Random Walk in Random Scenery (RWRS)
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Superprocesses and Mckean-Vlasov Equations with Creation of Mass
Sup erpro cesses and McKean-Vlasov equations with creation of mass L. Overb eck Department of Statistics, University of California, Berkeley, 367, Evans Hall Berkeley, CA 94720, y U.S.A. Abstract Weak solutions of McKean-Vlasov equations with creation of mass are given in terms of sup erpro cesses. The solutions can b e approxi- mated by a sequence of non-interacting sup erpro cesses or by the mean- eld of multityp e sup erpro cesses with mean- eld interaction. The lat- ter approximation is asso ciated with a propagation of chaos statement for weakly interacting multityp e sup erpro cesses. Running title: Sup erpro cesses and McKean-Vlasov equations . 1 Intro duction Sup erpro cesses are useful in solving nonlinear partial di erential equation of 1+ the typ e f = f , 2 0; 1], cf. [Dy]. Wenowchange the p oint of view and showhowtheyprovide sto chastic solutions of nonlinear partial di erential Supp orted byanFellowship of the Deutsche Forschungsgemeinschaft. y On leave from the Universitat Bonn, Institut fur Angewandte Mathematik, Wegelerstr. 6, 53115 Bonn, Germany. 1 equation of McKean-Vlasovtyp e, i.e. wewant to nd weak solutions of d d 2 X X @ @ @ + d x; + bx; : 1.1 = a x; t i t t t t t ij t @t @x @x @x i j i i=1 i;j =1 d Aweak solution = 2 C [0;T];MIR satis es s Z 2 t X X @ @ a f = f + f + d f + b f ds: s ij s t 0 i s s @x @x @x 0 i j i Equation 1.1 generalizes McKean-Vlasov equations of twodi erenttyp es. -
Partnership As Experimentation: Business Organization and Survival in Egypt, 1910–1949
Yale University EliScholar – A Digital Platform for Scholarly Publishing at Yale Discussion Papers Economic Growth Center 5-1-2017 Partnership as Experimentation: Business Organization and Survival in Egypt, 1910–1949 Cihan Artunç Timothy Guinnane Follow this and additional works at: https://elischolar.library.yale.edu/egcenter-discussion-paper-series Recommended Citation Artunç, Cihan and Guinnane, Timothy, "Partnership as Experimentation: Business Organization and Survival in Egypt, 1910–1949" (2017). Discussion Papers. 1065. https://elischolar.library.yale.edu/egcenter-discussion-paper-series/1065 This Discussion Paper is brought to you for free and open access by the Economic Growth Center at EliScholar – A Digital Platform for Scholarly Publishing at Yale. It has been accepted for inclusion in Discussion Papers by an authorized administrator of EliScholar – A Digital Platform for Scholarly Publishing at Yale. For more information, please contact [email protected]. ECONOMIC GROWTH CENTER YALE UNIVERSITY P.O. Box 208269 New Haven, CT 06520-8269 http://www.econ.yale.edu/~egcenter Economic Growth Center Discussion Paper No. 1057 Partnership as Experimentation: Business Organization and Survival in Egypt, 1910–1949 Cihan Artunç University of Arizona Timothy W. Guinnane Yale University Notes: Center discussion papers are preliminary materials circulated to stimulate discussion and critical comments. This paper can be downloaded without charge from the Social Science Research Network Electronic Paper Collection: https://ssrn.com/abstract=2973315 Partnership as Experimentation: Business Organization and Survival in Egypt, 1910–1949 Cihan Artunç⇤ Timothy W. Guinnane† This Draft: May 2017 Abstract The relationship between legal forms of firm organization and economic develop- ment remains poorly understood. Recent research disputes the view that the joint-stock corporation played a crucial role in historical economic development, but retains the view that the costless firm dissolution implicit in non-corporate forms is detrimental to investment. -
A Model of Gene Expression Based on Random Dynamical Systems Reveals Modularity Properties of Gene Regulatory Networks†
