Combinatorial Stochastic Processes

Coherent random allocations and the Ewens-Pitman formula. PDMI Preprint, Steklov Math. Institute, St. Petersburg, 1995. S. Kerov. The boundary of Young lattice and random Young tableaux. In Formal power series and algebraic ...

Author: Jim Pitman

Publisher: Springer Science & Business Media

ISBN: 9783540309901

Category: Mathematics

Page: 260

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The purpose of this text is to bring graduate students specializing in probability theory to current research topics at the interface of combinatorics and stochastic processes. There is particular focus on the theory of random combinatorial structures such as partitions, permutations, trees, forests, and mappings, and connections between the asymptotic theory of enumeration of such structures and the theory of stochastic processes like Brownian motion and Poisson processes.
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Semiclassical Analysis for Diffusions and Stochastic Processes

Analytic solutions to some linear PDE In this short Section we collect some general facts on analytic (or even formal power series) solutions to linear first order partial differential equations of the form 6S XS + (4.

Author: Vassili N. Kolokoltsov

Publisher: Springer

ISBN: 9783540465874

Category: Mathematics

Page: 356

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The monograph is devoted mainly to the analytical study of the differential, pseudo-differential and stochastic evolution equations describing the transition probabilities of various Markov processes. These include (i) diffusions (in particular,degenerate diffusions), (ii) more general jump-diffusions, especially stable jump-diffusions driven by stable Lévy processes, (iii) complex stochastic Schrödinger equations which correspond to models of quantum open systems. The main results of the book concern the existence, two-sided estimates, path integral representation, and small time and semiclassical asymptotics for the Green functions (or fundamental solutions) of these equations, which represent the transition probability densities of the corresponding random process. The boundary value problem for Hamiltonian systems and some spectral asymptotics ar also discussed. Readers should have an elementary knowledge of probability, complex and functional analysis, and calculus.
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Stochastic Processes Physics and Geometry New Interplays II

... on the linear space C TM ( M ) [ [ H ] ] of formal power series in ħ with coefficients in C® ( M ) . That means , in the language of formal deformation theory , one understands quantization as deformation of the Poisson - algebra C® ...

Author: Sergio Albeverio

Publisher: American Mathematical Soc.

ISBN: 0821819607

Category: Mathematics

Page: 333

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This volume and Stochastic Processes, Physics and Geometry: New Interplays. I present state-of-the-art research currently unfolding at the interface between mathematics and physics. Included are select articles from the international conference held in Leipzig (Germany) in honor of Sergio Albeverio's sixtieth birthday. The theme of the conference, ``Infinite Dimensional (Stochastic) Analysis and Quantum Physics'', was chosen to reflect Albeverio's wide-ranging scientific interests. The articles in these books reflect that broad range of interests and provide a detailed overview highlighting the deep interplay among stochastic processes, mathematical physics, and geometry. The contributions are written by internationally recognized experts in the fields of stochastic analysis, linear and nonlinear (deterministic and stochastic) PDEs, infinite dimensional analysis, functional analysis, commutative and noncommutative probability theory, integrable systems, quantum and statistical mechanics, geometric quantization, and neural networks. Also included are applications in biology and other areas. Most of the contributions are high-level research papers. However, there are also some overviews on topics of general interest. The articles selected for publication in these volumes were specifically chosen to introduce readers to advanced topics, to emphasize interdisciplinary connections, and to stress future research directions. Volume I contains contributions from invited speakers; Volume II contains additional contributed papers.
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Modelling and Application of Stochastic Processes

A. 277 ( 2 mo g(z) q' (z) and we may extend this definition by linearity to multiplication of g (2) by any power series f(z), at least formally. Notice that we have a unique additive decomposition: zg (z) = 2 * g (2) + g' (z) where the ...

