DOCSLIB.ORG

Probabilistic and Geometric Methods in Last Passage Percolation

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Probabilistic and geometric methods in last passage percolation by Milind Hegde A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Alan Hammond, Co‐chair Professor Shirshendu Ganguly, Co‐chair Professor Fraydoun Rezakhanlou Professor Alistair Sinclair Spring 2021 Probabilistic and geometric methods in last passage percolation Copyright 2021 by Milind Hegde 1 Abstract Probabilistic and geometric methods in last passage percolation by Milind Hegde Doctor of Philosophy in Mathematics University of California, Berkeley Professor Alan Hammond, Co‐chair Professor Shirshendu Ganguly, Co‐chair Last passage percolation (LPP) refers to a broad class of models thought to lie within the Kardar‐ Parisi‐Zhang universality class of one‐dimensional stochastic growth models. In LPP models, there is a planar random noise environment through which directed paths travel; paths are as‐ signed a weight based on their journey through the environment, usually by, in some sense, integrating the noise over the path. For given points y and x, the weight from y to x is defined by maximizing the weight over all paths from y to x. A path which achieves the maximum weight is called a geodesic. A few last passage percolation models are exactly solvable, i.e., possess what is called integrable structure. This gives rise to valuable information, such as explicit probabilistic resampling prop‐ erties, distributional convergence information, or one point tail bounds of the weight profile as the starting point y is fixed and the ending point x varies. However, much of the behaviour currently proved within only the special models with integrable structure is expected to hold in a much larger class of models; further, integrable techniques alone seem unable to obtain cer‐ tain types of information, such as process‐level properties of natural limiting objects. This thesis explores probabilistic and geometric approaches which assume limited integrable input to infer information unavailable via exactly solvable techniques or to develop methods which will prove be to robust and eventually applicable to a broader class of non‐integrable models. In the first part, we study the parabolic Airy2 process P1 and the parabolic Airy line ensemble P via a probabilistic resampling structure enjoyed by the latter. The parabolic Airy line ensemble is an infinite N‐indexed collection of random continuous non‐intersecting curves, whose top (and lowest indexed) curve is P1, which is a limiting weight profile in LPP when the starting point is fixed (which can be thought of as a particular form of initial data) and the ending point is al‐ lowed to vary along a line. The ensemble P possess a probabilistic resampling property known as the Brownian Gibbs property, which gives an explicit description of a certain conditional 2 distribution of P in terms of non‐intersecting Brownian bridges. This property gives a qualita‐ tive comparison—absolute continuity—of P1 to Brownian motion, as proved in [CH14]. Using a framework introduced in [Ham19a], we prove here a strong quantitative form of comparison, showing that an event which has probability " under the law of Brownian motion has probability 1−o(1) −1 5/6 at most " under the law of an increment of P1, with an explicit form of exp(O(1)(log " ) ) for the "−o(1) error factor. Up to the error factor, this is expected to be sharp. One consequence is that the Radon‐Nikodym derivative of the increment of P1 with respect to Brownian motion lies in all Lp spaces with p 2 [1; 1), i.e., has finite polynomial moments of all orders. The bounds also hold for lower curves of P and for weight profiles in an LPP model known as Brownian LPP. In the second part, we work in an LPP model on the lattice Z2 and make use of a geometric approach, i.e., we study properties of geodesics and other weight maximizing structures. The random environment is given by i.i.d. non‐negative random variables associated to the vertices of Z2, and the weight of an up‐right path is the sum of the random variables it passes through. Now, in exactly solvable models such as when the vertex weight distribution is geometric or exponential, the GUE Tracy‐Widom distribution is known to be a scaling limit of the last pas‐ sage value Xr from (1; 1) to (r; r). The GUE Tracy‐Widom distribution is well‐known to have 3 upper and lower tail exponents of 2 and 3, which is also known for the prelimiting Xr in the mentioned exactly solvable models. Here we work in a more general setup and adopt some nat‐ ural assumptions of curvature and weak one‐point upper and lower tail estimates on the weight profile—with no assumptions on the vertex weight distribution and, hence, a non‐integrable setting—which we bootstrap up to obtain the optimal tail exponents for both tails, in terms of both upper and lower bounds. We also obtain sharp upper and lower bounds for the lower tail of the maximum weight over all paths which are constrained to lie in a strip of given width, a result which was not previously known in even the integrable models and does not seem easily accessible via integrable techniques. Finally, in the third part, we combine the geometric and probabilistic approaches to obtain an estimate on a certain probability of the KPZ fixed point. The KPZ fixed point (t; x) 7! ht(x) is a space‐time process constructed in [MQR17] which should be thought of as a limiting object analogous to P1 under general initial data. We show that the event, for fixed t > 0, that ht has "‐twin peaks, i.e., there exists a point at a given distance away from the maximizer of ht at which ht comes within " of its maximum, has probability at least of order ". This served as an important input in [CHHM21] to prove that the Hausdorff dimension of the set of exceptional 2 times t where ht has multiple maximizers is almost surely 3 , on the positive‐probability event that the set is non‐empty. Geometric and probabilistic arguments are combined by making use of a remarkable identity of last passage values between an original and a certain transformed environment proven in [DOV18], which allows us to consider geodesics through an environment which itself has the Brownian Gibbs property. i To family, friends, mentors, and the simple things which suffice. ii Contents Contents ii List of Figures v I Introduction 1 1 The KPZ universality class 2 1.1 The Eden model and first passage percolation .................. 2 1.2 Integrability within KPZ .............................. 5 1.3 Last passage percolation .............................. 9 1.4 The KPZ equation and the KPZ fixed point .................... 17 2 Probabilistic and geometric viewpoints 21 2.1 Background on the probabilistic approach .................... 22 2.2 Results in the probabilistic stream: Brownian regularity ............. 28 2.3 Background on the geometric approach ...................... 31 2.4 Results in the geometric stream: Bootstrapping .................. 34 2.5 Results combining the perspectives: Fractal geometry .............. 36 2.6 Other graduate work ............................... 40 II Brownian regularity 43 3 Proof context 44 3.1 Pertinent recent work ............................... 44 3.2 Method of proof .................................. 47 3.3 Organization of Part II .............................. 49 4 Notation and setup 50 4.1 Notation, Brownian Gibbs, and regular ensembles ................ 50 4.2 An important example of regular ensembles: Brownian LPP weight profiles . 53 4.3 Main result .................................... 55 iii 5 Proof framework 57 5.1 The jump ensemble ................................ 57 5.2 A conceptual framework in terms of costs ..................... 76 6 Proving the density bounds 89 6.1 The easy case: Below the pole on both sides ................... 89 6.2 The moderate case: Above the pole on both sides ................ 91 6.3 The difficult case: Above and below the pole on either side ........... 102 6.4 When no pole is present .............................. 110 III Bootstrapping 114 7 The basic idea of bootstrapping 115 7.1 Introduction, the model, and assumptions .................... 115 7.2 Main results .................................... 121 7.3 The key ideas ................................... 125 7.4 Tails for constrained weight ............................ 131 7.5 Related work .................................... 132 7.6 A few important tools ............................... 133 7.7 Organization of Part III .............................. 135 8 Concentration tools and the bootstrap 136 9 The tail bound proofs 140 9.1 Upper tail bounds ................................. 140 9.2 Lower bound on upper tail ............................ 151 9.3 Lower tail and constrained lower tail bounds ................... 152 10 Constructing disjoint high weight paths 162 10.1 The construction in outline ............................ 163 10.2 The construction in detail ............................. 164 10.3 Bounding below the expected weight of the construction ............. 168 10.4 One‐sided concentration of the construction weight ............... 170 IV Twinpeaks for the KPZ fixed point 173 11 The lower bound on the twin peaks probability 174 11.1 Preliminaries .................................... 176 11.2 Locations of maximizers .............................. 181 11.3 The resampling framework ............................ 184 11.4 The Brownian Gibbs property ..........................
Recommended publications
  • The Distribution of Local Times of a Brownian Bridge

