Andreas Greven
University of Erlangen-Nuremberg
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Archive | 2005
Jean-Dominique Deuschel; Andreas Greven
Stochastic Methods in Statistical Physics.- Coarse-Graining Techniques for (Random) Kac Models.- Euclidean Gibbs Measures of Quantum Crystals: Existence, Uniqueness and a Priori Estimates.- Some Jump Processes in Quantum Field Theory.- Gibbs Measures on Brownian Paths: Theory and Applications.- Spectral Theory for Nonstationary Random Potentials.- A Survey of Rigorous Results on Random Schrodinger Operators for Amorphous Solids.- The Parabolic Anderson Model.- Random Spectral Distributions.- Stochastic in Population Models.- Renormalization and Universality for Multitype Population Models.- Stochastic Insertion-Deletion Processes and Statistical Sequence Alignment.- Branching Processes in Random Environment - A View on Critical and Subcritical Cases.- Stochastic Analysis.- Thin Points of Brownian Motion Intersection Local Times.- Coupling, Regularity and Curvature.- Two Mathematical Approaches to Stochastic Resonance.- Continuity Properties of Inertial Manifolds for Stochastic Retarded Semilinear Parabolic Equations.- The Random Walk Representation for Interacting Diffusion Processes.- Applications of Stochastic Analysis in Finance, Engineering and Algorithms.- On Worst-Case Investment with Applications in Finance and Insurance Mathematics.- Random Dynamical Systems Methods in Ship Stability: A Case Study.- Analysis of Algorithms by the Contraction Method: Additive and Max-recursive Sequences.
Annals of Probability | 2007
Andreas Greven; den WThF Frank Hollander
Let ({Xi(t)}i∈Zd)t≥0 be the system of interacting diffusions on [0,∞) defined by the following collection of coupled stochastic differential equations: dXi(t) = ∑ j∈Zd a(i, j)[Xj(t)−Xi(t)] dt+ √ bXi(t) dWi(t), i ∈ Z, t ≥ 0. Here, a(·, ·) is an irreducible random walk transition kernel on Z ×Z, b ∈ (0,∞) is a diffusion parameter, and ({Wi(t)}i∈Zd)t≥0 is a collection of independent standard Brownian motions on R. The initial condition is chosen such that {Xi(0)}i∈Zd is a shift-invariant and shift-ergodic random field on [0,∞) with mean Θ ∈ (0,∞) (the evolution preserves the mean). We show that the long-time behaviour of this system is the result of a delicate interplay between a(·, ·) and b, in contrast to systems where the diffusion function is subquadratic. In particular, let â(i, j) = 12 [a(i, j) + a(j, i)], i, j ∈ Z, denote the symmetrised transition kernel. We show that: (A) If â(·, ·) is recurrent, then for any b > 0 the system locally dies out. (B) If â(·, ·) is transient, then there exist b∗ ≥ b2 > 0 such that: (B1) The system converges to an equilibrium νΘ (with mean Θ) if 0 b∗. (B3) νΘ has a finite 2-nd moment if and only if 0 b2. The equilibrium νΘ is shown to be associated and mixing for all 0 b2. We further conjecture that the system locally dies out at b = b∗. For the case where a(·, ·) is symmetric and transient we further show that: (C) There exists a sequence b2 ≥ b3 ≥ b4 ≥ · · · > 0 such that: (C1) νΘ has a finite m-th moment if and only if 0 bm. (C3) b2 ≤ (m− 1)bm < 2. (C4) limm→∞(m− 1)bm = c = supm≥2(m− 1)bm. ∗Mathematisches Institut, Universitat Erlangen-Nurnberg, Bismarckstrasse 1 1 2 , D-91504 Erlangen, Germany, greven@mi.uni-erlangen.de †Mathematical Institute, Leiden University, P.O. Box 9512, 2300 RA Leiden, The Netherlands, denholla@math.leidenuniv.nl ‡EURANDOM, P.O.Box 513, 5600 MB Eindhoven, The Netherlands
Transactions of the American Mathematical Society | 1995
Donald A. Dawson; Andreas Greven; Jean Vaillancourt
In this paper of infinite systems of interacting measure-valued diffusions each with state space ¿^([O, 1]), the set of probability measures on [0, 1], is constructed and analysed (Fleming-Viot systems). These systems arise as diffusion limits of population genetics models with infinitely many possible types of individuals (labelled by [0, 1]), spatially distributed over a countable collection of sites and evolving as follows. Individuals can migrate between sites and after an exponential waiting time a colony replaces its population by a new generation where the types are assigned by resampling from the empirical distribution of types at this site. It is proved that, depending on recurrence versus transience properties of the migration mechanism, the system either clusters as r —> oo , that is, converges in distribution to a law concentrated on the states in which all components are equal to some Su , « £ [0, 1], or the system approaches a nontrivial equilibrium state. The properties of the equilibrium states, respectively the cluster formation, are studied by letting a parameter in the migration mechanism tend to infinity and explicitly identifying the limiting dynamics in a sequence of different space-time scales. These limiting dynamics have stationary states which are quasi-equiiibria of the original system, that is, change only in longer time scales. Properties of these quasi-equilibria are derived and related to the global equilibrium process for large N. Finally we establish that the Fleming-Viot systems are the unique dynamics which remain invariant under the associated space-time renormalization procedure. 0. Introduction (a) Background and motivation. In the present paper, we construct a system consisting of countably many interacting Fleming-Viot processes. Each component takes values in the space of probability measures on a compact space, say [0, 1 ]. This model arises as the diffusion limit of the following model from population genetics. The population is spatially distributed among a collection of colonies in which there are individuals of various genetic types and these types are labelled via values in [0,1]. The types of individuals in the next generation in each colony are obtained by sampling according to the empirical frequency of current types within the colony. In addition individuals can migrate between colonies. Received by the editors January 20, 1994. 1991 Mathematics Subject Classification. Primary 60K35; Secondary 60J70.
