DOCSLIB.ORG

Random Dynamical Systems and Rough Paths

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Random Dynamical Systems and Rough Paths Random dynamical systems and rough paths Sebastian Riedel Technische Universit¨atBerlin Workshop \Rough Paths in Toulouse" INSA Toulouse 19.10.2017 Random dynamical systems and rough paths S. Riedel 1/24 Outline 1 Random dynamical systems: Motivation 2 Random dynamical systems and rough paths 3 Invariant measures for RDEs Random dynamical systems and rough paths S. Riedel 2/24 Outline 1 Random dynamical systems: Motivation 2 Random dynamical systems and rough paths 3 Invariant measures for RDEs Random dynamical systems and rough paths S. Riedel 3/24 It proved to be useful in particular when studying the long time behaviour of stochastic systems. Main tool: Multiplicative Ergodic Theorem; yields existence of Lyapunovexponents for linear systems (Oseledets '68). For non-linear systems, one may look at their linearization, and the MET can still provide information about the local behaviour of the non-linear system (stable manifold theorem, Ruelle '79). RDS and MET The theory of Random dynamical systems (RDS) provides a formalism which can be used to describe a large class of stochastic systems which evolve in time (like dynamical systems do for deterministic time evolutions). Random dynamical systems and rough paths S. Riedel 4/24 Main tool: Multiplicative Ergodic Theorem; yields existence of Lyapunovexponents for linear systems (Oseledets '68). For non-linear systems, one may look at their linearization, and the MET can still provide information about the local behaviour of the non-linear system (stable manifold theorem, Ruelle '79). RDS and MET The theory of Random dynamical systems (RDS) provides a formalism which can be used to describe a large class of stochastic systems which evolve in time (like dynamical systems do for deterministic time evolutions). It proved to be useful in particular when studying the long time behaviour of stochastic systems. Random dynamical systems and rough paths S. Riedel 4/24 For non-linear systems, one may look at their linearization, and the MET can still provide information about the local behaviour of the non-linear system (stable manifold theorem, Ruelle '79). RDS and MET The theory of Random dynamical systems (RDS) provides a formalism which can be used to describe a large class of stochastic systems which evolve in time (like dynamical systems do for deterministic time evolutions). It proved to be useful in particular when studying the long time behaviour of stochastic systems. Main tool: Multiplicative Ergodic Theorem; yields existence of Lyapunovexponents for linear systems (Oseledets '68). Random dynamical systems and rough paths S. Riedel 4/24 RDS and MET The theory of Random dynamical systems (RDS) provides a formalism which can be used to describe a large class of stochastic systems which evolve in time (like dynamical systems do for deterministic time evolutions). It proved to be useful in particular when studying the long time behaviour of stochastic systems. Main tool: Multiplicative Ergodic Theorem; yields existence of Lyapunovexponents for linear systems (Oseledets '68). For non-linear systems, one may look at their linearization, and the MET can still provide information about the local behaviour of the non-linear system (stable manifold theorem, Ruelle '79). Random dynamical systems and rough paths S. Riedel 4/24 Differential of the flow solves a linear equation; existence of Lyanpunovexponents follow from the MET (under certain assumptions): 1 ξ(!) P − lim log DξYt v 2 fλl < : : : < λ1g; t!1 t m 1 ≤ l ≤ m, v 2 R . Possible to conclude existence of stable (or unstable) manifolds around stationary solutions for the original equation (following Ruelle's strategy). Stable manifolds for SDE Stable manifold theorem for SDE. (Mohammed-Scheutzow '99) Assume ξ ξ ξ ξ m dYt = b(Yt ) dt + σ(Yt ) ◦ dBt(!);Y0 = ξ 2 R induces stochastic (semi-)flow and solution admits an ergodic invariant measure. Random dynamical systems and rough paths S. Riedel 5/24 Possible to conclude existence of stable (or unstable) manifolds around stationary solutions for the original equation (following Ruelle's strategy). Stable manifolds for SDE Stable manifold theorem for SDE. (Mohammed-Scheutzow '99) Assume ξ ξ ξ ξ m dYt = b(Yt ) dt + σ(Yt ) ◦ dBt(!);Y0 = ξ 2 R induces stochastic (semi-)flow and solution admits an ergodic invariant measure. Differential of the flow solves a linear equation; existence of Lyanpunovexponents follow from the MET (under certain assumptions): 1 ξ(!) P − lim log DξYt v 2 fλl < : : : < λ1g; t!1 t m 1 ≤ l ≤ m, v 2 R . Random dynamical