Branching Brownian Motion with Selection Pascal Maillard
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Persistent Random Walks. II. Functional Scaling Limits
Persistent random walks. II. Functional Scaling Limits Peggy Cénac1. Arnaud Le Ny2. Basile de Loynes3. Yoann Offret1 1Institut de Mathématiques de Bourgogne (IMB) - UMR CNRS 5584 Université de Bourgogne Franche-Comté, 21000 Dijon, France 2Laboratoire d’Analyse et de Mathématiques Appliquées (LAMA) - UMR CNRS 8050 Université Paris Est Créteil, 94010 Créteil Cedex, France 3Ensai - Université de Bretagne-Loire, Campus de Ker-Lann, Rue Blaise Pascal, BP 37203, 35172 BRUZ cedex, France Abstract We give a complete and unified description – under some stability assumptions – of the functional scaling limits associated with some persistent random walks for which the recurrent or transient type is studied in [1]. As a result, we highlight a phase transition phenomenon with respect to the memory. It turns out that the limit process is either Markovian or not according to – to put it in a nutshell – the rate of decrease of the distribution tails corresponding to the persistent times. In the memoryless situation, the limits are classical strictly stable Lévy processes of infinite variations. However, we point out that the description of the critical Cauchy case fills some lacuna even in the closely related context of Directionally Reinforced Random Walks (DRRWs) for which it has not been considered yet. Besides, we need to introduced some relevant generalized drift – extended the classical one – in order to study the critical case but also the situation when the limit is no longer Markovian. It appears to be in full generality a drift in mean for the Persistent Random Walk (PRW). The limit processes keeping some memory – given by some variable length Markov chain – of the underlying PRW are called arcsine Lamperti anomalous diffusions due to their marginal distribution which are computed explicitly here. -
Poisson Representations of Branching Markov and Measure-Valued
The Annals of Probability 2011, Vol. 39, No. 3, 939–984 DOI: 10.1214/10-AOP574 c Institute of Mathematical Statistics, 2011 POISSON REPRESENTATIONS OF BRANCHING MARKOV AND MEASURE-VALUED BRANCHING PROCESSES By Thomas G. Kurtz1 and Eliane R. Rodrigues2 University of Wisconsin, Madison and UNAM Representations of branching Markov processes and their measure- valued limits in terms of countable systems of particles are con- structed for models with spatially varying birth and death rates. Each particle has a location and a “level,” but unlike earlier con- structions, the levels change with time. In fact, death of a particle occurs only when the level of the particle crosses a specified level r, or for the limiting models, hits infinity. For branching Markov pro- cesses, at each time t, conditioned on the state of the process, the levels are independent and uniformly distributed on [0,r]. For the limiting measure-valued process, at each time t, the joint distribu- tion of locations and levels is conditionally Poisson distributed with mean measure K(t) × Λ, where Λ denotes Lebesgue measure, and K is the desired measure-valued process. The representation simplifies or gives alternative proofs for a vari- ety of calculations and results including conditioning on extinction or nonextinction, Harris’s convergence theorem for supercritical branch- ing processes, and diffusion approximations for processes in random environments. 1. Introduction. Measure-valued processes arise naturally as infinite sys- tem limits of empirical measures of finite particle systems. A number of ap- proaches have been developed which preserve distinct particles in the limit and which give a representation of the measure-valued process as a transfor- mation of the limiting infinite particle system. -
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. -
Lectures on Lévy Processes, Stochastic Calculus and Financial