A Model of Gene Expression Based on Random Dynamical Systems Reveals Modularity Properties of Gene Regulatory Networks† Fernando Antoneli1,4,*, Renata C. Ferreira3, Marcelo R. S. Briones2,4 1 Departmento de Informática em Saúde, Escola Paulista de Medicina (EPM), Universidade Federal de São Paulo (UNIFESP), SP, Brasil 2 Departmento de Microbiologia, Imunologia e Parasitologia, Escola Paulista de Medicina (EPM), Universidade Federal de São Paulo (UNIFESP), SP, Brasil 3 College of Medicine, Pennsylvania State University (Hershey), PA, USA 4 Laboratório de Genômica Evolutiva e Biocomplexidade, EPM, UNIFESP, Ed. Pesquisas II, Rua Pedro de Toledo 669, CEP 04039-032, São Paulo, Brasil Abstract. Here we propose a new approach to modeling gene expression based on the theory of random dynamical systems (RDS) that provides a general coupling prescription between the nodes of any given regulatory network given the dynamics of each node is modeled by a RDS. The main virtues of this approach are the following: (i) it provides a natural way to obtain arbitrarily large networks by coupling together simple basic pieces, thus revealing the modularity of regulatory networks; (ii) the assumptions about the stochastic processes used in the modeling are fairly general, in the sense that the only requirement is stationarity; (iii) there is a well developed mathematical theory, which is a blend of smooth dynamical systems theory, ergodic theory and stochastic analysis that allows one to extract relevant dynamical and statistical information without solving -
POISSON PROCESSES 1.1. the Rutherford-Chadwick-Ellis
POISSON PROCESSES 1. THE LAW OF SMALL NUMBERS 1.1. The Rutherford-Chadwick-Ellis Experiment. About 90 years ago Ernest Rutherford and his collaborators at the Cavendish Laboratory in Cambridge conducted a series of pathbreaking experiments on radioactive decay. In one of these, a radioactive substance was observed in N = 2608 time intervals of 7.5 seconds each, and the number of decay particles reaching a counter during each period was recorded. The table below shows the number Nk of these time periods in which exactly k decays were observed for k = 0,1,2,...,9. Also shown is N pk where k pk = (3.87) exp 3.87 =k! {− g The parameter value 3.87 was chosen because it is the mean number of decays/period for Rutherford’s data. k Nk N pk k Nk N pk 0 57 54.4 6 273 253.8 1 203 210.5 7 139 140.3 2 383 407.4 8 45 67.9 3 525 525.5 9 27 29.2 4 532 508.4 10 16 17.1 5 408 393.5 ≥ This is typical of what happens in many situations where counts of occurences of some sort are recorded: the Poisson distribution often provides an accurate – sometimes remarkably ac- curate – fit. Why? 1.2. Poisson Approximation to the Binomial Distribution. The ubiquity of the Poisson distri- bution in nature stems in large part from its connection to the Binomial and Hypergeometric distributions. The Binomial-(N ,p) distribution is the distribution of the number of successes in N independent Bernoulli trials, each with success probability p. -
Lecture 19: Polymer Models: Persistence and Self-Avoidance
Lecture 19: Polymer Models: Persistence and Self-Avoidance Scribe: Allison Ferguson (and Martin Z. Bazant) Martin Fisher School of Physics, Brandeis University April 14, 2005 Polymers are high molecular weight molecules formed by combining a large number of smaller molecules (monomers) in a regular pattern. Polymers in solution (i.e. uncrystallized) can be characterized as long flexible chains, and thus may be well described by random walk models of various complexity. 1 Simple Random Walk (IID Steps) Consider a Rayleigh random walk (i.e. a fixed step size a with a random angle θ). Recall (from Lecture 1) that the probability distribution function (PDF) of finding the walker at position R~ after N steps is given by −3R2 2 ~ e 2a N PN (R) ∼ d (1) (2πa2N/d) 2 Define RN = |R~| as the end-to-end distance of the polymer. Then from previous results we q √ 2 2 ¯ 2 know hRN i = Na and RN = hRN i = a N. We can define an additional quantity, the radius of gyration GN as the radius from the centre of mass (assuming the polymer has an approximately 1 2 spherical shape ). Hughes [1] gives an expression for this quantity in terms of hRN i as follows: 1 1 hG2 i = 1 − hR2 i (2) N 6 N 2 N As an example of the type of prediction which can be made using this type of simple model, consider the following experiment: Pull on both ends of a polymer and measure the force f required to stretch it out2. Note that the force will be given by f = −(∂F/∂R) where F = U − TS is the Helmholtz free energy (T is the temperature, U is the internal energy, and S is the entropy). -
