A CLASS OF WEAKLY SELF-AVOIDING WALKS (2009)
Abstract: We define a class of weakly self-avoiding walks on the integers by conditioning a simple random walk of length n to have a p-fold self-intersection local time smaller than n β, where 1...
Ageing in the parabolic Anderson model (2009)
Mörters, Peter, Ortgiese, Marcel, Sidorova, Nadia
The parabolic Anderson model is the Cauchy problem for the heat equation with a random potential. We consider this model in a setting which is continuous in time and discrete in space, and focus on...
Random networks with sublinear preferential attachment: Degree evolutions (2009)
Dereich, Steffen; Technische Universität Berlin; Dereich@math.tu-berlin.de, Mörters, Peter; University Of Bath; Maspm@bath.ac.uk
We define a dynamic model of random networks, where new vertices are connected to old ones with a probability proportional to a sublinear function of their degree. We first give a strong limit law...
Wolfgang König, Peter Mörters, Nadia Sidorova
Abstract: The parabolic Anderson problem is the Cauchy problem for the heat equation ∂tu(t, z) = ∆u(t, z) + ξ(z)u(t, z) on (0, ∞) × Z d with random potential (ξ(z): z ∈ Z d). We consider...
Upper tails for intersection local times of random walks in supercritical dimensions (2009)
We determine the precise asymptotics of the logarithmic upper tail probability of the total intersection local time of p independent random walks in ℤd under the assumption p(d−2) > d. Our...
A TWO CITIES THEOREM FOR THE PARABOLIC ANDERSON MODEL (2008)
Wolfgang König, Hubert Lacoin, Peter Mörters, Nadia Sidorova
Abstract: The parabolic Anderson problem is the Cauchy problem for the heat equation ∂tu(t, z) = ∆u(t, z) + ξ(z)u(t, z) on (0, ∞) × Z d with random potential (ξ(z): z ∈ Z d). We consider...
Multiple intersection exponents (2008)
Let $p\ge2$, $n_1\le...\le n_p$ be positive integers and $B_1^1, ..., B_{n_1}^1; ...; B_1^p, ..., B_{n_p}^{p}$ be independent planar Brownian motions started uniformly on the boundary of the unit...
Small value probabilities via the branching tree heuristic (2008)
Peter Mörters, Marcel Ortgiese
Abstract: In the first part of this paper we give easy and intuitive proofs for the small value probabilities of the martingale limit of a supercritical Galton-Watson process in both the Schröder...
A TWO CITIES THEOREM FOR THE PARABOLIC ANDERSON MODEL (2008)
Wolfgang König, Hubert Lacoin, Peter Mörters, Nadia Sidorova
Abstract: The parabolic Anderson problem is the Cauchy problem for the heat equation ∂tu(t, z) = ∆u(t, z) + ξ(z)u(t, z) on (0, ∞) × Z d with random potential (ξ(z): z ∈ Z d). We consider...
The exact packing measure of Brownian double points (2008)
Peter Mörters, Narn-rueih Shieh
Abstract: Let D ⊂ R3 be the set of double points of a 3-dimensional Brownian motion. We show that, if ξ = ξ3(2, 2) is the intersection exponent of two packets of two independent Brownian motions,...
Moderate deviations of random walk in random scenery (2008)
Klaus Fleischmann, Klaus Fleischmann, Peter Mörters, Peter Mörters, Vitali Wachtel, Vitali Wachtel
Abstract: We investigate the cumulative scenery process associated with random walks in independent, identically distributed random sceneries under the assumption that the scenery variables satisfy...
UPPER TAILS FOR INTERSECTION LOCAL TIMES OF RANDOM WALKS IN SUPERCRITICAL DIMENSIONS (2008)
Abstract. We determine the precise asymptotics of the logarithmic upper tail probability of the total intersection local time of p independent random walks in Z d under the assumption p(d − 2)>...
Complete localisation in the parabolic Anderson model with Pareto-distributed potential (2008)
Wolfgang König, Peter Mörters, Nadia Sidorova
The parabolic Anderson problem is the Cauchy problem for the heat equation ∂tu(t, z) = ∆u(t, z) + ξ(z)u(t, z) on (0, ∞) × Z d with random potential (ξ(z): z ∈ Z d). We consider independent...
Large Deviation Principles for Empirical Measures of Coloured Random Graphs (2008)
Kwabena Doku-amponsah, Peter Mörters
For any finite coloured graph we define the empirical neighbourhood measure, which counts the number of vertices of a given colour connected to a given number of vertices of each colour, and the...
Large Deviation Principles for Empirical Measures of Coloured Random Graphs (2008)
Kwabena Doku-amponsah, Peter Mörters
For any finite coloured graph we define the empirical neighbourhood measure, which counts the number of vertices of a given colour connected to a given number of vertices of each colour, and the...
