Stochastic ordinary and stochastic partial differential equations : transition from microscopic to macroscopic equations / Peter Kotelenez.

Format:

Book

Author/Creator:

Kotelenez, P. (Peter), 1943-

Contributor:

Class of 1924 Book Fund.

Series:

SpringerLink.

Stochastic modelling and applied probability ; 58.

Stochastic modelling and applied probability ; 58

Language:

English

Subjects (All):

Stochastic differential equations.

Stochastic partial differential equations.

Physical Description:

x, 458 pages ; 25 cm.

Place of Publication:

New York : Springer Science+Business Media, [2008]

Summary:

This book provides the first rigorous derivation of mesoscopic and macroscopic equations from a deterministic system of microscopic equations. The microscopic equations are cast in the form of a deterministic (Newtonian) system of coupled non-linear oscillators for N large particles and infinitely many small particles. The mesoscopic equations are stochastic ordinary differential equations (SODEs) and stochastic partial differential equations (SPDEs), and the macroscopic limit is described by a parabolic partial differential equation.

A detailed analysis of the SODEs and (quasi-linear) SPDEs is presented. Semi-linear (parabolic) SPDEs are represented as first order stochastic transport equations driven by Stratonovich differentials. The time evolution of correlated Brownian motions is shown to be consistent with the depletion phenomena experimentally observed in colloids. A covariance analysis of the random processes and random fields as well as a review section of various approaches to SPDEs are also provided. Scientists and graduate students in probability theory (stochastic analysis), mathematical/theoretical physics, and mathematical biology will find this book useful.

Contents:

Part I From Microscopic Dynamics to Mesoscopic Kinematics

1 Heuristics: Microscopic Model and Space-Time Scales 9

2 Deterministic Dynamics in a Lattice Model and a Mesoscopic (Stochastic) Limit 15

3 Proof of the Mesoscopic Limit Theorem 31

Part II Mesoscopic A: Stochastic Ordinary Differential Equations

4 Stochastic Ordinary Differential Equations: Existence, Uniqueness, and Flows Properties 59

4.2 The Governing Stochastic Ordinary Differential Equations 64

4.3 Equivalence in Distribution and Flow Properties for SODEs 73

5 Qualitative Behavior of Correlated Brownian Motions 85

5.1 Uncorrelated and Correlated Brownian Motions 85

5.2 Shift and Rotational Invariance of w(dq, dt) 92

5.3 Separation and Magnitude of the Separation of Two Correlated Brownian Motions with Shift-Invariant and Frame-Indifferent Integral Kernels 94

5.4 Asymptotics of Two Correlated Brownian Motions with Shift-Invariant and Frame-Indifferent Integral Kernels 105

5.5 Decomposition of a Diffusion into the Flux and a Symmetric Diffusion 110

5.6 Local Behavior of Two Correlated Brownian Motions with Shift-Invariant and Frame-Indifferent Integral Kernels 116

5.7 Examples and Additional Remarks 121

5.8 Asymptotics of Two Correlated Brownian Motions with Shift-Invariant Integral Kernels 128

6 Proof of the Flow Property 133

6.1 Proof of Statement 3 of Theorem 4.5 133

6.2 Smoothness of the Flow 138

7 Comments on SODEs: A Comparison with Other Approaches 151

7.1 Preliminaries and a Comparison with Kunita's Model 151

7.2 Examples of Correlation Functions 156

Part III Mesoscopic B: Stochastic Partial Differential Equations

8 Stochastic Partial Differential Equations: Finite Mass and Extensions 163

8.2 A Priori Estimates 171

8.3 Noncoercive SPDEs 174

8.4 Coercive and Noncoercive SPDEs 189

8.5 General SPDEs 197

8.6 Semilinear Stochastic Partial Differential Equations in Stratonovich Form 198

9 Stochastic Partial Differential Equations: Infinite Mass 203

9.1 Noncoercive Quasilinear SPDEs for Infinite Mass Evolution 203

9.2 Noncoercive Semilinear SPDEs for Infinite Mass Evolution in Stratonovich Form 219

10 Stochastic Partial Differential Equations: Homogeneous and Isotropic Solutions 221

11 Proof of Smoothness, Integrability, and Ito's Formula 229

11.1 Basic Estimates and State Spaces 229

11.2 Proof of Smoothness of (8.25) and (8.73) 246

11.3 Proof of the Ito formula (8.42) 269

12 Proof of Uniqueness 273

13 Comments on Other Approaches to SPDEs 291

13.1 Classification 291

13.1.1 Linear SPDEs 294

13.1.2 Bilinear SPDEs 297

13.1.3 Semilinear SPDEs 299

13.1.4 Quasilinear SPDEs 301

13.1.5 Nonlinear SPDEs 301

13.1.6 Stochastic Wave Equations 302

13.2 Models 302

13.2.1 Nonlinear Filtering 302

13.2.2 SPDEs for Mass Distributions 303

13.2.3 Fluctuation Limits for Particles 304

13.2.4 SPDEs in Genetics 305

13.2.5 SPDEs in Neuroscience 305

13.2.6 SPDEs in Euclidean Field Theory 306

13.2.7 SPDEs in Fluid Mechanics 306

13.2.8 SPDEs in Surface Physics/Chemistry 308

13.2.9 SPDEs for Strings 308

13.3 Books on SPDEs 308

Part IV Macroscopic: Deterministic Partial Differential Equations

14 Partial Differential Equations as a Macroscopic Limit 313

14.1 Limiting Equations and Hypotheses 313

14.2 The Macroscopic Limit for d [greater than or equal] 2 316

14.4 A Remark on d = 1 330

14.5 Convergence of Stochastic Transport Equations to Macroscopic Parabolic Equations 331

15.1 Analysis 335

15.1.1 Metric Spaces: Extension by Continuity, Contraction Mappings, and Uniform Boundedness 335

15.1.2 Some Classical Inequalities 336

15.1.3 The Schwarz Space 340

15.1.4 Metrics on Spaces of Measures 348

15.1.5 Riemann Stieltjes Integrals 357

15.1.6 The Skorokhod Space D([0, [infinity]); B) 359

15.2 Stochastics 362

15.2.1 Relative Compactness and Weak Convergence 362

15.2.2 Regular and Cylindrical Hilbert Space-Valued Brownian Motions 366

15.2.3 Martingales, Quadratic Variation, and Inequalities 371

15.2.4 Random Covariance and Space-time Correlations for Correlated Brownian Motions 380

15.2.5 Stochastic Ito Integrals 387

15.2.6 Stochastic Stratonovich Integrals 403

15.2.7 Markov-Diffusion Processes 411

15.2.8 Measure-Valued Flows: Proof of Proposition 4.3 418

15.3 The Fractional Step Method 422

15.4 Mechanics: Frame-Indifference 424.

Notes:

Includes bibliographical references (pages 445-458) and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Class of 1924 Book Fund.

ISBN:

9780387743165

0387743162

OCLC:

271718592

Access Restriction:

Electronic version is restricted to subscribing institutions.