Lectures on Self-Avoiding Walks
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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. -
Brownian Motion and Relativity Brownian Motion 55
54 My God, He Plays Dice! Chapter 7 Chapter Brownian Motion and Relativity Brownian Motion 55 Brownian Motion and Relativity In this chapter we describe two of Einstein’s greatest works that have little or nothing to do with his amazing and deeply puzzling theories about quantum mechanics. The first,Brownian motion, provided the first quantitative proof of the existence of atoms and molecules. The second,special relativity in his miracle year of 1905 and general relativity eleven years later, combined the ideas of space and time into a unified space-time with a non-Euclidean 7 Chapter curvature that goes beyond Newton’s theory of gravitation. Einstein’s relativity theory explained the precession of the orbit of Mercury and predicted the bending of light as it passes the sun, confirmed by Arthur Stanley Eddington in 1919. He also predicted that galaxies can act as gravitational lenses, focusing light from objects far beyond, as was confirmed in 1979. He also predicted gravitational waves, only detected in 2016, one century after Einstein wrote down the equations that explain them.. What are we to make of this man who could see things that others could not? Our thesis is that if we look very closely at the things he said, especially his doubts expressed privately to friends, today’s mysteries of quantum mechanics may be lessened. As great as Einstein’s theories of Brownian motion and relativity are, they were accepted quickly because measurements were soon made that confirmed their predictions. Moreover, contemporaries of Einstein were working on these problems. Marion Smoluchowski worked out the equation for the rate of diffusion of large particles in a liquid the year before Einstein. -
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). -
And Nanoemulsions As Vehicles for Essential Oils: Formulation, Preparation and Stability
nanomaterials Review An Overview of Micro- and Nanoemulsions as Vehicles for Essential Oils: Formulation, Preparation and Stability Lucia Pavoni, Diego Romano Perinelli , Giulia Bonacucina, Marco Cespi * and Giovanni Filippo Palmieri School of Pharmacy, University of Camerino, 62032 Camerino, Italy; [email protected] (L.P.); [email protected] (D.R.P.); [email protected] (G.B.); gianfi[email protected] (G.F.P.) * Correspondence: [email protected] Received: 23 December 2019; Accepted: 10 January 2020; Published: 12 January 2020 Abstract: The interest around essential oils is constantly increasing thanks to their biological properties exploitable in several fields, from pharmaceuticals to food and agriculture. However, their widespread use and marketing are still restricted due to their poor physico-chemical properties; i.e., high volatility, thermal decomposition, low water solubility, and stability issues. At the moment, the most suitable approach to overcome such limitations is based on the development of proper formulation strategies. One of the approaches suggested to achieve this goal is the so-called encapsulation process through the preparation of aqueous nano-dispersions. Among them, micro- and nanoemulsions are the most studied thanks to the ease of formulation, handling and to their manufacturing costs. In this direction, this review intends to offer an overview of the formulation, preparation and stability parameters of micro- and nanoemulsions. Specifically, recent literature has been examined in order to define the most common practices adopted (materials and fabrication methods), highlighting their suitability and effectiveness. Finally, relevant points related to formulations, such as optimization, characterization, stability and safety, not deeply studied or clarified yet, were discussed. -
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. -
Long Memory and Self-Similar Processes
ANNALES DE LA FACULTÉ DES SCIENCES Mathématiques GENNADY SAMORODNITSKY Long memory and self-similar processes Tome XV, no 1 (2006), p. 107-123. <http://afst.cedram.org/item?id=AFST_2006_6_15_1_107_0> © Annales de la faculté des sciences de Toulouse Mathématiques, 2006, tous droits réservés. L’accès aux articles de la revue « Annales de la faculté des sci- ences de Toulouse, Mathématiques » (http://afst.cedram.org/), implique l’accord avec les conditions générales d’utilisation (http://afst.cedram. org/legal/). Toute reproduction en tout ou partie cet article sous quelque forme que ce soit pour tout usage autre que l’utilisation à fin strictement personnelle du copiste est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/ Annales de la Facult´e des Sciences de Toulouse Vol. XV, n◦ 1, 2006 pp. 107–123 Long memory and self-similar processes(∗) Gennady Samorodnitsky (1) ABSTRACT. — This paper is a survey of both classical and new results and ideas on long memory, scaling and self-similarity, both in the light-tailed and heavy-tailed cases. RESUM´ E.´ — Cet article est une synth`ese de r´esultats et id´ees classiques ou nouveaux surla longue m´emoire, les changements d’´echelles et l’autosimi- larit´e, `a la fois dans le cas de queues de distributions lourdes ou l´eg`eres. 1. Introduction The notion of long memory (or long range dependence) has intrigued many at least since B. -