Lectures on L¶evyProcesses, Stochastic Calculus and Financial Applications, Ovronnaz September 2005 David Applebaum Probability and Statistics Department, University of She±eld, Hicks Building, Houns¯eld Road, She±eld, England, S3 7RH e-mail: D.Applebaum@she±eld.ac.uk Introduction A L¶evyprocess is essentially a stochastic process with stationary and in- dependent increments. The basic theory was developed, principally by Paul L¶evyin the 1930s. In the past 15 years there has been a renaissance of interest and a plethora of books, articles and conferences. Why ? There are both theoretical and practical reasons. Theoretical ² There are many interesting examples - Brownian motion, simple and compound Poisson processes, ®-stable processes, subordinated processes, ¯nancial processes, relativistic process, Riemann zeta process . ² L¶evyprocesses are simplest generic class of process which have (a.s.) continuous paths interspersed with random jumps of arbitrary size oc- curring at random times. ² L¶evyprocesses comprise a natural subclass of semimartingales and of Markov processes of Feller type. ² Noise. L¶evyprocesses are a good model of \noise" in random dynamical systems. 1 Input + Noise = Output Attempts to describe this di®erentially leads to stochastic calculus.A large class of Markov processes can be built as solutions of stochastic di®erential equations driven by L¶evynoise. L¶evydriven stochastic partial di®erential equations are beginning to be studied with some intensity. ² Robust structure. Most applications utilise L¶evyprocesses taking val- ues in Euclidean space but this can be replaced by a Hilbert space, a Banach space (these are important for spdes), a locally compact group, a manifold. Quantised versions are non-commutative L¶evyprocesses on quantum groups. -
Local Conditioning in Dawson–Watanabe Superprocesses
The Annals of Probability 2013, Vol. 41, No. 1, 385–443 DOI: 10.1214/11-AOP702 c Institute of Mathematical Statistics, 2013 LOCAL CONDITIONING IN DAWSON–WATANABE SUPERPROCESSES By Olav Kallenberg Auburn University Consider a locally finite Dawson–Watanabe superprocess ξ =(ξt) in Rd with d ≥ 2. Our main results include some recursive formulas for the moment measures of ξ, with connections to the uniform Brown- ian tree, a Brownian snake representation of Palm measures, continu- ity properties of conditional moment densities, leading by duality to strongly continuous versions of the multivariate Palm distributions, and a local approximation of ξt by a stationary clusterη ˜ with nice continuity and scaling properties. This all leads up to an asymptotic description of the conditional distribution of ξt for a fixed t> 0, given d that ξt charges the ε-neighborhoods of some points x1,...,xn ∈ R . In the limit as ε → 0, the restrictions to those sets are conditionally in- dependent and given by the pseudo-random measures ξ˜ orη ˜, whereas the contribution to the exterior is given by the Palm distribution of ξt at x1,...,xn. Our proofs are based on the Cox cluster representa- tions of the historical process and involve some delicate estimates of moment densities. 1. Introduction. This paper may be regarded as a continuation of [19], where we considered some local properties of a Dawson–Watanabe super- process (henceforth referred to as a DW-process) at a fixed time t> 0. Recall that a DW-process ξ = (ξt) is a vaguely continuous, measure-valued diffu- d ξtf µvt sion process in R with Laplace functionals Eµe− = e− for suitable functions f 0, where v = (vt) is the unique solution to the evolution equa- 1 ≥ 2 tion v˙ = 2 ∆v v with initial condition v0 = f. -
Lectures on Multiparameter Processes
Lecture Notes on Multiparameter Processes: Ecole Polytechnique Fed´ erale´ de Lausanne, Switzerland Davar Khoshnevisan Department of Mathematics University of Utah Salt Lake City, UT 84112–0090 [email protected] http://www.math.utah.edu/˜davar April–June 2001 ii Contents Preface vi 1 Examples from Markov chains 1 2 Examples from Percolation on Trees and Brownian Motion 7 3ProvingLevy’s´ Theorem and Introducing Martingales 13 4 Preliminaries on Ortho-Martingales 19 5 Ortho-Martingales and Intersections of Walks and Brownian Motion 25 6 Intersections of Brownian Motion, Multiparameter Martingales 35 7 Capacity, Energy and Dimension 43 8 Frostman’s Theorem, Hausdorff Dimension and Brownian Motion 49 9 Potential Theory of Brownian Motion and Stable Processes 55 10 Brownian Sheet and Kahane’s Problem 65 Bibliography 71 iii iv Preface These are the notes for a one-semester course based on ten lectures given at the Ecole Polytechnique Fed´ erale´ de Lausanne, April–June 2001. My goal has been to illustrate, in some detail, some of the salient features of the theory of multiparameter processes and in particular, Cairoli’s theory of multiparameter mar- tingales. In order to get to the heart of the matter, and develop a kind of intuition at the same time, I have chosen the simplest topics of random walks, Brownian motions, etc. to highlight the methods. The full theory can be found in Multi-Parameter Processes: An Introduction to Random Fields (henceforth, referred to as MPP) which is to be published by Springer-Verlag, although these lectures also contain material not covered in the mentioned book. -