    The Distribution of Local Times of a Brownian Bridge

    The distribution of lo cal times of a Brownian bridge by Jim Pitman Technical Rep ort No. 539 Department of Statistics University of California 367 Evans Hall 3860 Berkeley, CA 94720-3860 Nov. 3, 1998 1 The distribution of lo cal times of a Brownian bridge Jim Pitman Department of Statistics, University of California, 367 Evans Hall 3860, Berkeley, CA 94720-3860, USA [email protected] 1 Intro duction x Let L ;t 0;x 2 R denote the jointly continuous pro cess of lo cal times of a standard t one-dimensional Brownian motion B ;t 0 started at B = 0, as determined bythe t 0 o ccupation density formula [19 ] Z Z t 1 x dx f B ds = f xL s t 0 1 for all non-negative Borel functions f . Boro din [7, p. 6] used the metho d of Feynman- x Kac to obtain the following description of the joint distribution of L and B for arbitrary 1 1 xed x 2 R: for y>0 and b 2 R 1 1 2 x jxj+jbxj+y 2 p P L 2 dy ; B 2 db= jxj + jb xj + y e dy db: 1 1 1 2 This formula, and features of the lo cal time pro cess of a Brownian bridge describ ed in the rest of this intro duction, are also implicitinRay's description of the jointlawof x ;x 2 RandB for T an exp onentially distributed random time indep endentofthe L T T Brownian motion [18, 22 , 6 ]. See [14, 17 ] for various characterizations of the lo cal time pro cesses of Brownian bridge and Brownian excursion, and further references.
  • Derivatives of Self-Intersection Local Times