Probability Theory and Related Fields | 1996
J. Theodore Cox; Klaus Fleischmann; Andreas Greven
SummaryA general comparison argument for expectations of certain multitime functionals of infinite systems of linearly interacting diffusions differing in the diffusion coefficient is derived. As an application we prove clustering occurs in the case when the symmetrized interaction kernel is recurrent, and the components take values in an interval bounded on one side. The technique also gives an alternative proof of clustering in the case of compact intervals.
Probability Theory and Related Fields | 1994
Klaus Fleischmann; Andreas Greven
SummaryWe study a countable system of interacting diffusions on the interval [0,1], indexed by a hierarchical group. A particular choice of the interaction guaranties, we are in the diffusive clustering regime. This means clusters of components with values either close to 0 or close to 1 grow on various different scales. However, single components oscillate infinitely often between values close to 0 and close to 1 in such a way that they spend fraction one of their time together and close to the boundary. The processes in the whole class considered and starting with a shift-ergodic initial law have the same qualitative properties (universality).
Probability Theory and Related Fields | 1990
J. T. Cox; Andreas Greven
SummaryWe consider the problem of comparing large finite and infinite systems with locally interacting components, and present a general comparison scheme for the case when the infinite system is nonergodic. We show that this scheme holds for some specific models. One of these is critical branching random walk onZd. Letηt denote this system, and letηtN denote a finite version ofηt defined on the torus [−N,N]d∩Zd. Ford≧3 we prove that for stationary, shift ergodic initial measures with density θ, that ifT(N)→∞ andT(N)/(2N+1)d →s∈[0,∞] asN→∞, then {vθ}, θ≧0 is the set of extremal invariant measures for the infinite systemηt andQs is the transition function of Fellers branching diffusion. We prove several extensions and refinements of this result. The other systems we consider are the voter model and the contact process.
Annals of Applied Probability | 2012
Andrej Depperschmidt; Andreas Greven; Peter Pfaffelhuber
The Fleming-Viot measure-valued diffusion is a Markov process describing the evolution of (allelic) types under mutation, selection and random reproduction. We enrich this process by genealogical relations of individuals so that the random type distribution as well as the genealogical distances in the population evolve stochastically. The state space of this tree-valued enrichment of the Fleming-Viot dynamics with mutation and selection (TFVMS) consists of marked ultrametric measure spaces, equipped with the marked Gromov-weak topology and a suitable notion of polynomials as a separating algebra of test functions. The construction and study of the TFVMS is based on a well-posed martingale problem. For existence, we use approximating finite population models, the tree-valued Moran models, while uniqueness follows from duality to a function-valued process. Path properties of the resulting process carry over from the neutral case due to absolute continuity, given by a new Girsanov-type theorem on marked metric measure spaces. To study the long-time behavior of the process, we use a duality based on ideas from Dawson and Greven [On the effects of migration in spatial Fleming-Viot models with selection and mutation (2011c) Unpublished manuscript] and prove ergodicity of the TFVMS if the Fleming-Viot measure-valued diffusion is ergodic. As a further application, we consider the case of two allelic types and additive selection. For small selection strength, we give an expansion of the Laplace transform of genealogical distances in equilibrium, which is a first step in showing that distances are shorter in the selective case.
Probability Theory and Related Fields | 1993
Donald A. Dawson; Andreas Greven
SummaryThe purpose of this paper is to explore the connection between multiple space-time scale behaviour for block averages and phase transitions, respectively formation of clusters, in infinite systems with locally interacting components. The essential object is the associated Markov chain which describes the joint distribution of the block averages at different time scales. A fixed-point and stability property of a particular dynamical system under a renormalisation procedure is used to explain this pattern of cluster formation and the fact that the longtime behaviour is universal in entire classes of evolutions.
Stochastic Processes and their Applications | 1995
M. Cranston; Andreas Greven
Consider two transient Markov processes (Xvt)t[epsilon]R·, (X[mu]t)t[epsilon]R· with the same transition semigroup and initial distributions v and [mu]. The probability spaces supporting the processes each are also assumed to support an exponentially distributed random variable independent of the process. We show that there exist (randomized) stopping times S for (Xvt), T for (X[mu]t) with common final distribution, L(XvSS
Probability Theory and Related Fields | 1995
J. T. Cox; Andreas Greven; Tokuzo Shiga
SummaryWe study the problem of relating the long time behavior of finite and infinite systems of locally interacting components. We consider in detail a class of lincarly interacting diffusionsx(t)={xi(t),i ∈ ℤd} in the regime where there is a one-parameter family of nontrivial invariant measures. For these systems there are naturally defined corresponding finite systems,