Large deviation theory and applications (2008)
Large deviation theory deals with the decay of the probability of increasingly unlikely events. It is one of the key techniques of modern probability, a role which is emphasised by the recent award...
How fast are the particles of super-Brownian motion? (2007)
In this paper we investigate fast particles in the range and support of super-Brownian motion in the historical setting. In this setting each particle of superBrownian motion alive at time t is...
Self-Similar Random Measures and Tangent Measure Distributions (2007)
> (E) = (u +E) : Unfortunately, it turns out that there are no nontrivial examples of probability distributions on M(R d ) that are invariant with respect to these operators. To obtain a...
Almost sure Kallianpur-Robbins laws for Brownian motion in the plane (2007)
. The Kallianpur-Robbins law describes the long term asymptotic behaviour of integrable additive functionals of Brownian motion in the plane. In this paper we prove an almost sure version of this...
A Pathwise Version of Spitzer's Law (2007)
. Spitzer's law describes the long term asymptotic behaviour of the distribution of the winding number of a Brownian motion in the plane. The pathwise result of this paper shows that an...
Average Densities And Linear Rectifiability Of Measures (2007)
: We show that a measure on IR d is linearly rectifiable if and only if the lower 1-density is positive and finite and agrees with the lower average 1-density almost everywhere. 1 Introduction Let ¯...
Brownian Intersection Local Times: Exponential Moments and Law of Large Masses (2007)
Wolfgang König, Wolfgang K Onig, Peter Mörters
Consider p independent Brownian motions in R , each running up to its first exit time from an open domain B, and their intersection local time # as a measure on B. We give a sharp criterion for the...
Thick Points of Super-Brownian Motion (2007)
We determine for a super-Brownian motion fX t : t 0g in R , d 3, the precise gauge function ' such that, almost surely on survival up to time t, '(r) < 1; improving a result of Barlow,...
Intersection Exponents and the Multifractal Spectrum for Measures on Brownian Paths (2007)
The aim of this paper is to give a survey of recent developments in the multifractal analysis of measures arising in the study of Brownian motion.
Why study multifractal spectra? (2007)
We show by three simple examples how multifractal spectra can enrich our understanding of stochastic processes. The first example concerns the problem of describing the speed of fragmentation in a...
Hydrodynamic limit fluctuations of super-Brownian motion with a stable catalyst (2006)
Fleischmann, Klaus; Weierstrass Institute For Applied Analysis And Stochastics, Berlin; Fleischm@wias-berlin.de, Mörters, Peter; University Of Bath; Maspm@bath.ac.uk, Wachtel, Vitali; Weierstrass Institute For Applied Analysis And Stochastics, Berlin; Vakhtel@wias-berlin.de
We consider the behaviour of a continuous super-Brownian motion catalysed by a random medium with infinite overall density under the hydrodynamic scaling of mass, time, and space. We show that, in...
Weak and almost sure limits for the parabolic Anderson model with heavy tailed potentials (2006)
Van Der Hofstad, Remco, Mörters, Peter, Sidorova\tsup, Nadia
We study the parabolic Anderson problem, that is, the heat equation $\partial_tu=\Delta u+\xi u$ on $(0,\infty)\times{\mathbb{Z}}^d$ with independent identically distributed random potential...
Sh67] L.A. Shepp. A first passage problem for the Wiener process. Ann. Math. Statist. 38, 1912--1914. [Sh98] Z. Shi. Windings of Brownian motion and random walks in the plane. Ann. Probab. 26...
Weak and almost sure limits for the parabolic Anderson model with heavy tailed potentials (2006)
We study the parabolic Anderson problem, i.e., the heat equation ∂tu = ∆u + ξu on (0, ∞) × Z d with independent identically distributed random potential {ξ(z): z ∈ Z d} and localised...
The multifractal spectrum of Brownian intersection local times (2005)
Let ℓ be the projected intersection local time of two independent Brownian paths in ℝd for d=2,3. We determine the lower tail of the random variable $\ell(\mathbb {U})$ , where $\mathbb {U}$ is...
On the multifractal spectrum of the branching measure of a Galton-Watson tree (2004)
Mörters, Peter, Shieh, Narn-Rueih
Suppose that μ is the branching measure on the boundary of a supercritical Galton-Watson tree with offspring distribution N such that E[N log N] < ∞ and P{N = 1} > 0. We determine the multifractal...
The Multifractal Spectrum of Brownian Intersection Local Times (2004)
Introduction and main results 1.1 Aims of the paper Intersections of Brownian motion or random walk paths have been studied for quite a long time in probability theory and statistical mechanics. One...
Five Lectures on Hausdorff Dimension, Random Trees and Brownian Motion (2003)
Dimensions are a tool to measure the size of mathematical objects...
Brownian intersection local times: Upper tail asymptotics and thick points (2002)
König, Wolfgang, Mörters, Peter
We equip the intersection of p independent Brownian paths in $\mathbb{R}^d$, $d\ge 2$, with the natural measure $\ell$ defined by projecting the intersection local time measure via one of the...