    Derivatives of Self-Intersection Local Times

    Derivatives of self-intersection local times Jay Rosen? Department of Mathematics College of Staten Island, CUNY Staten Island, NY 10314 e-mail: [email protected] Summary. We show that the renormalized self-intersection local time γt(x) for both the Brownian motion and symmetric stable process in R1 is differentiable in 0 the spatial variable and that γt(0) can be characterized as the continuous process of zero quadratic variation in the decomposition of a natural Dirichlet process. This Dirichlet process is the potential of a random Schwartz distribution. Analogous results for fractional derivatives of self-intersection local times in R1 and R2 are also discussed. 1 Introduction In their study of the intrinsic Brownian local time sheet and stochastic area integrals for Brownian motion, [14, 15, 16], Rogers and Walsh were led to analyze the functional Z t A(t, Bt) = 1[0,∞)(Bt − Bs) ds (1) 0 where Bt is a 1-dimensional Brownian motion. They showed that A(t, Bt) is not a semimartingale, and in fact showed that Z t Bs A(t, Bt) − Ls dBs (2) 0 x has finite non-zero 4/3-variation. Here Ls is the local time at x, which is x R s formally Ls = 0 δ(Br − x) dr, where δ(x) is Dirac’s ‘δ-function’. A formal d d2 0 application of Ito’s lemma, using dx 1[0,∞)(x) = δ(x) and dx2 1[0,∞)(x) = δ (x), yields Z t 1 Z t Z s Bs 0 A(t, Bt) − Ls dBs = t + δ (Bs − Br) dr ds (3) 0 2 0 0 ? This research was supported, in part, by grants from the National Science Foun- dation and PSC-CUNY.
  • An Excursion Approach to Ray-Knight Theorems for Perturbed Brownian Motion

    An Excursion Approach to Ray-Knight Theorems for Perturbed Brownian Motion

    stochastic processes and their applications ELSEVIER Stochastic Processes and their Applications 63 (1996) 67 74 An excursion approach to Ray-Knight theorems for perturbed Brownian motion Mihael Perman* L"nil,ersi O, q/ (~mt)ridge, Statistical Laboratory. 16 Mill Lane. Cambridge CB2 15B. ~ 'h" Received April 1995: levised November 1995 Abstract Perturbed Brownian motion in this paper is defined as X, = IB, I - p/~ where B is standard Brownian motion, (/t: t >~ 0) is its local time at 0 and H is a positive constant. Carmona et al. (1994) have extended the classical second Ray --Knight theorem about the local time processes in the space variable taken at an inverse local time to perturbed Brownian motion with the resulting Bessel square processes having dimensions depending on H. In this paper a proof based on splitting the path of perturbed Brownian motion at its minimum is presented. The derivation relies mostly on excursion theory arguments. K<rwordx: Excursion theory; Local times: Perturbed Brownian motion; Ray Knight theorems: Path transfl)rmations 1. Introduction The classical Ray-Knight theorems assert that the local time of Brownian motion as a process in the space variable taken at appropriate stopping time can be shown to be a Bessel square process of a certain dimension. There are many proofs of these theorems of which perhaps the ones based on excursion theory are the simplest ones. There is a substantial literature on Ray Knight theorems and their generalisations; see McKean (1975) or Walsh (1978) for a survey. Carmona et al. (1994) have extended the second Ray Knight theorem to perturbed BM which is defined as X, = IB, I - Iz/, where B is standard BM (/,: I ~> 0) is its local time at 0 and tl is an arbitrary positive constant.
  • Processes on Complex Networks. Percolation