Introduction to Lévy Processes
Introduction to L´evyprocesses Graduate lecture 22 January 2004 Matthias Winkel Departmental lecturer (Institute of Actuaries and Aon lecturer in Statistics) 1. Random walks and continuous-time limits 2. Examples 3. Classification and construction of L´evy processes 4. Examples 5. Poisson point processes and simulation 1 1. Random walks and continuous-time limits 4 Definition 1 Let Yk, k ≥ 1, be i.i.d. Then n X 0 Sn = Yk, n ∈ N, k=1 is called a random walk. -4 0 8 16 Random walks have stationary and independent increments Yk = Sk − Sk−1, k ≥ 1. Stationarity means the Yk have identical distribution. Definition 2 A right-continuous process Xt, t ∈ R+, with stationary independent increments is called L´evy process. 2 Page 1 What are Sn, n ≥ 0, and Xt, t ≥ 0? Stochastic processes; mathematical objects, well-defined, with many nice properties that can be studied. If you don’t like this, think of a model for a stock price evolving with time. There are also many other applications. If you worry about negative values, think of log’s of prices. What does Definition 2 mean? Increments , = 1 , are independent and Xtk − Xtk−1 k , . , n , = 1 for all 0 = . Xtk − Xtk−1 ∼ Xtk−tk−1 k , . , n t0 < . < tn Right-continuity refers to the sample paths (realisations). 3 Can we obtain L´evyprocesses from random walks? What happens e.g. if we let the time unit tend to zero, i.e. take a more and more remote look at our random walk? If we focus at a fixed time, 1 say, and speed up the process so as to make n steps per time unit, we know what happens, the answer is given by the Central Limit Theorem: 2 Theorem 1 (Lindeberg-L´evy) If σ = V ar(Y1) < ∞, then Sn − (Sn) √E → Z ∼ N(0, σ2) in distribution, as n → ∞. -
Part C Lévy Processes and Finance
Part C Levy´ Processes and Finance Matthias Winkel1 University of Oxford HT 2007 1Departmental lecturer (Institute of Actuaries and Aon Lecturer in Statistics) at the Department of Statistics, University of Oxford MS3 Levy´ Processes and Finance Matthias Winkel – 16 lectures HT 2007 Prerequisites Part A Probability is a prerequisite. BS3a/OBS3a Applied Probability or B10 Martin- gales and Financial Mathematics would be useful, but are by no means essential; some material from these courses will be reviewed without proof. Aims L´evy processes form a central class of stochastic processes, contain both Brownian motion and the Poisson process, and are prototypes of Markov processes and semimartingales. Like Brownian motion, they are used in a multitude of applications ranging from biology and physics to insurance and finance. Like the Poisson process, they allow to model abrupt moves by jumps, which is an important feature for many applications. In the last ten years L´evy processes have seen a hugely increased attention as is reflected on the academic side by a number of excellent graduate texts and on the industrial side realising that they provide versatile stochastic models of financial markets. This continues to stimulate further research in both theoretical and applied directions. This course will give a solid introduction to some of the theory of L´evy processes as needed for financial and other applications. Synopsis Review of (compound) Poisson processes, Brownian motion (informal), Markov property. Connection with random walks, [Donsker’s theorem], Poisson limit theorem. Spatial Poisson processes, construction of L´evy processes. Special cases of increasing L´evy processes (subordinators) and processes with only positive jumps. -
Non-Local Branching Superprocesses and Some Related Models