systems and rough paths S. Riedel 5/24 Stable manifolds for SDE Stable manifold theorem for SDE. (Mohammed-Scheutzow '99) Assume ξ ξ ξ ξ m dYt = b(Yt ) dt + σ(Yt ) ◦ dBt(!);Y0 = ξ 2 R induces stochastic (semi-)flow and solution admits an ergodic invariant measure. Differential of the flow solves a linear equation; existence of Lyanpunovexponents follow from the MET (under certain assumptions): 1 ξ(!) P − lim log DξYt v 2 fλl < : : : < λ1g; t!1 t m 1 ≤ l ≤ m, v 2 R . Possible to conclude existence of stable (or unstable) manifolds around stationary solutions for the original equation (following Ruelle's strategy). Random dynamical systems and rough paths S. Riedel 5/24 It¯o-theory not really necessary; only used to make sense of initial equation. Markov property used only to speak about invariant measures. In fact, not really necessary, too, more general concept of invariant measures available in the theory of RDS (cf. below). One of our goals: Build a bridge between the \two cultures" (L. Arnold) Dynamical systems and Stochastic analysis. Remarks Remark. Random dynamical systems and rough paths S. Riedel 6/24 Markov property used only to speak about invariant measures. In fact, not really necessary, too, more general concept of invariant measures available in the theory of RDS (cf. below). One of our goals: Build a bridge between the \two cultures" (L. Arnold) Dynamical systems and Stochastic analysis. Remarks Remark. It¯o-theory not really necessary; only used to make sense of initial equation. Random dynamical systems and rough paths S. Riedel 6/24 One of our goals: Build a bridge between the \two cultures" (L. Arnold) Dynamical systems and Stochastic analysis. Remarks Remark. It¯o-theory not really necessary; only used to make sense of initial equation. Markov property used only to speak about invariant measures. In fact, not really necessary, too, more general concept of invariant measures available in the theory of RDS (cf. below). Random dynamical systems and rough paths S. Riedel 6/24 Remarks Remark. It¯o-theory not really necessary; only used to make sense of initial equation. Markov property used only to speak about invariant measures. In fact, not really necessary, too, more general concept of invariant measures available in the theory of RDS (cf. below). One of our goals: Build a bridge between the \two cultures" (L. Arnold) Dynamical systems and Stochastic analysis. Random dynamical systems and rough paths S. Riedel 6/24 Moreover, we will be interested in: equilibrium of the system, invariant measures long time behaviour (convergence towards equilibrium, attractors,...) qualitative difference between random and deterministic system (stabilization, synchronization by noise, ...) Remark: In general, solution Y to (1) not Markovian. Long term goal: Prove more general stable manifold theorem for dYt = b(Yt) dt + σ(Yt) dXt(!) (1) with t 7! Xt rough paths lift of a stochastic process (e.g. fBm). Random dynamical systems and rough paths S. Riedel 7/24 equilibrium of the system, invariant measures long time behaviour (convergence towards equilibrium, attractors,...) qualitative difference between random and deterministic system (stabilization, synchronization by noise, ...) Remark: In general, solution Y to (1) not Markovian. Long term goal: Prove more general stable manifold theorem for dYt = b(Yt) dt + σ(Yt) dXt(!) (1) with t 7! Xt rough paths lift of a stochastic process (e.g. fBm). Moreover, we will be interested in: Random dynamical systems and rough paths S. Riedel 7/24 long time behaviour (convergence towards equilibrium, attractors,...) qualitative difference between random and deterministic system (stabilization, synchronization by noise, ...) Remark: In general, solution Y to (1) not Markovian. Long term goal: Prove more general stable manifold theorem for dYt = b(Yt) dt + σ(Yt) dXt(!) (1) with t 7! Xt rough paths lift of a stochastic process (e.g. fBm). Moreover, we will be interested in: equilibrium of the system, invariant measures Random dynamical systems and rough paths S. Riedel 7/24 qualitative difference between random and deterministic system (stabilization, synchronization by noise, ...) Remark: In general, solution Y to (1) not Markovian. Long term goal: Prove more general stable manifold theorem for dYt = b(Yt) dt + σ(Yt) dXt(!) (1) with t 7! Xt rough paths lift of a stochastic process (e.g. fBm). Moreover, we will be interested in: equilibrium of the system, invariant measures long time behaviour (convergence towards equilibrium, attractors,...) Random dynamical systems and rough paths S. Riedel 7/24 Remark: In general, solution Y to (1) not Markovian. Long term goal: Prove more general stable