8 Convergence of Loop-Erased Random Walk to SLE2 8.1 Comments on the Paper
8 Convergence of loop-erased random walk to SLE2 8.1 Comments on the paper The proof that the loop-erased random walk converges to SLE2 is in the paper \Con- formal invariance of planar loop-erased random walks and uniform spanning trees" by Lawler, Schramm and Werner. (arxiv.org: math.PR/0112234). This section contains some comments to give you some hope of being able to actually make sense of this paper. In the first sections of the paper, δ is used to denote the lattice spacing. The main theorem is that as δ 0, the loop erased random walk converges to SLE2. HOWEVER, beginning in section !2.2 the lattice spacing is 1, and in section 3.2 the notation δ reappears but means something completely different. (δ also appears in the proof of lemma 2.1 in section 2.1, but its meaning here is restricted to this proof. The δ in this proof has nothing to do with any δ's outside the proof.) After the statement of the main theorem, there is no mention of the lattice spacing going to zero. This is hidden in conditions on the \inner radius." For a domain D containing 0 the inner radius is rad0(D) = inf z : z = D (364) fj j 2 g Now fix a domain D containing the origin. Let be the conformal map of D onto U. We want to show that the image under of a LERW on a lattice with spacing δ converges to SLE2 in the unit disc. In the paper they do this in a slightly different way. -
Pdf File of Second Edition, January 2018
Probability on Graphs Random Processes on Graphs and Lattices Second Edition, 2018 GEOFFREY GRIMMETT Statistical Laboratory University of Cambridge copyright Geoffrey Grimmett Geoffrey Grimmett Statistical Laboratory Centre for Mathematical Sciences University of Cambridge Wilberforce Road Cambridge CB3 0WB United Kingdom 2000 MSC: (Primary) 60K35, 82B20, (Secondary) 05C80, 82B43, 82C22 With 56 Figures copyright Geoffrey Grimmett Contents Preface ix 1 Random Walks on Graphs 1 1.1 Random Walks and Reversible Markov Chains 1 1.2 Electrical Networks 3 1.3 FlowsandEnergy 8 1.4 RecurrenceandResistance 11 1.5 Polya's Theorem 14 1.6 GraphTheory 16 1.7 Exercises 18 2 Uniform Spanning Tree 21 2.1 De®nition 21 2.2 Wilson's Algorithm 23 2.3 Weak Limits on Lattices 28 2.4 Uniform Forest 31 2.5 Schramm±LownerEvolutionsÈ 32 2.6 Exercises 36 3 Percolation and Self-Avoiding Walks 39 3.1 PercolationandPhaseTransition 39 3.2 Self-Avoiding Walks 42 3.3 ConnectiveConstantoftheHexagonalLattice 45 3.4 CoupledPercolation 53 3.5 Oriented Percolation 53 3.6 Exercises 56 4 Association and In¯uence 59 4.1 Holley Inequality 59 4.2 FKG Inequality 62 4.3 BK Inequalitycopyright Geoffrey Grimmett63 vi Contents 4.4 HoeffdingInequality 65 4.5 In¯uenceforProductMeasures 67 4.6 ProofsofIn¯uenceTheorems 72 4.7 Russo'sFormulaandSharpThresholds 80 4.8 Exercises 83 5 Further Percolation 86 5.1 Subcritical Phase 86 5.2 Supercritical Phase 90 5.3 UniquenessoftheIn®niteCluster 96 5.4 Phase Transition 99 5.5 OpenPathsinAnnuli 103 5.6 The Critical Probability in Two Dimensions 107 -
Mean Field Methods for Classification with Gaussian Processes
Mean field methods for classification with Gaussian processes Manfred Opper Neural Computing Research Group Division of Electronic Engineering and Computer Science Aston University Birmingham B4 7ET, UK. opperm~aston.ac.uk Ole Winther Theoretical Physics II, Lund University, S6lvegatan 14 A S-223 62 Lund, Sweden CONNECT, The Niels Bohr Institute, University of Copenhagen Blegdamsvej 17, 2100 Copenhagen 0, Denmark winther~thep.lu.se Abstract We discuss the application of TAP mean field methods known from the Statistical Mechanics of disordered systems to Bayesian classifi cation models with Gaussian processes. In contrast to previous ap proaches, no knowledge about the distribution of inputs is needed. Simulation results for the Sonar data set are given. 