Lewis Fry Richardson: Scientist, Visionary and Pacifist
Lett Mat Int DOI 10.1007/s40329-014-0063-z Lewis Fry Richardson: scientist, visionary and pacifist Angelo Vulpiani Ó Centro P.RI.ST.EM, Universita` Commerciale Luigi Bocconi 2014 Abstract The aim of this paper is to present the main The last of seven children in a thriving English Quaker aspects of the life of Lewis Fry Richardson and his most family, in 1898 Richardson enrolled in Durham College of important scientific contributions. Of particular importance Science, and two years later was awarded a grant to study are the seminal concepts of turbulent diffusion, turbulent at King’s College in Cambridge, where he received a cascade and self-similar processes, which led to a profound diverse education: he read physics, mathematics, chemis- transformation of the way weather forecasts, turbulent try, meteorology, botany, and psychology. At the beginning flows and, more generally, complex systems are viewed. of his career he was quite uncertain about which road to follow, but he later resolved to work in several areas, just Keywords Lewis Fry Richardson Á Weather forecasting Á like the great German scientist Helmholtz, who was a Turbulent diffusion Á Turbulent cascade Á Numerical physician and then a physicist, but following a different methods Á Fractals Á Self-similarity order. In his words, he decided ‘‘to spend the first half of my life under the strict discipline of physics, and after- Lewis Fry Richardson (1881–1953), while undeservedly wards to apply that training to researches on living things’’. little known, had a fundamental (often posthumous) role in In addition to contributing important results to meteo- twentieth-century science. -
Random Walk in Random Scenery (RWRS)
IMS Lecture Notes–Monograph Series Dynamics & Stochastics Vol. 48 (2006) 53–65 c Institute of Mathematical Statistics, 2006 DOI: 10.1214/074921706000000077 Random walk in random scenery: A survey of some recent results Frank den Hollander1,2,* and Jeffrey E. Steif 3,† Leiden University & EURANDOM and Chalmers University of Technology Abstract. In this paper we give a survey of some recent results for random walk in random scenery (RWRS). On Zd, d ≥ 1, we are given a random walk with i.i.d. increments and a random scenery with i.i.d. components. The walk and the scenery are assumed to be independent. RWRS is the random process where time is indexed by Z, and at each unit of time both the step taken by the walk and the scenery value at the site that is visited are registered. We collect various results that classify the ergodic behavior of RWRS in terms of the characteristics of the underlying random walk (and discuss extensions to stationary walk increments and stationary scenery components as well). We describe a number of results for scenery reconstruction and close by listing some open questions. 1. Introduction Random walk in random scenery is a family of stationary random processes ex- hibiting amazingly rich behavior. We will survey some of the results that have been obtained in recent years and list some open questions. Mike Keane has made funda- mental contributions to this topic. As close colleagues it has been a great pleasure to work with him. We begin by defining the object of our study. Fix an integer d ≥ 1. -
Particle Filtering Within Adaptive Metropolis Hastings Sampling
Particle filtering within adaptive Metropolis-Hastings sampling Ralph S. Silva Paolo Giordani School of Economics Research Department University of New South Wales Sveriges Riksbank [email protected] [email protected] Robert Kohn∗ Michael K. Pitt School of Economics Economics Department University of New South Wales University of Warwick [email protected] [email protected] October 29, 2018 Abstract We show that it is feasible to carry out exact Bayesian inference for non-Gaussian state space models using an adaptive Metropolis-Hastings sampling scheme with the likelihood approximated by the particle filter. Furthermore, an adaptive independent Metropolis Hastings sampler based on a mixture of normals proposal is computation- ally much more efficient than an adaptive random walk Metropolis proposal because the cost of constructing a good adaptive proposal is negligible compared to the cost of approximating the likelihood. Independent Metropolis-Hastings proposals are also arXiv:0911.0230v1 [stat.CO] 2 Nov 2009 attractive because they are easy to run in parallel on multiple processors. We also show that when the particle filter is used, the marginal likelihood of any model is obtained in an efficient and unbiased manner, making model comparison straightforward. Keywords: Auxiliary variables; Bayesian inference; Bridge sampling; Marginal likeli- hood. ∗Corresponding author. 1 Introduction We show that it is feasible to carry out exact Bayesian inference on the parameters of a general state space model by using the particle filter to approximate the likelihood and adaptive Metropolis-Hastings sampling to generate unknown parameters. The state space model can be quite general, but we assume that the observation equation can be evaluated analytically and that it is possible to generate from the state transition equation. -
