hardy space and other holomorphic Besov-Sobolev spaces on the unit ball in Cn\mathbb{C}^{n}

We prove that the multiplier algebra of the Drury-Arveson Hardy space $H_{n}^{2}$ on the unit ball in $\mathbb{C}^{n}$ has no corona in its maximal ideal space, thus generalizing the famous Corona Theorem of L. Carleson to higher dimensions. This result is obtained as a corollary of the Toeplitz corona theorem and a new Banach space result: the Besov-Sobolev space $B_{p}^{\sigma}$ has the "baby corona property" for all $\sigma \geq 0$ and $1<p<\infty$. In addition we obtain infinite generator and semi-infinite matrix versions of these theorems.Comment: v1: 70 pgs; v2: 73 pgs.; introduction expanded, clarified. v3: 73 pgs.; restriction in main result removed (see 9.2), arguments expanded (see 8.1.1). v4: 74 pgs.; systematic arithmetic misprints fixed on pgs. 37-48. v5: 76 pgs.; incorrect embedding corrected via Proposition 4. v6: 80 pgs.; main result extended to vector-valued setting. v7: 82 pgs.; typos removed

Harnack Inequality and Regularity for a Product of Symmetric Stable Process and Brownian Motion

In this paper, we consider a product of a symmetric stable process in $\mathbb{R}^d$ and a one-dimensional Brownian motion in $\mathbb{R}^+$. Then we define a class of harmonic functions with respect to this product process. We show that bounded non-negative harmonic functions in the upper-half space satisfy Harnack inequality and prove that they are locally H\"older continuous. We also argue a result on Littlewood-Paley functions which are obtained by the $\alpha$-harmonic extension of an $L^p(\mathbb{R}^d)$ function.Comment: 23 page

Uncertainty inequalities on groups and homogeneous spaces via isoperimetric inequalities

We prove a family of $L^p$ uncertainty inequalities on fairly general groups and homogeneous spaces, both in the smooth and in the discrete setting. The crucial point is the proof of the $L^1$ endpoint, which is derived from a general weak isoperimetric inequality.Comment: 17 page

Random walks of Wilson loops in the screening regime

Dynamics of Wilson loops in pure Yang-Mills theories is analyzed in terms of random walks of the holonomies of the gauge field on the gauge group manifold. It is shown that such random walks should necessarily be free. The distribution of steps of these random walks is related to the spectrum of string tensions of the theory and to certain cumulants of Yang-Mills curvature tensor. It turns out that when colour charges are completely screened, the holonomies of the gauge field can change only by the elements of the group center, which indicates that in the screening regime confinement persists due to thin center vortices. Thick center vortices are also considered and the emergence of such stepwise changes in the limits of infinitely thin vortices and infinitely large loops is demonstrated.Comment: Major revision of the previous version, to appear in Nucl. Phys. B (10 pages RevTeX, 3 figures

LpL^p-Spectral theory of locally symmetric spaces with QQ-rank one

We study the $L^p$-spectrum of the Laplace-Beltrami operator on certain complete locally symmetric spaces $M=\Gamma\backslash X$ with finite volume and arithmetic fundamental group $\Gamma$ whose universal covering $X$ is a symmetric space of non-compact type. We also show, how the obtained results for locally symmetric spaces can be generalized to manifolds with cusps of rank one

When a totally bounded group topology is the Bohr Topology of a LCA group

We look at the Bohr topology of maximally almost periodic groups (MAP, for short). Among other results, we investigate when a totally bounded abelian group $(G,w)$ is the Bohr reflection of a locally compact abelian group. Necessary and sufficient conditions are established in terms of the inner properties of $w$. As an application, an example of a MAP group $(G,t)$ is given such that every closed, metrizable subgroup $N$ of $bG$ with $N \cap G = \{0\}$ preserves compactness but $(G,t)$ does not strongly respects compactness. Thereby, we respond to Questions 4.1 and 4.3 in [comftrigwu]