Strong clumping of super-Brownian motion in a stable catalytic medium (2002)
Dawson, Donald A., Fleischmann, Klaus, Mörters, Peter
A typical feature of the long time behavior of continuous super-Brownian motion in a stable catalytic medium is the development of large mass clumps (or clusters) at spatially rare sites. We describe...
A set with finite curvature and projections of zero length (2000)
A compact subset E of the complex plane is called removable if all bounded analytic functions on its complement are constant or, equivalently, i f its analytic capacity vanishes. The problem of...
Answering a question by Bedford and Fisher we show that for every Radon measure on the line with positive and finite lower and upper densities the one-sided average densities always agree with one...
Tangent measure distributions of hyperbolic Cantor sets (2000)
Mörters, Peter, Krieg, Daniela
Tangent measure distributions were introduced by Bandt and Graf as a means to describe the local geometry of self-similar sets generated by iteration of contractive similitudes. In this paper we...
Average densities and linear rectifiability of measures (2000)
We show that a measure in a Euclidean space is linearly rectifiable if and only if the lower 1-density is positive and finite and agrees with the lower average 1-density almost everywhere.
Tangent measure distributions of fractal measures (2000)
Tangent measure distributions are a natural tool to describe the local geometry of arbitrary measures of any dimension. We show that for every measure on a Euclidean space and every s, at almost...
The average density of planar Brownian motion (2000)
We show that the occupation measure on the path of a planar Brownian motion run for an arbitrary finite time intervalhas an average density of order three with respect to thegauge function t^2...
Density theorems for the intersection local times of planar Brownian motion (2000)
We show that the intersection local times $mu_p$ on the intersection of $p$ independent planar Brownian paths have an average density of order three with respect to the gauge function $r^2picdot...
Pathwise Kallianpur-Robbins laws for Brownian motion in the plane (2000)
The Kallianpur-Robbins law describes the long term asymptotic behaviour of the distribution of the occupation measure of a Brownian motion in the plane. In this paper we show that this behaviour can...
The average density of super-Brownian motion (2000)
Peter Mörters, Kaiserslautern Germany
. In this paper we prove the existence of average densities for the support of a superBrownian motion at a xed time. Our result establishes a dimension-dependent fractal parameter for super-Brownian...
Lecture Notes on Stochastic Analysis (2000)
Contents 1 Martingales and local martingales 7 1.1 Denition and examples of martingales . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 The optional stopping theorem . . . . . . . . . . . . . ....
A pathwise version of Spitzer's law (1999)
Spitzer's law describes the long term asymptotic behaviour of the distribution of the winding number of a Brownian motion in the plane. The pathwise result of this paper shows that an analogous...
A Uniform Dimension Result For The Brownian Snake (1999)
Peter Mörters, Graduiertenkolleg \stochastische Prozesse
this paper.
The average density of super-Brownian motion (1999)
Peter Mörters, Graduiertenkolleg \stochastische Prozesse
. In this paper we prove the existence of average densities for the support of a superBrownian motion at a xed time. Our result establishes a dimension-dependent fractal parameter for super-Brownian...
Small scale limit theorems for the intersection local times of Brownian motion (1999)
Peter Mörters, Narn-rueih Shieh
In this paper we contribute to the investigation of the fractal nature of the intersection local time measure on the intersection of independent Brownian paths. We particularly point out the...
The tangent measure distributions of hyperbolic Cantor sets (1998)
Tangent measure distributions were introduced by Bandt [2] and Graf [8] as a means to describe the local geometry of self-similar sets generated by iteration of contractive similitudes. In this paper...
Tangent Measure Distributions of Hyperbolic Cantor Sets (1998)
: Tangent measure distributions were introduced by Bandt [2] and Graf [8] as a means to describe the local geometry of self-similar sets generated by iteration of contractive similitudes. In this...
A set with finite curvature and projections of zero length (1997)
A compact subset E of the complex plane is called removable if all bounded analytic functions on its complement are constant or, equivalently, if its analytic capacity vanishes. The problem of...
On One-Sided Average Densities of Fractal Measures on the Line (1996)
this paper to show that the lower one-sided average densities of these measures do not vanish, and therefore the concept of average densities is able to reveal some of the local symmetry a measure...
Tangent Measure Distributions of Fractal Measures (1996)
this paper we show that for any dimension ff and
Thin and thick points for branching measure on a Galton-Watson tree
Mörters, Peter, Shieh, Narn-Rueih
Suppose that [mu] is the branching measure on the boundary of a supercritical Galton-Watson tree with offspring distribution N such that E[N log+ N]
Pathwise Kallianpur-Robbins laws for Brownian motion in the plane
The Kallianpur-Robbins law describes the long term asymptotic behaviour of the distribution of the occupation measure of a Brownian motion in the plane. In this paper we show that this behaviour can...