1 Modeling with Gaussian Processes Bayesian models which are based on Gaussian prior distributions on function spaces are promising non-parametric statistical tools. They have been recently introduced into the Neural Computation community (Neal 1996, Williams & Rasmussen 1996, Mackay 1997). To give their basic definition, we assume that the likelihood of the output or target variable T for a given input s E RN can be written in the form p(Tlh(s)) where h : RN --+ R is a priori assumed to be a Gaussian random field. If we assume fields with zero prior mean, the statistics of h is entirely defined by the second order correlations C(s, S') == E[h(s)h(S')], where E denotes expectations 310 M Opper and 0. Winther with respect to the prior. Interesting examples are C(s, s') (1) C(s, s') (2) The choice (1) can be motivated as a limit of a two-layered neural network with infinitely many hidden units with factorizable input-hidden weight priors (Williams 1997). -
Contents-Preface
Stochastic Processes From Applications to Theory CHAPMAN & HA LL/CRC Texts in Statis tical Science Series Series Editors Francesca Dominici, Harvard School of Public Health, USA Julian J. Faraway, University of Bath, U K Martin Tanner, Northwestern University, USA Jim Zidek, University of Br itish Columbia, Canada Statistical !eory: A Concise Introduction Statistics for Technology: A Course in Applied F. Abramovich and Y. Ritov Statistics, !ird Edition Practical Multivariate Analysis, Fifth Edition C. Chat!eld A. A!!, S. May, and V.A. Clark Analysis of Variance, Design, and Regression : Practical Statistics for Medical Research Linear Modeling for Unbalanced Data, D.G. Altman Second Edition R. Christensen Interpreting Data: A First Course in Statistics Bayesian Ideas and Data Analysis: An A.J.B. Anderson Introduction for Scientists and Statisticians Introduction to Probability with R R. Christensen, W. Johnson, A. Branscum, K. Baclawski and T.E. Hanson Linear Algebra and Matrix Analysis for Modelling Binary Data, Second Edition Statistics D. Collett S. Banerjee and A. Roy Modelling Survival Data in Medical Research, Mathematical Statistics: Basic Ideas and !ird Edition Selected Topics, Volume I, D. Collett Second Edition Introduction to Statistical Methods for P. J. Bickel and K. A. Doksum Clinical Trials Mathematical Statistics: Basic Ideas and T.D. Cook and D.L. DeMets Selected Topics, Volume II Applied Statistics: Principles and Examples P. J. Bickel and K. A. Doksum D.R. Cox and E.J. Snell Analysis of Categorical Data with R Multivariate Survival Analysis and Competing C. R. Bilder and T. M. Loughin Risks Statistical Methods for SPC and TQM M. -
Probability on Graphs Random Processes on Graphs and Lattices
Probability on Graphs Random Processes on Graphs and Lattices GEOFFREY GRIMMETT Statistical Laboratory University of Cambridge c G. R. Grimmett 1/4/10, 17/11/10, 5/7/12 Geoffrey Grimmett Statistical Laboratory Centre for Mathematical Sciences University of Cambridge Wilberforce Road Cambridge CB3 0WB United Kingdom 2000 MSC: (Primary) 60K35, 82B20, (Secondary) 05C80, 82B43, 82C22 With 44 Figures c G. R. Grimmett 1/4/10, 17/11/10, 5/7/12 Contents Preface ix 1 Random walks on graphs 1 1.1 RandomwalksandreversibleMarkovchains 1 1.2 Electrical networks 3 1.3 Flowsandenergy 8 1.4 Recurrenceandresistance 11 1.5 Polya's theorem 14 1.6 Graphtheory 16 1.7 Exercises 18 2 Uniform spanning tree 21 2.1 De®nition 21 2.2 Wilson's algorithm 23 2.3 Weak limits on lattices 28 2.4 Uniform forest 31 2.5 Schramm±LownerevolutionsÈ 32 2.6 Exercises 37 3 Percolation and self-avoiding walk 39 3.1 Percolationandphasetransition 39 3.2 Self-avoiding walks 42 3.3 Coupledpercolation 45 3.4 Orientedpercolation 45 3.5 Exercises 48 4 Association and in¯uence 50 4.1 Holley inequality 50 4.2 FKGinequality 53 4.3 BK inequality 54 4.4 Hoeffdinginequality 56 c G. R. Grimmett 1/4/10, 17/11/10, 5/7/12 vi Contents 4.5 In¯uenceforproductmeasures 58 4.6 Proofsofin¯uencetheorems 63 4.7 Russo'sformulaandsharpthresholds 75 4.8 Exercises 78 5 Further percolation 81 5.1 Subcritical phase 81 5.2 Supercritical phase 86 5.3 Uniquenessofthein®nitecluster 92 5.4 Phase transition 95 5.5 Openpathsinannuli 99 5.6 The critical probability in two dimensions 103 5.7 Cardy's formula 110 5.8 The -
