Average Densities,Tangent Measures and Rectifiability

Periodica Mathematica Hungarica -     

Tangent measure distributions of hyperbolic Cantor sets

Tangent measure distributions were introduced byBandt [2] andGraf [8] as a means to describe the local geometry of self-similar sets generated by iteration of contractive similitudes. In this paper we study the tangent measure distributions of hyperbolic Cantor sets generated by certain contractive mappings, which are not necessarily similitudes. We show that the tangent measure distributions of these sets equipped with either Hausdorff- or Gibbs measure are unique almost everywhere and give an explicit formula describing them as probability distributions on the set of limit models ofBedford andFisher [5].     

Brownian intersection local times: Exponential moments and law of large masses

Consider independent Brownian motions in , each running up to its first exit time from an open domain , and their intersection local time as a measure on . We give a sharp criterion for the finiteness of exponential moments,

where are nonnegative, bounded functions with compact support in . We also derive a law of large numbers for intersection local time conditioned to have large total mass.

     

The Semi-infinite Asymmetric Exclusion Process: Large Deviations via Matrix Products

We study the totally asymmetric exclusion process on the positive integers with a single particle source at the origin. Liggett (Trans. Am. Math. Soc. 213, 237–261, 1975) has shown that the long term behaviour of this process has a phase transition: If the particle production rate at the source and the original density are below a critical value, the stationary measure is a product measure, otherwise the stationary measure is spatially correlated. Following the approach of Derrida et al. (J. Phys. A 26(7), 1493, 1993) it was shown by Grosskinsky (2004) that these correlations can be described by means of a matrix product representation. In this paper we derive a large deviation principle with explicit rate function for the particle density in a macroscopic box based on this representation. The novel and rigorous technique we develop for this problem combines spectral theoretical and combinatorial ideas and is potentially applicable to other models described by matrix products.     

Skorokhod embeddings for two-sided Markov chains

Let \((X_n :n\in \mathbb {Z})\) be a two-sided recurrent Markov chain with fixed initial state \(X_0\) and let \(\nu \) be a probability measure on its state space. We give a necessary and sufficient criterion for the existence of a non-randomized time T such that \((X_{T+n} :n\in \mathbb {Z})\) has the law of the same Markov chain with initial distribution \(\nu \). In the case when our criterion is satisfied we give an explicit solution, which is also a stopping time, and study its moment properties. We show that this solution minimizes the expectation of \(\psi (T)\) in the class of all non-negative solutions, simultaneously for all non-negative concave functions \(\psi \).     

Multiple Intersection Exponents for Planar Brownian Motion

Let p≥2, n ₁≤⋅⋅⋅≤n _p be positive integers and be independent planar Brownian motions started uniformly on the boundary of the unit circle. We define a p-fold intersection exponent ς _p (n ₁,…,n _p ), as the exponential rate of decay of the probability that the packets , i=1,…,p, have no joint intersection. The case p=2 is well-known and, following two decades of numerical and mathematical activity, Lawler et al. (Acta Math. 187:275–308, 2001) rigorously identified precise values for these exponents. The exponents have not been investigated so far for p>2. We present an extensive mathematical and numerical study, leading to an exact formula in the case n ₁=1, n ₂=2, and several interesting conjectures for other cases.     

How fast are the particles of super-Brownian motion?

In this paper we investigate fast particles in the range and support ofsuper-Brownian motion in the historical setting. In this setting eachparticle of super-Brownian motion alive at time t is represented by apath w:[0,t]→ℝ ^d and the state of historical super-Brownian motionis a measure on the set of paths. Typical particles have Brownian paths,however in the uncountable collection of particles in the range of asuper-Brownian motion there are some which at exceptional times movefaster than Brownian motion. We determine the maximal speed of allparticles during a given time period E, which turns out to be afunction of the packing dimension of E. A path w in the support ofhistorical super-Brownian motion at time t is called a-fast if . Wecalculate the Hausdorff dimension of the set of a-fast paths in thesupport and the range of historical super-Brownian motion. A valuabletool in the proofs is a uniform dimension formula for the Browniansnake, which reduces dimension problems in the space of stopped paths to dimension problems on the line. Received: 27 January 2000 / Revised version: 28 August 2000 / Published online: 24 July 2001     

The Hausdorff Dimension of the Double Points on the Brownian Frontier

The frontier of a planar Brownian motion is the boundary of the unbounded component of the complement of its range. In this paper, we find the Hausdorff dimension of the set of double points on the frontier.     

Thick points of super-Brownian motion

We determine for a super-Brownian motion {X_t:t0} in ^d, d3, the precise gauge function such that, almost surely on survival up to time t, improving a result of Barlow, Evans and Perkins about the most visited sites of super-Brownian motion. We also determine upper and lower bounds for the Hausdorff dimension spectrum of thick points refining the multifractal analysis of super-Brownian motion by Taylor and Perkins. The upper bound, conjectured to be sharp, involves a constant which can be characterized in terms of the upper tails of the associated equilibrium Palm distribution.Acknowledgements. This work is part of the DFG project Dimensionsspektren für die Super-Brownsche Bewegung at Universität Kaiserslautern, and we would like to thank the Deutsche Forschungsgemeinschaft DFG for the support. We also thank the London Mathematical Society for a travel grant, which facilitated our collaboration. We owe thanks to an anonymous referee who pointed out substantial errors in a previous version of the paper. Last but not least we would like to thank David Hobson, Alison Etheridge and Heinrich von Weizsäcker for helpful discussions.     

The age-dependent random connection model

Queueing Systems - We investigate a class of growing graphs embedded into the d-dimensional torus where new vertices arrive according to a Poisson process in time, are randomly placed in space and...