496 research outputs found
New Bounds for Edge-Cover by Random Walk
We show that the expected time for a random walk on a (multi-)graph to
traverse all edges of , and return to its starting point, is at most
; if each edge must be traversed in both directions, the bound is .
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
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
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 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 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
The Corona Theorem for the Drury-Arveson Hardy space and other holomorphic Besov-Sobolev spaces on the unit ball in
We prove that the multiplier algebra of the Drury-Arveson Hardy space
on the unit ball in 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
has the "baby corona property" for all and
. 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
Generalized uncertainty inequalities
In this paper, Heisenberg-Pauli-Weyl-type uncertainty inequalities are
obtained for a pair of positive-self adjoint operators on a Hilbert space,
whose spectral projectors satisfy a ``balance condition'' involving certain
operator norms. This result is then applied to obtain uncertainty inequalities
on Riemannian manifolds, Riemannian symmetric spaces of non-compact type,
homogeneous graphs and unimodular Lie groups with sublaplacians.Comment: 19 page
On gradient bounds for the heat kernel on the Heisenberg group
It is known that the couple formed by the two dimensional Brownian motion and
its L\'evy area leads to the heat kernel on the Heisenberg group, which is one
of the simplest sub-Riemannian space. The associated diffusion operator is
hypoelliptic but not elliptic, which makes difficult the derivation of
functional inequalities for the heat kernel. However, Driver and Melcher and
more recently H.-Q. Li have obtained useful gradient bounds for the heat kernel
on the Heisenberg group. We provide in this paper simple proofs of these
bounds, and explore their consequences in terms of functional inequalities,
including Cheeger and Bobkov type isoperimetric inequalities for the heat
kernel.Comment: Minor correction
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