manifold theorem for dYt = b(Yt) dt + σ(Yt) dXt(!) (1) with t 7! Xt rough paths lift of a stochastic process (e.g. fBm). Moreover, we will be interested in: equilibrium of the system, invariant measures long time behaviour (convergence towards equilibrium, attractors,...) qualitative difference between random and deterministic system (stabilization, synchronization by noise, ...) Random dynamical systems and rough paths S. Riedel 7/24 Long term goal: Prove more general stable manifold theorem for dYt = b(Yt) dt + σ(Yt) dXt(!) (1) with t 7! Xt rough paths lift of a stochastic process (e.g. fBm). Moreover, we will be interested in: equilibrium of the system, invariant measures long time behaviour (convergence towards equilibrium, attractors,...) qualitative difference between random and deterministic system (stabilization, synchronization by noise, ...) Remark: In general, solution Y to (1) not Markovian.
Load more

Recommended publications
  • Data Assimilation and Parameter Estimation for a Multiscale Stochastic System with Α-Stable Lévy Noise
    Data assimilation and parameter estimation for a multiscale stochastic system with α-stable L´evy noise Yanjie Zhang1, Zhuan Cheng2, Xinyong Zhang3, Xiaoli Chen1, Jinqiao Duan2 and Xiaofan Li2 1Center for Mathematical Sciences & School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China 2Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA 3Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China E-mail: [email protected], [email protected], [email protected], [email protected], [email protected] and [email protected] . ∗This work was partly supported by the NSF grant 1620449, and NSFC grants 11531006, 11371367, and 11271290. Abstract. This work is about low dimensional reduction for a slow-fast data assimilation system with non-Gaussian α stable L´evy noise via stochastic averaging. − When the observations are only available for slow components, we show that the averaged, low dimensional filter approximates the original filter, by examining the corresponding Zakai stochastic partial differential equations. Furthermore, we demonstrate that the low dimensional slow system approximates the slow dynamics of the original system, by examining parameter estimation and most probable paths. Keywords: Multiscale systems, non-Gaussian L´evy noise, averaging principle, Zakai equation, parameter estimation, most probable paths arXiv:1801.02846v1 [math.DS] 9 Jan 2018 1. Introduction Data assimilation is a procedure to extract system state information with the help of observations [18]. The state evolution and the observations are usually under random fluctuations. The general idea is to gain the best estimate for the true system state, in terms of the probability distribution for the system state, given only some noisy observations of the system.
    [Show full text]
  • Limit Theorems for Non-Stationary and Random
    LIMIT THEOREMS FOR NON-STATIONARY AND RANDOM DYNAMICAL SYSTEMS by Yaofeng Su A dissertation submitted to the Department of Mathematics, College of Natural Sciences and Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics Chair of Committee: Andrew T¨or¨ok Committee Member: Vaughn Climenhaga Committee Member: Gemunu Gunaratne Committee Member: Matthew Nicol University of Houston May 2020 Copyright 2020, Yaofeng Su ACKNOWLEDGMENTS This dissertation would not have been possible without the guidance and the help of several in- dividuals who in one way or another contributed and extended their valuable assistance in the preparation and completion of this study. First, my utmost gratitude to my advisor Dr. Andrew T¨or¨okfor the support and help he provided me throughout my graduate education; my commit- tee members, Dr. Vaughn Climenhaga, Dr. Gemunu Gunaratne, and Dr. Matthew Nicol for their participation and their thoughtful comments; all professors in the dynamical system research group of University of Houston, Dr. Vaughn Climenhaga, Dr. Alan Haynes, Dr. Matthew Nicol, Dr. William Ott, and Dr. Andrew T¨or¨okfor organizing the wonderful \Houston Summer School on Dynamical Systems", where I learned a lot about dynamical systems, ergodic theory, number theory, and quantum mechanics. iii ABSTRACT We study the limit behavior of the non-stationary/random chaotic dynamical systems and prove a strong statistical limit theorem: (vector-valued) almost sure invariance principle for the non- stationary dynamical systems and quenched (vector-valued) almost sure invariance principle for the random dynamical systems. It is a matching of the trajectories of the dynamical system with a Brownian motion in such a way that the error is negligible in comparison with the Birkhoff sum.
