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Author: J. T. Cox
Publisher: American Mathematical Soc.
ISBN: 0821835424
Category : Mathematics
Languages : en
Pages : 114
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Book Description
Studies the evolution of the large finite spatial systems in size-dependent time scales and compare them with the behavior of the infinite systems, which amounts to establishing the so-called finite system scheme. This title introduces the concept of a continuum limit in the hierarchical mean field limit.
Author: J. T. Cox
Publisher: American Mathematical Soc.
ISBN: 0821835424
Category : Mathematics
Languages : en
Pages : 114
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Book Description
Studies the evolution of the large finite spatial systems in size-dependent time scales and compare them with the behavior of the infinite systems, which amounts to establishing the so-called finite system scheme. This title introduces the concept of a continuum limit in the hierarchical mean field limit.
Author: J. T. Cox Donald Andrew Dawson Andreas Greven
Publisher: American Mathematical Soc.
ISBN: 9780821865316
Category :
Languages : en
Pages : 116
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Book Description
We study features of the longtime behavior and the spatial continuum limit for the diffusion limit of the following particle model. Consider populations consisting of two types of particles located on sites labeled by a countable group. The populations of each of the types evolve as follows: Each particle performs a random walk and dies or splits in two with probability $\frac{1}{2}$ and the branching rates of a particle of each type at a site $x$ at time $t$ is proportional to the size of the population at $x$ at time $t$ of the other type. The diffusion limit of ``small mass, large number of initial particles'' is a pair of two coupled countable collections of interacting diffusions, the mutually catalytic super branching random walk. Consider now increasing sequences of finite subsets of sites and define the corresponding finite versions of the process. We study the evolution of these large finite spatial systems in size-dependent time scales and compare them with the behavior of the infinite systems, which amounts to establishing the so-called finite system scheme. A dichotomy is known between transient and recurrent symmetrized migrations for the infinite system, namely, between convergence to equilibria allowing for coexistence in the first case and concentration on monotype configurations in the second case. Correspondingly we show (i) in the recurrent case both large finite and infinite systems behave similar in all time scales, (ii) in the transient case we see for small time scales a behavior resembling the one of the infinite system, whereas for large time scales the system behaves as in the finite case with fixed size and finally in intermediate scales interesting behavior is exhibited, the system diffuses through the equilibria of the infinite system which are indexed by the pair of intensities and this diffusion process can be described as mutually catalytic diffusion on $(\mathbb{R}^+)^2$. At the same time, the above finite system asymptotics can be applied to mean-field systems of $N$ exchangeable mutually catalytic diffusions. This is the building block for a renormalization analysis of the spatially infinite hierarchical model and leads to an association of this system with the so-called interaction chain, which reflects the behavior of the process on large space-time scales. Similarly we introduce the concept of a continuum limit in the hierarchical mean field limit and show that this limit always exists and that the small-scale properties are described by another Markov chain called small scale characteristics. Both chains are analyzed in detail and exhibit the following interesting effects. The small scale properties of the continuum limit exhibit the dichotomy, overlap or segregation of densities of the two populations, as a function of the underlying random walk kernel. A corresponding concept to study hot spots is presented. Next we look in the transient regime for global equilibria and their equilibrium fluctuations and in the recurrent regime on the formation of monotype regions. For particular migration kernels in the recurrent regime we exhibit diffusive clustering, which means that the sizes (suitably defined) of monotype regions have a random order of magnitude as time proceeds and its distribution is explicitly identifiable. On the other hand in the regime of very large clusters we identify the deterministic order of magnitude of monotype regions and determine the law of the random size. These two regimes occur for different migration kernels than for the cases of ordinary branching or Fisher-Wright diffusion. Finally we find a third regime of very rapid deterministic spatial cluster growth which is not present in other models just mentioned. A further consequence of the analysis is that mutually catalytic branching has a fixed point property under renormalization and gives a natural example different from the trivial case of multitype models consisting of two independent versions of the fixed points for the one type case.
Author: Janos Englander
Publisher: Springer Science & Business Media
ISBN: 1461462401
Category : Mathematics
Languages : en
Pages : 129
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Book Description
Sergei Kuznetsov is one of the top experts on measure valued branching processes (also known as “superprocesses”) and their connection to nonlinear partial differential operators. His research interests range from stochastic processes and partial differential equations to mathematical statistics, time series analysis and statistical software; he has over 90 papers published in international research journals. His most well known contribution to probability theory is the "Kuznetsov-measure." A conference honoring his 60th birthday has been organized at Boulder, Colorado in the summer of 2010, with the participation of Sergei Kuznetsov’s mentor and major co-author, Eugene Dynkin. The conference focused on topics related to superprocesses, branching diffusions and nonlinear partial differential equations. In particular, connections to the so-called “Kuznetsov-measure” were emphasized. Leading experts in the field as well as young researchers contributed to the conference. The meeting was organized by J. Englander and B. Rider (U. of Colorado).