Analytic and Geometric Background of Recurrence and Non-Explosion of the Brownian Motion on Riemannian Manifolds
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 36, Number 2, Pages 135–249 S 0273-0979(99)00776-4 Article electronically published on February 19, 1999 ANALYTIC AND GEOMETRIC BACKGROUND OF RECURRENCE AND NON-EXPLOSION OF THE BROWNIAN MOTION ON RIEMANNIAN MANIFOLDS ALEXANDER GRIGOR’YAN Abstract. We provide an overview of such properties of the Brownian mo- tion on complete non-compact Riemannian manifolds as recurrence and non- explosion. It is shown that both properties have various analytic characteri- zations, in terms of the heat kernel, Green function, Liouville properties, etc. On the other hand, we consider a number of geometric conditions such as the volume growth, isoperimetric inequalities, curvature bounds, etc., which are related to recurrence and non-explosion. Contents 1. Introduction 136 Acknowledgments 141 Notation 141 2. Heat semigroup on Riemannian manifolds 142 2.1. Laplace operator of the Riemannian metric 142 2.2. Heat kernel and Brownian motion on manifolds 143 2.3. Manifolds with boundary 145 3. Model manifolds 145 3.1. Polar coordinates 145 3.2. Spherically symmetric manifolds 146 4. Some potential theory 149 4.1. Harmonic functions 149 4.2. Green function 150 4.3. Capacity 152 4.4. Massive sets 155 4.5. Hitting probabilities 159 4.6. Exterior of a compact 161 5. Equivalent definitions of recurrence 164 6. Equivalent definitions of stochastic completeness 170 7. Parabolicity and volume growth 174 7.1. Upper bounds of capacity 174 7.2. Sufficient conditions for parabolicity 177 8. Transience and isoperimetric inequalities 181 9. Non-explosion and volume growth 184 Received by the editors October 1, 1997, and, in revised form, September 2, 1998. -
A Guide to Brownian Motion and Related Stochastic Processes
Vol. 0 (0000) A guide to Brownian motion and related stochastic processes Jim Pitman and Marc Yor Dept. Statistics, University of California, 367 Evans Hall # 3860, Berkeley, CA 94720-3860, USA e-mail: [email protected] Abstract: This is a guide to the mathematical theory of Brownian mo- tion and related stochastic processes, with indications of how this theory is related to other branches of mathematics, most notably the classical the- ory of partial differential equations associated with the Laplace and heat operators, and various generalizations thereof. As a typical reader, we have in mind a student, familiar with the basic concepts of probability based on measure theory, at the level of the graduate texts of Billingsley [43] and Durrett [106], and who wants a broader perspective on the theory of Brow- nian motion and related stochastic processes than can be found in these texts. Keywords and phrases: Markov process, random walk, martingale, Gaus- sian process, L´evy process, diffusion. AMS 2000 subject classifications: Primary 60J65. Contents 1 Introduction................................. 3 1.1 History ................................ 3 1.2 Definitions............................... 4 2 BM as a limit of random walks . 5 3 BMasaGaussianprocess......................... 7 3.1 Elementarytransformations . 8 3.2 Quadratic variation . 8 3.3 Paley-Wiener integrals . 8 3.4 Brownianbridges........................... 10 3.5 FinestructureofBrownianpaths . 10 arXiv:1802.09679v1 [math.PR] 27 Feb 2018 3.6 Generalizations . 10 3.6.1 FractionalBM ........................ 10 3.6.2 L´evy’s BM . 11 3.6.3 Browniansheets ....................... 11 3.7 References............................... 11 4 BMasaMarkovprocess.......................... 12 4.1 Markovprocessesandtheirsemigroups. 12 4.2 ThestrongMarkovproperty. 14 4.3 Generators ............................. -
A Course in Interacting Particle Systems
A Course in Interacting Particle Systems J.M. Swart January 14, 2020 arXiv:1703.10007v2 [math.PR] 13 Jan 2020 2 Contents 1 Introduction 7 1.1 General set-up . .7 1.2 The voter model . .9 1.3 The contact process . 11 1.4 Ising and Potts models . 14 1.5 Phase transitions . 17 1.6 Variations on the voter model . 20 1.7 Further models . 22 2 Continuous-time Markov chains 27 2.1 Poisson point sets . 27 2.2 Transition probabilities and generators . 30 2.3 Poisson construction of Markov processes . 31 2.4 Examples of Poisson representations . 33 3 The mean-field limit 35 3.1 Processes on the complete graph . 35 3.2 The mean-field limit of the Ising model . 36 3.3 Analysis of the mean-field model . 38 3.4 Functions of Markov processes . 42 3.5 The mean-field contact process . 47 3.6 The mean-field voter model . 49 3.7 Exercises . 51 4 Construction and ergodicity 53 4.1 Introduction . 53 4.2 Feller processes . 54 4.3 Poisson construction . 63 4.4 Generator construction . 72 4.5 Ergodicity . 79 4.6 Application to the Ising model . 81 4.7 Further results . 85 5 Monotonicity 89 5.1 The stochastic order . 89 5.2 The upper and lower invariant laws . 94 5.3 The contact process . 97 5.4 Other examples . 100 3 4 CONTENTS 5.5 Exercises . 101 6 Duality 105 6.1 Introduction . 105 6.2 Additive systems duality . 106 6.3 Cancellative systems duality . 113 6.4 Other dualities .