    [Show full text]
  • Math.DS] 5 Aug 2014 Etns E PK O Opeesv Uvyo H Ujc.I Subject
    SYNCHRONIZATION PROPERTIES OF RANDOM PIECEWISE ISOMETRIES VICTOR KLEPTSYN AND ANTON GORODETSKI Abstract. We study the synchronization properties of the random double rotations on tori. We give a criterion that show when synchronization is present in the case of random double rotations on the circle and prove that it is always absent in dimensions two and higher. 1. Introduction The observation of a synchronization effect goes back at least to 17th century, when Huygens [Hu] discovered the synchronization of two linked pendulums. Since then, synchronization phenomena have been observed in numerous systems and settings, see [PRK] for a comprehensive survey of the subject. In the theory of dynamical systems, synchronization usually refers to random dynamical system trajectories of different initial points converging to each other under the applica- tion of a sequence of random transformations. A first such result is the famous Furstenberg’s Theorem [Fur1], stating that under some very mild assumptions, the angle (mod π) between the images of any two vectors under a long product of random matrices (exponentially) tends to zero. Projectivizing the dynamics, it is easy to see that this theorem in fact states that random trajectories of the quotient system on the projective space (exponentially) approach each other. For random dynamical systems on the circle, several results are known. Surely, in random projective dynamics there is a synchronization due to the simplest pos- sible case of Furstenberg’s Theorem. In 1984, this result was generalized to the setting of homeomorphisms (with some very mild and natural assumptions of min- imality of the action and the presence of a North-South map) in the work of a physicist V.
    [Show full text]
  • 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
    [Show full text]
  • The Mathematics of Critical Shifts in Ecological Networks with Alternative Stable State Theory, a Potential Framework for Early Warning Indicators
    The Mathematics of Critical Shifts in Ecological Networks with Alternative Stable State Theory, A Potential Framework for Early Warning Indicators by Michael Kupperman A THESIS submitted to Oregon State University Honors College in partial fulfillment of the requirements for the degree of Honors Bachelors of Science in Mathematics (Honors Scholar) Honors Bachelors of Science in Biochemistry & Molecular Biology (Honors Scholar) Presented June 14, 2019 Commencement June 2020 AN ABSTRACT OF THE THESIS OF Michael Kupperman for the degree of Honors Bachelors of Science in Mathematics and Honors Bachelors of Science in Biochemistry & Molecular Biology presented on June 14, 2019. Title: The Mathematics of Critical Shifts in Ecological Networks with Alternative Stable State Theory, A Potential Framework for Early Warning Indicators Abstract approved: David Koslicki A long running problem in mathematical biology is the prediction of extinction events, a specialized case of the larger global stability problem found in differential equations and dynamical systems theory. A central technical question is how to introduce the randomness observed in real ecological systems not accounted for in deterministic models. This work introduces the SP-system as a new mathematical object in which ecological parameters are treated as sequences of random variables that attain values over intervals of random lengths of time. The SP-system characterization of ecological networks leads to two differ- ent novel approaches for the simulation and extinction prediction. The first approach uses a construct new to modeling literature to describe the probability of a parameter transition event into an extinction event. The second approach utilizes Markov chains for both sim- ulation and extinction prediction.
