414,018 research outputs found

### Recurrence of cocycles and stationary random walks

We survey distributional properties of $\mathbb{R}^d$-valued cocycles of
finite measure preserving ergodic transformations (or, equivalently, of
stationary random walks in $\mathbb{R}^d$) which determine recurrence or
transience.Comment: Published at http://dx.doi.org/10.1214/074921706000000112 in the IMS
Lecture Notes--Monograph Series
(http://www.imstat.org/publications/lecnotes.htm) by the Institute of
Mathematical Statistics (http://www.imstat.org

### New exact solutions for power-law inflation Friedmann models

We consider the spatially flat Friedmann model. For a(t) = t^p, especially,
if p is larger or equal to 1, this is called power-law inflation. For the
Lagrangian L = R^m with p = - (m - 1)(2m - 1)/(m - 2), power-law inflation is
an exact solution, as it is for Einstein gravity with a minimally coupled
scalar field Phi in an exponential potential V(Phi) = exp(mu Phi) and also for
the higher-dimensional Einstein equation with a special Kaluza-Klein ansatz.
The synchronized coordinates are not adapted to allow a closed-form solution,
so we use another gauge. Finally, special solutions for the closed and open
Friedmann model are found.Comment: 9 pages, LaTeX, reprinted from Astron. Nachr. 311 (1990) 16

### The Parabolic Anderson Model with Acceleration and Deceleration

We describe the large-time moment asymptotics for the parabolic Anderson
model where the speed of the diffusion is coupled with time, inducing an
acceleration or deceleration. We find a lower critical scale, below which the
mass flow gets stuck. On this scale, a new interesting variational problem
arises in the description of the asymptotics. Furthermore, we find an upper
critical scale above which the potential enters the asymptotics only via some
average, but not via its extreme values. We make out altogether five phases,
three of which can be described by results that are qualitatively similar to
those from the constant-speed parabolic Anderson model in earlier work by
various authors. Our proofs consist of adaptations and refinements of their
methods, as well as a variational convergence method borrowed from finite
elements theory.Comment: 19 page

### Central limit theorems for Poisson hyperplane tessellations

We derive a central limit theorem for the number of vertices of convex
polytopes induced by stationary Poisson hyperplane processes in $\mathbb{R}^d$.
This result generalizes an earlier one proved by Paroux [Adv. in Appl. Probab.
30 (1998) 640--656] for intersection points of motion-invariant Poisson line
processes in $\mathbb{R}^2$. Our proof is based on Hoeffding's decomposition of
$U$-statistics which seems to be more efficient and adequate to tackle the
higher-dimensional case than the ``method of moments'' used in [Adv. in Appl.
Probab. 30 (1998) 640--656] to treat the case $d=2$. Moreover, we extend our
central limit theorem in several directions. First we consider $k$-flat
processes induced by Poisson hyperplane processes in $\mathbb{R}^d$ for $0\le
k\le d-1$. Second we derive (asymptotic) confidence intervals for the
intensities of these $k$-flat processes and, third, we prove multivariate
central limit theorems for the $d$-dimensional joint vectors of numbers of
$k$-flats and their $k$-volumes, respectively, in an increasing spherical
region.Comment: Published at http://dx.doi.org/10.1214/105051606000000033 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org

### Limit theorems for functionals on the facets of stationary random tessellations

We observe stationary random tessellations $X=\{\Xi_n\}_{n\ge1}$ in
$\mathbb{R}^d$ through a convex sampling window $W$ that expands unboundedly
and we determine the total $(k-1)$-volume of those $(k-1)$-dimensional manifold
processes which are induced on the $k$-facets of $X$ ($1\le k\le d-1$) by their
intersections with the $(d-1)$-facets of independent and identically
distributed motion-invariant tessellations $X_n$ generated within each cell
$\Xi_n$ of $X$. The cases of $X$ being either a Poisson hyperplane tessellation
or a random tessellation with weak dependences are treated separately. In both
cases, however, we obtain that all of the total volumes measured in $W$ are
approximately normally distributed when $W$ is sufficiently large. Structural
formulae for mean values and asymptotic variances are derived and explicit
numerical values are given for planar Poisson--Voronoi tessellations (PVTs) and
Poisson line tessellations (PLTs).Comment: Published at http://dx.doi.org/10.3150/07-BEJ6131 in the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

### Tame class field theory for arithmetic schemes

We extend the unramified class field theory for arithmetic schemes of K. Kato
and S. Saito to the tame case. Let $X$ be a regular proper arithmetic scheme
and let $D$ be a divisor on $X$ whose vertical irreducible components are
normal schemes.
Theorem: There exists a natural reciprocity isomorphism \rec_{X,D}:
\CH_0(X,D) \liso \tilde \pi_1^t(X,D)^\ab\. Both groups are finite.
This paper corrects and generalizes my paper "Relative K-theory and class
field theory for arithmetic surfaces" (math.NT/0204330

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