Arxiv:1504.02898V2 [Cond-Mat.Stat-Mech] 7 Jun 2015 Keywords: Percolation, Explosive Percolation, SLE, Ising Model, Earth Topography
Recent advances in percolation theory and its applications Abbas Ali Saberi aDepartment of Physics, University of Tehran, P.O. Box 14395-547,Tehran, Iran bSchool of Particles and Accelerators, Institute for Research in Fundamental Sciences (IPM) P.O. Box 19395-5531, Tehran, Iran Abstract Percolation is the simplest fundamental model in statistical mechanics that exhibits phase transitions signaled by the emergence of a giant connected component. Despite its very simple rules, percolation theory has successfully been applied to describe a large variety of natural, technological and social systems. Percolation models serve as important universality classes in critical phenomena characterized by a set of critical exponents which correspond to a rich fractal and scaling structure of their geometric features. We will first outline the basic features of the ordinary model. Over the years a variety of percolation models has been introduced some of which with completely different scaling and universal properties from the original model with either continuous or discontinuous transitions depending on the control parameter, di- mensionality and the type of the underlying rules and networks. We will try to take a glimpse at a number of selective variations including Achlioptas process, half-restricted process and spanning cluster-avoiding process as examples of the so-called explosive per- colation. We will also introduce non-self-averaging percolation and discuss correlated percolation and bootstrap percolation with special emphasis on their recent progress. Directed percolation process will be also discussed as a prototype of systems displaying a nonequilibrium phase transition into an absorbing state. In the past decade, after the invention of stochastic L¨ownerevolution (SLE) by Oded Schramm, two-dimensional (2D) percolation has become a central problem in probability theory leading to the two recent Fields medals. -
A Representation for Functionals of Superprocesses Via Particle Picture Raisa E
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector stochastic processes and their applications ELSEVIER Stochastic Processes and their Applications 64 (1996) 173-186 A representation for functionals of superprocesses via particle picture Raisa E. Feldman *,‘, Srikanth K. Iyer ‘,* Department of Statistics and Applied Probability, University oj’ Cull~ornia at Santa Barbara, CA 93/06-31/O, USA, and Department of Industrial Enyineeriny and Management. Technion - Israel Institute of’ Technology, Israel Received October 1995; revised May 1996 Abstract A superprocess is a measure valued process arising as the limiting density of an infinite col- lection of particles undergoing branching and diffusion. It can also be defined as a measure valued Markov process with a specified semigroup. Using the latter definition and explicit mo- ment calculations, Dynkin (1988) built multiple integrals for the superprocess. We show that the multiple integrals of the superprocess defined by Dynkin arise as weak limits of linear additive functionals built on the particle system. Keywords: Superprocesses; Additive functionals; Particle system; Multiple integrals; Intersection local time AMS c.lassijication: primary 60555; 60H05; 60580; secondary 60F05; 6OG57; 60Fl7 1. Introduction We study a class of functionals of a superprocess that can be identified as the multiple Wiener-It6 integrals of this measure valued process. More precisely, our objective is to study these via the underlying branching particle system, thus providing means of thinking about the functionals in terms of more simple, intuitive and visualizable objects. This way of looking at multiple integrals of a stochastic process has its roots in the previous work of Adler and Epstein (=Feldman) (1987), Epstein (1989), Adler ( 1989) Feldman and Rachev (1993), Feldman and Krishnakumar ( 1994).