    [Show full text]
  • Engel-M-2018-Phd-Thesis.Pdf
    Imperial College London, Department of Mathematics Local phenomena in random dynamical systems: bifurcations, synchronisation, and quasi-stationary dynamics A thesis presented for the degree of Doctor of Philosophy of Imperial College London by Maximilian Engel Supervisors: Prof. Jeroen S. W. Lamb, Dr. Martin Rasmussen January 2018 Declaration I declare that this thesis is entirely my own work and that where any material could be construed as the work of others, it is fully cited and referenced, and/or with appropriate acknowledgement given. Maximilian Engel 2 Copyright The copyright of this thesis rests with the author and is made available under a Creative Commons Attribution Non-Commercial No Derivatives licence. Researchers are free to copy, distribute or transmit the thesis on the condition that they attribute it, that they do not use it for commercial purposes and that they do not alter, transform or build upon it. For any reuse or redistribution, researchers must make clear to others the licence terms of this work. 3 Acknowledgements I would like to express my gratitude to the following people who have supported me in accomplishing the present work. First of all, I would like to thank my supervisors Jeroen Lamb and Martin Rasmussen for the numerous passionate discussions we have had throughout the past three years, for encouraging me in difficult times and being enthusiastic but still critical in times of apparent breakthroughs. I am extremely grateful for their guidance, support and trust in me as a PhD student. In particular, I have to emphasize their willingness to enable many inspiring research connections and visits to different parts of the world.
    [Show full text]
  • Stability and Random Attractors for a Reaction-Diffusion Equation with Multiplicative Noise
    DISCRETE AND CONTINUOUS Website: http://AIMsciences.org DYNAMICAL SYSTEMS Volume 6, Number 4, October 2000 pp. 875{892 STABILITY AND RANDOM ATTRACTORS FOR A REACTION-DIFFUSION EQUATION WITH MULTIPLICATIVE NOISE Tomas¶ Caraballo & Jose¶ A. Langa Dpto. Ecuaciones Diferenciales y An¶alisisNum¶erico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla, Spain. James C. Robinson Mathematics Institute, University of Warwick Coventry CV4 7AL, U.K. Abstract. We study the asymptotic behaviour of a reaction-di®usion equation, and prove that the addition of multiplicative white noise (in the sense of It^o)stabilizes the stationary solution x ´ 0. We show in addition that this stochastic equation has a ¯nite-dimensional random attractor, and from our results conjecture a possible bifurcation scenario. 1. Introduction. The study of the asymptotic behaviour of evolution equations is one of the most important problems in the ¯eld of di®erential equations, as the vast literature on the subject shows. Perhaps the ¯rst step in this programme is to investigate the stability of the stationary solutions, and this is at present a well-developed branch of the theory of stochastic ordinary and partial di®erential equations (see, among others, Has'minskii [20], Mao [23], [24], Caraballo and Liu [5]). Such a stability analysis is essentially a local study, as we obtain information on the dynamics only around these points. A more complete analysis of the quali- tative properties of the system as a whole requires information relating not only to the stationary points but also, for example, to the stability or instability of the as- sociated invariant manifolds of these points.
    [Show full text]
  • On the Computation of Invariant Measures in Random Dynamical Systems∗
    On the computation of invariant measures in random dynamical systems¤ Peter Imkeller Peter Kloeden Institut f¨urMathematik Fachbereich Mathematik Humboldt-Universit¨atzu Berlin J-W von Goethe-Universit¨at Unter den Linden 6 Postfach 11 19 32 10099 Berlin 60054 Frankfurt am Main Germany Germany May 27, 2003 Abstract Invariant measures of dynamical systems generated e. g. by difference equa- tions can be computed by discretizing the originally continuum state space, and replacing the action of the generator by the transition mechanism of a Markov chain. In fact they are approximated by stationary vectors of these Markov chains. Here we extend this well known approximation result and the underlying algorithm to the setting of random dynamical systems, i.e. dynamical systems on the skew product of a probability space carrying the underlying stationary stochasticity and the state space, a particular non-autonomous framework. The systems are generated by difference equations driven by stationary random pro- cesses modelled on a metric dynamical system. The approximation algorithm involves spatial discretizations and the definition of appropriate random Markov chains with stationary vectors converging to the random invariant measure of the system. 2000 AMS subject classifications: 60 G, 60 H 25, 60 J 10, 65 C 30, 65 Q 05, 37 H 10, 37 M 25. Key words and phrases: dynamical system; random dynamical system; approx- imation; invariant measure; Markov chain; random Markov chain; interval matrix; spatial discretization. ¤This work was partially supported by the DFG research center ‘Mathematics for key technologies’ (FZT 86) in Berlin. 1 1 Introduction Computational methods are a basic tool in the investigation of dynamical systems, both to explore what may happen and to approximate specific dynamical features such as limit cycles or attractors and, more generally, invariant measures; see, e.g., [5, 20, 23].