Author: Uday B. Desai

Publisher: Springer Science & Business Media

ISBN: 9781461322672

Category: Science

Page: 288

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The subject of modelling and application of stochastic processes is too vast to be exhausted in a single volume. In this book, attention is focused on a small subset of this vast subject. The primary emphasis is on realization and approximation of stochastic systems. Recently there has been considerable interest in the stochastic realization problem, and hence, an attempt has been made here to collect in one place some of the more recent approaches and algorithms for solving the stochastic realiza tion problem. Various different approaches for realizing linear minimum-phase systems, linear nonminimum-phase systems, and bilinear systems are presented. These approaches range from time-domain methods to spectral-domain methods. An overview of the chapter contents briefly describes these approaches. Also, in most of these chapters special attention is given to the problem of developing numerically ef ficient algorithms for obtaining reduced-order (approximate) stochastic realizations. On the application side, chapters on use of Markov random fields for modelling and analyzing image signals, use of complementary models for the smoothing problem with missing data, and nonlinear estimation are included. Chapter 1 by Klein and Dickinson develops the nested orthogonal state space realization for ARMA processes. As suggested by the name, nested orthogonal realizations possess two key properties; (i) the state variables are orthogonal, and (ii) the system matrices for the (n + l)st order realization contain as their "upper" n-th order blocks the system matrices from the n-th order realization (nesting property).
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Stochastic Processes and their Applications

In applications it may happen that the expectations of the random operators under consideration all commute. ... Since the GNS representation it cannot be used, we employ formal power series as in Section 2 of Ref. 1, X 4 a. X ... a.

Author: Sergio Albeverio

Publisher: Springer Science & Business Media

ISBN: 9789400921177

Category: Mathematics

Page: 403

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'Et moi ..., si j'avait su comment en revenIT, One service mathematics has rendered the je n'y serais point allt\.' human race. It has put common sense back where it belongs, on the topmost shelf next Jules Verne to the dusty canister labelled 'discarded non- The series is divergent; therefore we may be sense'. able to do something with it. Eric T. Bell O. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. :; 'One service logic has rendered com puter science .. :; 'One service category theory has rendered mathematics .. :. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series.
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Stochastic Processes and Random Matrices

Continuing this process with A3, ..., AN, we obtain for any 6 × 0 and N > 1: R. v. " k; (Al...AN) = YE ** IIT,(A)(A, ... Alternatively, it is enough to think that we are working with formal power series. Fig. 2.26 A possible choice of ...

Author: Grégory Schehr

Publisher: Oxford University Press

ISBN: 9780198797319

Category: Mathematics

Page: 672

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The field of stochastic processes and Random Matrix Theory (RMT) has been a rapidly evolving subject during the last fifteen years. The continuous development and discovery of new tools, connections and ideas have led to an avalanche of new results. These breakthroughs have been made possible thanks, to a large extent, to the recent development of various new techniques in RMT. Matrix models have been playing an important role in theoretical physics for a long time and they are currently also a very active domain of research in mathematics. An emblematic example of these recent advances concerns the theory of growth phenomena in the Kardar-Parisi-Zhang (KPZ) universality class where the joint efforts of physicists and mathematicians during the last twenty years have unveiled the beautiful connections between this fundamental problem of statistical mechanics and the theory of random matrices, namely the fluctuations of the largest eigenvalue of certain ensembles of random matrices. This text not only covers this topic in detail but also presents more recent developments that have emerged from these discoveries, for instance in the context of low dimensional heat transport (on the physics side) or integrable probability (on the mathematical side).
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High dimensional Nonlinear Diffusion Stochastic Processes

as the solution of problem ( 1.12.16 ) , ( 1.5.7 ) in the form of formal power series with respect to A ( see ( 5 ) in Roy and Spanos , 1993 ) . In so doing , the rigorous mathematical justification is not attempted ( see the text just ...