Published in: Acta Applicandae Mathematicae 74 (2002), 93–112. Non-local Branching Superprocesses and Some Related Models Donald A. Dawson1 School of Mathematics and Statistics, Carleton University, 1125 Colonel By Drive, Ottawa, Canada K1S 5B6 E-mail: [email protected] Luis G. Gorostiza2 Departamento de Matem´aticas, Centro de Investigaci´ony de Estudios Avanzados, A.P. 14-740, 07000 M´exicoD. F., M´exico E-mail: [email protected] Zenghu Li3 Department of Mathematics, Beijing Normal University, Beijing 100875, P.R. China E-mail: [email protected] Abstract A new formulation of non-local branching superprocesses is given from which we derive as special cases the rebirth, the multitype, the mass- structured, the multilevel and the age-reproduction-structured superpro- cesses and the superprocess-controlled immigration process. This unified treatment simplifies considerably the proof of existence of the old classes of superprocesses and also gives rise to some new ones. AMS Subject Classifications: 60G57, 60J80 Key words and phrases: superprocess, non-local branching, rebirth, mul- titype, mass-structured, multilevel, age-reproduction-structured, superprocess- controlled immigration. 1Supported by an NSERC Research Grant and a Max Planck Award. 2Supported by the CONACYT (Mexico, Grant No. 37130-E). 3Supported by the NNSF (China, Grant No. 10131040). 1 1 Introduction Measure-valued branching processes or superprocesses constitute a rich class of infinite dimensional processes currently under rapid development. Such processes arose in appli- cations as high density limits of branching particle systems; see e.g. Dawson (1992, 1993), Dynkin (1993, 1994), Watanabe (1968). The development of this subject has been stimu- lated from different subjects including branching processes, interacting particle systems, stochastic partial differential equations and non-linear partial differential equations. -
Levy Processes
LÉVY PROCESSES, STABLE PROCESSES, AND SUBORDINATORS STEVEN P.LALLEY 1. DEFINITIONSAND EXAMPLES d Definition 1.1. A continuous–time process Xt = X(t ) t 0 with values in R (or, more generally, in an abelian topological groupG ) isf called a Lévyg ≥ process if (1) its sample paths are right-continuous and have left limits at every time point t , and (2) it has stationary, independent increments, that is: (a) For all 0 = t0 < t1 < < tk , the increments X(ti ) X(ti 1) are independent. − (b) For all 0 s t the··· random variables X(t ) X−(s ) and X(t s ) X(0) have the same distribution.≤ ≤ − − − The default initial condition is X0 = 0. A subordinator is a real-valued Lévy process with nondecreasing sample paths. A stable process is a real-valued Lévy process Xt t 0 with ≥ initial value X0 = 0 that satisfies the self-similarity property f g 1/α (1.1) Xt =t =D X1 t > 0. 8 The parameter α is called the exponent of the process. Example 1.1. The most fundamental Lévy processes are the Wiener process and the Poisson process. The Poisson process is a subordinator, but is not stable; the Wiener process is stable, with exponent α = 2. Any linear combination of independent Lévy processes is again a Lévy process, so, for instance, if the Wiener process Wt and the Poisson process Nt are independent then Wt Nt is a Lévy process. More important, linear combinations of independent Poisson− processes are Lévy processes: these are special cases of what are called compound Poisson processes: see sec. -
Thick Points for the Cauchy Process
Ann. I. H. Poincaré – PR 41 (2005) 953–970 www.elsevier.com/locate/anihpb Thick points for the Cauchy process Olivier Daviaud 1 Department of Mathematics, Stanford University, Stanford, CA 94305, USA Received 13 June 2003; received in revised form 11 August 2004; accepted 15 October 2004 Available online 23 May 2005 Abstract Let µ(x, ) denote the occupation measure of an interval of length 2 centered at x by the Cauchy process run until it hits −∞ − ]∪[ ∞ 2 → → ( , 1 1, ). We prove that sup|x|1 µ(x,)/((log ) ) 2/π a.s. as 0. We also obtain the multifractal spectrum 2 for thick points, i.e. the Hausdorff dimension of the set of α-thick points x for which lim→0 µ(x,)/((log ) ) = α>0. 2005 Elsevier SAS. All rights reserved. Résumé Soit µ(x, ) la mesure d’occupation de l’intervalle [x − ,x + ] parleprocessusdeCauchyarrêtéàsasortiede(−1, 1). 2 → → Nous prouvons que sup|x|1 µ(x, )/((log ) ) 2/π p.s. lorsque 0. Nous obtenons également un spectre multifractal 2 de points épais en montrant que la dimension de Hausdorff des points x pour lesquels lim→0 µ(x, )/((log ) ) = α>0est égale à 1 − απ/2. 2005 Elsevier SAS. All rights reserved. MSC: 60J55 Keywords: Thick points; Multi-fractal analysis; Cauchy process 1. Introduction Let X = (Xt ,t 0) be a Cauchy process on the real line R, that is a process starting at 0, with stationary independent increments with the Cauchy distribution: s dx P X + − X ∈ (x − dx,x + dx) = ,s,t>0,x∈ R. -
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).