    Processes on Complex Networks. Percolation

    Chapter 5 Processes on complex networks. Percolation 77 Up till now we discussed the structure of the complex networks. The actual reason to study this structure is to understand how this structure influences the behavior of random processes on networks. I will talk about two such processes. The first one is the percolation process. The second one is the spread of epidemics. There are a lot of open problems in this area, the main of which can be innocently formulated as: How the network topology influences the dynamics of random processes on this network. We are still quite far from a definite answer to this question. 5.1 Percolation 5.1.1 Introduction to percolation Percolation is one of the simplest processes that exhibit the critical phenomena or phase transition. This means that there is a parameter in the system, whose small change yields a large change in the system behavior. To define the percolation process, consider a graph, that has a large connected component. In the classical settings, percolation was actually studied on infinite graphs, whose vertices constitute the set Zd, and edges connect each vertex with nearest neighbors, but we consider general random graphs. We have parameter ϕ, which is the probability that any edge present in the underlying graph is open or closed (an event with probability 1 − ϕ) independently of the other edges. Actually, if we talk about edges being open or closed, this means that we discuss bond percolation. It is also possible to talk about the vertices being open or closed, and this is called site percolation.
  • Stochastic Analysis in Continuous Time

    Stochastic Analysis in Continuous Time

    Stochastic Analysis in Continuous Time Stochastic basis with the usual assumptions: F = (Ω; F; fFtgt≥0; P), where Ω is the probability space (a set of elementary outcomes !); F is the sigma-algebra of events (sub-sets of Ω that can be measured by P); fFtgt≥0 is the filtration: Ft is the sigma-algebra of events that happened by time t so that Fs ⊆ Ft for s < t; P is probability: finite non-negative countably additive measure on the (Ω; F) normalized to one (P(Ω) = 1). The usual assumptions are about the filtration: F0 is P-complete, that is, if A 2 F0 and P(A) = 0; then every sub-set of A is in F0 [this fF g minimises the technicalT difficulties related to null sets and is not hard to achieve]; and t t≥0 is right-continuous, that is, Ft = Ft+", t ≥ 0 [this simplifies some further technical developments, like stopping time vs. optional ">0 time and might be hard to achieve, but is assumed nonetheless.1] Stopping time τ on F is a random variable with values in [0; +1] and such that, for every t ≥ 0, the setTf! : 2 τ(!) ≤ tg is in Ft. The sigma algebra Fτ consists of all the events A from F such that, for every t ≥ 0, A f! : τ(!) ≤ tg 2 Ft. In particular, non-random τ = T ≥ 0 is a stopping time and Fτ = FT . For a stopping time τ, intervals such as (0; τ) denote a random set f(!; t) : 0 < t < τ(!)g. A random/stochastic process X = X(t) = X(!; t), t ≥ 0, is a collection of random variables3.
  • Probability on Graphs Random Processes on Graphs and Lattices

    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

    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

    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).
  • Universal Scaling Behavior of Non-Equilibrium Phase Transitions

    Universal Scaling Behavior of Non-Equilibrium Phase Transitions

    Universal scaling behavior of non-equilibrium phase transitions Sven L¨ubeck Theoretische Physik, Univerit¨at Duisburg-Essen, 47048 Duisburg, Germany, [email protected] December 2004 arXiv:cond-mat/0501259v1 [cond-mat.stat-mech] 11 Jan 2005 Summary Non-equilibrium critical phenomena have attracted a lot of research interest in the recent decades. Similar to equilibrium critical phenomena, the concept of universality remains the major tool to order the great variety of non-equilibrium phase transitions systematically. All systems belonging to a given universality class share the same set of critical exponents, and certain scaling functions become identical near the critical point. It is known that the scaling functions vary more widely between different uni- versality classes than the exponents. Thus, universal scaling functions offer a sensitive and accurate test for a system’s universality class. On the other hand, universal scaling functions demonstrate the robustness of a given universality class impressively. Unfor- tunately, most studies focus on the determination of the critical exponents, neglecting the universal scaling functions. In this work a particular class of non-equilibrium critical phenomena is considered, the so-called absorbing phase transitions. Absorbing phase transitions are expected to occur in physical, chemical as well as biological systems, and a detailed introduc- tion is presented. The universal scaling behavior of two different universality classes is analyzed in detail, namely the directed percolation and the Manna universality class. Especially, directed percolation is the most common universality class of absorbing phase transitions. The presented picture gallery of universal scaling functions includes steady state, dynamical as well as finite size scaling functions.
  • Long Time Behaviour and Mean-Field Limit of Atlas Models Julien Reygner