    [Show full text]
  • An Introduction to Random Dynamical Systems for Climate
    An Introduction to Random Dynamical Systems for Climate Franco Flandoli and Elisa Tonello Septemebr 2019 2 Contents 0.1 Preface . 4 1 Non-Autonomous Dynamical Systems 1 1.1 Introduction . 1 1.2 Non autonomous dynamical systems: definition and examples . 2 1.3 Invariance. Criticalities of the concept for NADS . 5 1.4 Global attractor and omega-limit sets . 7 1.5 A criterium for existence of global attractor . 9 1.6 On the uniqueness of the global attractor . 13 1.7 Invariant measures: attempts of qualifications to reduce non-uniqueness . 14 1.8 Separation between non-autonomous input and autonomous dynamics: co- cycles . 16 1.9 Improvement of the invariance concept for cocycles . 18 1.10 Non-autonomous RDS (naRDS) . 21 2 Generalities on RDS 25 2.1 Random dynamical systems . 25 2.2 Definitions . 25 2.3 RDS from Stochastic Differential Equations . 29 2.3.1 Additive noise . 34 2.3.2 A finite dimensional model . 35 2.3.3 n-point motion . 38 2.4 RDS as products of i.i.d. random mappings . 40 2.5 An example in infinite dimensions: the stochastic Navier-Stokes equations . 40 2.6 Multiplicative ergodic theorem and Lyapunov exponents . 43 3 Random Attractors and Random Invariant Measures 45 3.1 Attractors for random dynamical systems . 45 3.1.1 Attractor for the finite dimensional approximation of SNSE . 48 3.2 Random Measures . 50 3 4 CONTENTS 3.3 Random measures and random dynamical systems . 51 3.3.1 Support of a random measure . 54 3.3.2 Markov measures . 56 3.4 From invariant measures to invariant random measures .
    [Show full text]
  • PDF (9459 Kbyte)
    Institut fiir A:r:igewandte Analysis und Stochastik im Forschungsverbund Berlin e.V. Proceedings of the 1st Workshop on Stochastic Numerics E. Platen 1• 2 (Editor) submitted: 24th November 1992 1 Australian National University 2 Institut fiir Angewandte Analysis Institute of Advanced Studies und Stochastik SRS, GPO Box 4, Canberra Hausvogteiplatz 5 - 7 A.C.T: 2601 D - 0 1086 Berlin Australia Germany Preprint No. 21 Berlin 1992 1991 Mathematics Subject Classification. 60H10. Key words and phrases. stochastic differential equations, numerical solution, statistical methods, simulation, limit theorems, financial mathematics. Herausgegeben vom Institut fiir Angewandte Analysis und Stochastik Hausvogteiplatz 5- 7 D - 0 1086 Berlin Fax: + 49 30 2004975 e-Mail (X.400): c=de;a=dbp;p=iaas-berlin;s=preprint e-Mail (Internet): [email protected] PROCEEDINGS of the 1st Workshop on STOCHASTIC NUMERICS September 7 - 12, 1992 in Gosen (near'Berlin) organized by the Institute of Applied Analysis and Stochastics Berlin together with the Humboldt University Berlin Chairman: E. Platen (Canberra/Berlin) Scientific Program Committee: L. Arnold (Bremen), S. Artemev (Novosibirsk), P. Deuflhard (Berlin), J. Gartner (Berlin), B. Grigelionis (Vilnius), J. Jacod (Paris), P. Kloeden (Geelong), U. Kuchler (Berlin), G.N. Milstein (Ekaterinburg), T. Mitsui (Nagoya), H. Neunzert (Kaiserslautern) P. Protter (Purdue), R. Rebolledo (Santiago), D. Talay (Sophia Antipolis), Local Organizing Committee: Mrs. M. Teuchert, IAAS, Hausvogteiplatz 5-7, D-0-1086 Berlin, Fax: (030) 2004975, Tel: (030) 20377 594 Forward One aim of the pt Workshop on Stochastic Numerics was to bring together for the first time leading experts on this field from the eastern countries with those from the west.