Author: Jean-Dominique Deuschel
Publisher: Springer Science & Business Media
ISBN: 3540271104
Category : Mathematics
Languages : en
Pages : 443
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Book Description
Core papers emanating from the research network, DFG-Schwerpunkt: Interacting stochastic systems of high complexity.
Author: M. Csörgö
Publisher: American Mathematical Soc.
ISBN: 0821835610
Category : Mathematics
Languages : en
Pages : 546
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Book Description
Honoring over forty years of Miklos Csorgo's work in probability and statistics, this title shows the state of the research. This book covers such topics as: path properties of stochastic processes, weak convergence of random size sums, almost sure stability of weighted maxima, and procedures for detecting changes in statistical models.
Author: Amadeu Delshams
Publisher: American Mathematical Soc.
ISBN: 0821838245
Category : Mathematics
Languages : en
Pages : 158
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Book Description
Beginning by introducing a geometric mechanism for diffusion in a priori unstable nearly integrable dynamical systems. This book is based on the observation that resonances, besides destroying the primary KAM tori, create secondary tori and tori of lower dimension. It argues that these objects created by resonances can be incorporated in transition chains taking the place of the destroyed primary KAM tori.The authors establish rigorously the existence of this mechanism in a simplemodel that has been studied before. The main technique is to develop a toolkit to study, in a unified way, tori of different topologies and their invariant manifolds, their intersections as well as shadowing properties of these bi-asymptotic orbits. This toolkit is based on extending and unifyingstandard techniques. A new tool used here is the scattering map of normally hyperbolic invariant manifolds.The model considered is a one-parameter family, which for $\varepsilon = 0$ is an integrable system. We give a small number of explicit conditions the jet of order $3$ of the family that, if verified imply diffusion. The conditions are just that some explicitely constructed functionals do not vanish identically or have non-degenerate critical points, etc.An attractive feature of themechanism is that the transition chains are shorter in the places where the heuristic intuition and numerical experimentation suggests that the diffusion is strongest.
Author: Ottmar Loos
Publisher: American Mathematical Soc.
ISBN: 0821835467
Category : Mathematics
Languages : en
Pages : 232
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Book Description
We develop the basic theory of root systems $R$ in a real vector space $X$ which are defined in analogy to the usual finite root systems, except that finiteness is replaced by local finiteness: the intersection of $R$ with every finite-dimensional subspace of $X$ is finite. The main topics are Weyl groups, parabolic subsets and positive systems, weights, and gradings.
Author: Jean-Dominique Deuschel
Publisher: Springer Science & Business Media
ISBN: 3642238114
Category : Mathematics
Languages : en
Pages : 518
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Book Description
Probabilistic approaches have played a prominent role in the study of complex physical systems for more than thirty years. This volume collects twenty articles on various topics in this field, including self-interacting random walks and polymer models in random and non-random environments, branching processes, Parisi formulas and metastability in spin glasses, and hydrodynamic limits for gradient Gibbs models. The majority of these articles contain original results at the forefront of contemporary research; some of them include review aspects and summarize the state-of-the-art on topical issues – one focal point is the parabolic Anderson model, which is considered with various novel aspects including moving catalysts, acceleration and deceleration and fron propagation, for both time-dependent and time-independent potentials. The authors are among the world’s leading experts. This Festschrift honours two eminent researchers, Erwin Bolthausen and Jürgen Gärtner, whose scientific work has profoundly influenced the field and all of the present contributions.
Author: Guy Métivier
Publisher: American Mathematical Soc.
ISBN: 0821836498
Category : Mathematics
Languages : en
Pages : 122
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Book Description
Studies two types of integral transformation associated with fractional Brownian motion, that are applied to construct approximation schemes for fractional Brownian motion by polygonal approximation of standard Brownian motion. This approximation is the best in the sense that it minimizes the mean square error.
Author: K. R. Goodearl
Publisher: American Mathematical Soc.
ISBN: 0821837168
Category : Mathematics
Languages : en
Pages : 134
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Book Description
Introduction Partial commutative monoids Continuous dimension scales Espaliers Classes of espaliers Bibliography Index