    Long Time Behaviour and Mean-Field Limit of Atlas Models Julien Reygner

    Long time behaviour and mean-field limit of Atlas models Julien Reygner To cite this version: Julien Reygner. Long time behaviour and mean-field limit of Atlas models. ESAIM: Proceedings and Surveys, EDP Sciences, 2017, Journées MAS 2016 de la SMAI – Phénomènes complexes et hétérogènes, 60, pp.132-143. 10.1051/proc/201760132. hal-01525360 HAL Id: hal-01525360 https://hal.archives-ouvertes.fr/hal-01525360 Submitted on 19 May 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Long time behaviour and mean-field limit of Atlas models Julien Reygner ABSTRACT. This article reviews a few basic features of systems of one-dimensional diffusions with rank-based characteristics. Such systems arise in particular in the modelling of financial mar- kets, where they go by the name of Atlas models. We mostly describe their long time and large scale behaviour, and lay a particular emphasis on the case of mean-field interactions. We finally present an application of the reviewed results to the modelling of capital distribution in systems with a large number of agents. 1. Introduction The term Atlas model was originally introduced in Fernholz’ monograph on Stochastic Portfolio Theory [13] to describe a stock market in which the asset prices evolve according to independent and identically distributed processes, except the smallest one which undergoes an additional up- ward push.
  • Gaussian Processes and the Local Times of Symmetric Lévy

    Gaussian Processes and the Local Times of Symmetric Lévy

    GAUSSIAN PROCESSES AND THE LOCAL TIMES OF SYMMETRIC LEVY´ PROCESSES MICHAEL B. MARCUS AND JAY ROSEN CITY COLLEGE AND COLLEGE OF STATEN ISLAND Abstract. We give a relatively simple proof of the necessary and sufficient condition for the joint continuity of the local times of symmetric L´evypro- cesses. This result was obtained in 1988 by M. Barlow and J. Hawkes without requiring that the L´evyprocesses be symmetric. In 1992 the authors used a very different approach to obtain necessary and sufficient condition for the joint continuity of the local times of strongly symmetric Markov processes, which includes symmetric L´evyprocesses. Both the 1988 proof and the 1992 proof are long and difficult. In this paper the 1992 proof is significantly simplified. This is accomplished by using two recent isomorphism theorems, which relate the local times of strongly symmetric Markov processes to certain Gaussian processes, one due to N. Eisenbaum alone and the other to N. Eisenbaum, H. Kaspi, M.B. Marcus, J. Rosen and Z. Shi. Simple proofs of these isomorphism theorems are given in this paper. 1. Introduction We give a relatively simple proof of the necessary and sufficient condition for the joint continuity of the local times of symmetric L´evyprocesses. Let X = {X(t), t ∈ R+} be a symmetric L´evyprocess with values in R and characteristic function (1.1) EeiξX(t) = e−tψ(ξ). x x Let Lt denote the local time of X at x ∈ R. Heuristically, Lt is the amount of time that the process spends at x, up to time t.
  • Percolation Theory Are Well-Suited To

    Percolation Theory Are Well-Suited To

    Models of Disordered Media and Predictions of Associated Hydraulic Conductivity A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science By L AARON BLANK B.S., Wright State University, 2004 2006 Wright State University WRIGHT STATE UNIVERSITY SCHOOL OF GRADUATE STUDIES Novermber 6, 2006 I HEREBY RECOMMEND THAT THE THESIS PREPARED UNDER MY SUPERVISION BY L Blank ENTITLED Models of Disordered Media and Predictions of Associated Hydraulic Conductivity BE ACCEPTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF Master of Science. _______________________ Allen Hunt, Ph.D. Thesis Advisor _______________________ Lok Lew Yan Voon, Ph.D. Department Chair _______________________ Joseph F. Thomas, Jr., Ph.D. Dean of the School of Graduate Studies Committee on Final Examination ____________________ Allen Hunt, Ph.D. ____________________ Brent D. Foy, Ph.D. ____________________ Gust Bambakidis, Ph.D. ____________________ Thomas Skinner, Ph.D. Abstract In the late 20th century there was a spill of Technetium in eastern Washington State at the US Department of Energy Hanford site. Resulting contamination of water supplies would raise serious health issues for local residents. Therefore, the ability to predict how these contaminants move through the soil is of great interest. The main contribution to contaminant transport arises from being carried along by flowing water. An important control on the movement of the water through the medium is the hydraulic conductivity, K, which defines the ease of water flow for a given pressure difference (analogous to the electrical conductivity). The overall goal of research in this area is to develop a technique which accurately predicts the hydraulic conductivity as well as its distribution, both in the horizontal and the vertical directions, for media representative of the Hanford subsurface.