Author: Yevgeny Mamontov

Publisher: World Scientific

ISBN: 9812810544

Category: Mathematics

Page: 324

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Annotation This book is one of the first few devoted to high-dimensional diffusion stochastic processes with nonlinear coefficients. These processes are closely associated with large systems of Ito's stochastic differential equations and with discretized-in-the-parameter versions of Ito's stochastic differential equations that are nonlocally dependent on the parameter. The latter models include Ito's stochastic integro-differential, partial differential and partial integro-differential equations.The book presents the new analytical treatment which can serve as the basis of a combined, analytical -- numerical approach to greater computational efficiency. Some examples of the modelling of noise in semiconductor devices are provided
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Advances in Inequalities for Series

Theorem 1.3 may be useful where generating functions or formal power series are utilized such as in enumerative combinatorics and stochastic processes ( cf. Wilf [ 48 ] , Feller [ 22 ] , Kijima [ 26 ] , Heathcote ( 24 ] , Kendall [ 25 ] ...

Author: Sever Silvestru Dragomir

Publisher: Nova Publishers

ISBN: 1600219209

Category: Mathematics

Page: 233

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This research monograph, deals with identities and inequalities relating to series and their application. This is the first volume of research monographs on advances in inequalities for series. All of the papers in this volume have been fully peer reviewed. Some papers in this volume appear in print for the first time, detailing many technical results and some other papers offer a review of a number of recently published results. The papers appear in author alphabetical order and not in mathematics subject classification. There are fifteen diverse papers in this volume each with its own speciality. An important issue in many applications of Probability Theory is finding an approximate measure of distance, or discrimination, between two probability distributions. A number of divergence measures for this purpose have been proposed.
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Stochastic Processes and Orthogonal Polynomials

4 Sheffer Systems Lévy processes appear in many areas, such as in models for queues, insurance risks, ... m > 0} when both functions u(z) and f(z) can be expanded in a formal power series and if u(0) = 0, w'(0) # 0, and f(0) # 0.

Author: Wim Schoutens

Publisher: Springer Science & Business Media

ISBN: 9781461211709

Category: Mathematics

Page: 184

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The book offers an accessible reference for researchers in the probability, statistics and special functions communities. It gives a variety of interdisciplinary relations between the two main ingredients of stochastic processes and orthogonal polynomials. It covers topics like time dependent and asymptotic analysis for birth-death processes and diffusions, martingale relations for Lévy processes, stochastic integrals and Stein's approximation method. Almost all well-known orthogonal polynomials, which are brought together in the so-called Askey Scheme, come into play. This volume clearly illustrates the powerful mathematical role of orthogonal polynomials in the analysis of stochastic processes and is made accessible for all mathematicians with a basic background in probability theory and mathematical analysis. Wim Schoutens is a Postdoctoral Researcher of the Fund for Scientific Research-Flanders (Belgium). He received his PhD in Science from the Catholic University of Leuven, Belgium.
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Multidimensional Stochastic Processes as Rough Paths

Integration of paths—a faithful representation of paths by non-commutative formal power series. Trans. Amer. Math. Soc., 89:395–407, 1958. [30] L.Coutin, P.Friz and N.Victoir. Good rough path sequences and applications to anticipating ...

Author: Peter K. Friz

Publisher: Cambridge University Press

ISBN: 9781139487214

Category: Mathematics

Page:

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Rough path analysis provides a fresh perspective on Ito's important theory of stochastic differential equations. Key theorems of modern stochastic analysis (existence and limit theorems for stochastic flows, Freidlin-Wentzell theory, the Stroock-Varadhan support description) can be obtained with dramatic simplifications. Classical approximation results and their limitations (Wong-Zakai, McShane's counterexample) receive 'obvious' rough path explanations. Evidence is building that rough paths will play an important role in the future analysis of stochastic partial differential equations and the authors include some first results in this direction. They also emphasize interactions with other parts of mathematics, including Caratheodory geometry, Dirichlet forms and Malliavin calculus. Based on successful courses at the graduate level, this up-to-date introduction presents the theory of rough paths and its applications to stochastic analysis. Examples, explanations and exercises make the book accessible to graduate students and researchers from a variety of fields.
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