    [Show full text]
  • Random Periodic Solutions and Ergodicity for Stochastic Differential
    Random periodic solutions and ergodicity for stochastic differential equations Kenneth Uda1 and Huaizhong Zhao2 1School of Mathematics, University of Edinburgh, UK 2Department of Mathematical Sciences, Loughborough University, UK Abstract In this paper, we establish some sufficient conditions for the existence of stable random periodic solutions of stochastic differential equations on Rd and ergodicity in the random periodic regime. The techniques involve the existence of Lyapunov type function, using two- point generator of the stochastic flow map, strong Feller argument and weak convergence. Keywords: Strong Feller property; periodic measures; PS-ergodicity; random periodic solu- tions, two-point generator. 1 Introduction Random dynamical system (RDS) is a dynamical system with some randomness, its idea was discussed in 1945 by Ulam and von Nuemann [34] and few years later by Kakutani [22] and continued in the 1970s in the framework of ergodic theory. Series of works by Elworthy, Meyer, Baxendale, Bismut, Ikeda, Kunita, Watanabe and others [7, 8, 10, 14, 21, 23, 24] showing that stochastic differential equations (SDEs) induce stochastic flows, gave a substantial push to the subject. Towards late 1980s, it became clear that the techniques from dynamical systems and probability theory could produce a theory of RDS. It was extensively developed by Arnold [3] and his ”Bremen group” based on two-parameter stochastic flows generated by stochastic differential equations due to Kunita [23, 24] and others. To investigate the long time behaviour of RDS is of great interests both in applications and theory. There are two main issues that motivate the behaviour of a mathematical model in arXiv:1811.05503v1 [math.PR] 13 Nov 2018 the long run with theoretical and practical consequences.
    [Show full text]
  • SEMI-KOLMOGOROV MODELS for PREDATION with INDIRECT EFFECTS in RANDOM ENVIRONMENTS Tomás Caraballo & Renato Colucci Xiaoying
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by idUS. Depósito de Investigación Universidad de Sevilla DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS SERIES B Volume 00, Number 0, Xxxx XXXX pp. 000{000 SEMI-KOLMOGOROV MODELS FOR PREDATION WITH INDIRECT EFFECTS IN RANDOM ENVIRONMENTS Tomas´ Caraballo & Renato Colucci1 Departamento de Ecuaciones Diferenciales y An´alisisNum´erico Universidad de Sevilla Apdo. de Correos 1160, 41080 Sevilla, Spain Xiaoying Han Department of Mathematics and Statistics Auburn University Auburn, AL 36849, USA (Communicated by the associate editor name) Abstract. In this work we study semi-Kolmogorov models for predation with both the carrying capacities and the indirect effects varying with respect to randomly fluctuating environments. In particular, we consider one random semi-Kolmogorov system involving random and essentially bounded param- eters, and one stochastic semi-Kolmogorov system involving white noise and stochastic parameters defined upon a continuous-time Markov chain. For both systems we investigate the existence and uniqueness of solutions, as well as positiveness and boundedness of solutions. For the random semi-Kolmogorov system we also obtain sufficient conditions for the existence of a global random attractor. 1. Introduction. Kolmogorov's predator-prey model refers to the general model describing dynamics of interacting populations: dx = xf(x; y); dt dy = yg(x; y); dt where f and g denote the respective per capita growth rates of the two species satisfying @yf(x; y) < 0 and @xg(x; y) > 0, respectively. One simple example of Kolmogorov-type system is the following plakton food web model dZ = Z(−ν + c P ); dt z 3 dP = P (c I − c Z − ν P ); dt 1 0 2 p 2010 Mathematics Subject Classification.
    [Show full text]