Hydrodynamic limit fluctuations of super-Brownian motion with a stable catalyst

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 supercritical dimensions, the scaled process converges to a macroscopic heat flow, and the appropriately rescaled random fluctuations around this macroscopic flow are asymptotically bounded, in the sense of log-Laplace transforms, by generalised stable Ornstein-Uhlenbeck processes. The most interesting new effect we observe is the occurrence of an index-jump from a 'Gaussian' situation to stable fluctuations of index 1+gamma, where gamma is an index associated to the medium.Comment: 40 page

Tangent measure distributions and the geometry of measures

In this thesis we investigate the geometry of measures in Euclidean spaces by means of their average densities, average tangent measures and tangent measure distributions. These notions were recently introduced into geometric measure theory by Bedford and Fisher, Bandt, Graf and others, as tools for the study of non-rectifiable measures. Our main result yields a connection between tangent measure distributions of measures on the line and Palm distributions: Let [alpha] be a measure on the line with positive and finite [alpha]-densities almost everywhere. Then at almost all points all tangent measure distributions are Palm distributions. Therefore the tangent measure distributions define a-self similar random measures in the axiomatic sense of U. Zahle. This result enables us to give a complete description of the one-sided average [alpha]-densities of the measure in terms of its lower and upper circular average [alpha]-densities. It also enables us to give an example of a measure with positive and finite [alpha]-densities which has unique average tangent measures but non-unique tangent measure distributions almost everywhere. If [mu] is a measure on n-dimensional Euclidean space with positive and finite [alpha]-densities almost everywhere we show that at almost all points the unique tangent measure distribution, if it exists, is a Palm distribution. We illustrate the limitations of tangent measure distributions by means of an example of a non-zero measure that has no non-trivial tangent measure distributions almost everywhere. Such measures can be studied by means of normalized tangent measure distributions and we prove an existence and a shift-invariance result for these distributions

METASTABILITY OF THE CONTACT PROCESS ON FAST EVOLVING SCALE-FREE NETWORKS

We study the contact process in the regime of small infection rates on finite scale-free networks with stationary dynamics based on simultaneous updating of all connections of a vertex. We allow the update rates of individual vertices to increase with the strength of a vertex, leading to a fast evolution of the network. We first develop an approach for inhomogeneous networks with general kernel and then focus on two canonical cases, the factor kernel and the preferential attachment kernel. For these specific networks, we identify and analyse four possible strategies how the infection can survive for a long time. We show that there is fast extinction of the infection when neither of the strategies is successful, otherwise there is slow extinction and the most successful strategy determines the asymptotics of the metastable density as the infection rate goes to zero. We identify the domains in which these strategies dominate in terms of phase diagrams for the exponent describing the decay of the metastable density

The Shape of the Emerging Condensate in Effective Models of Condensation

We consider effective models of condensation where the condensation occurs as time t goes to infinity. We provide natural conditions under which the buildup of the condensate occurs on a spatial scale of 1 / t and has the universal form of a Gamma density. The exponential parameter of this density is determined only by the equation and the total mass of the condensate, while the power law parameter may in addition depend on the decay properties of the initial condition near the condensation point. We apply our results to some examples, including simple models of Bose-Einstein condensation

Competing growth processes with random growth rates and random birth times

Comparing individual contributions in a strongly interacting system of stochastic growth processes can be a very difficult problem. This is particularly the case when new growth processes are initiated depending on the state of previous ones and the growth rates of the individual processes are themselves random. We propose a novel technique to deal with such problems and show how it can be applied to a broad range of examples where it produces new insight and surprising results. The method relies on two steps: In the first step, which is highly problem dependent, the growth processes are jointly embedded into continuous time so that their evolutions after initiation become approximately independent while we retain some control over the initiation times. Once such an embedding is achieved, the second step is to apply a Poisson limit theorem that enables a comparison of the state of the processes initiated in a critical window and therefore allows an asymptotic description of the extremal process. In this paper we prove a versatile limit theorem of this type and show how this tool can be applied to obtain novel asymptotic results for a variety of interesting stochastic processes. These include (a) the maximal degree in different types of preferential attachment networks with fitnesses like the well-known Bianconi-Barabasi tree and a network model of Dereich, (b) the most successful mutant in branching processes evolving by selection and mutation, and (c) the ratio between the largest and second largest cycles in a random permutation with random cycle weights, which can also be interpreted as a disordered version of Pitman's Chinese restaurant process. (C) 2021 Elsevier B.V. All rights reserved

Near Critical Preferential Attachment Networks have Small Giant Components

Preferential attachment networks with power law exponent tau > 3 are known to exhibit a phase transition. There is a value rho(c) > 0 such that, for small edge densities rho rho(c) there is a unique giant component comprising an asymptotically positive proportion of vertices. In this paper we study the decay in the size of the giant component as the critical edge density is approached from above. We show that the size decays very rapidly, like exp(-c/root rho - rho(c)) for an explicit constant c > 0 depending on the model implementation. This result is in contrast to the behaviour of the class of rank-one models of scale-free networks, including the configuration model, where the decay is polynomial. Our proofs rely on the local neighbourhood approximations of Dereich and Morters (Ann Probab 41(1): 329-384, 2013) and recent progress in the theory of branching random walks (Gantert et al. in Ann Inst Henri Poincare Probab Stat 47(1): 111-129, 2011)

The age-dependent random connection model

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 connect to existing vertices with a probability depending on time, their spatial distance and their relative birth times. This simple model for a scale-free network is called the age-based spatial preferential attachment network and is based on the idea of preferential attachment with spatially induced clustering. We show that the graphs converge weakly locally to a variant of the random connection model, which we call the age-dependent random connection model. This is a natural infinite graph on a Poisson point process where points are marked by a uniformly distributed age and connected with a probability depending on their spatial distance and both ages. We use the limiting structure to investigate asymptotic degree distribution, clustering coefficients and typical edge lengths in the age-based spatial preferential attachment network