Ebook: Diffusion Processes and Related Problems in Analysis, Volume II: Stochastic Flows

During the weekend of March 16-18, 1990 the University of North Carolina at Charlotte hosted a conference on the subject of stochastic flows, as part of a Special Activity Month in the Department of Mathematics. This conference was supported jointly by a National Science Foundation grant and by the University of North Carolina at Charlotte. Originally conceived as a regional conference for researchers in the Southeastern United States, the conference eventually drew participation from both coasts of the U. S. and from abroad. This broad-based par­ ticipation reflects a growing interest in the viewpoint of stochastic flows, particularly in probability theory and more generally in mathematics as a whole. While the theory of deterministic flows can be considered classical, the stochastic counterpart has only been developed in the past decade, through the efforts of Harris, Kunita, Elworthy, Baxendale and others. Much of this work was done in close connection with the theory of diffusion processes, where dynamical systems implicitly enter probability theory by means of stochastic differential equations. In this regard, the Charlotte conference served as a natural outgrowth of the Conference on Diffusion Processes, held at Northwestern University, Evanston Illinois in October 1989, the proceedings of which has now been published as Volume I of the current series. Due to this natural flow of ideas, and with the assistance and support of the Editorial Board, it was decided to organize the present two-volume effort.

---

While the theory of deterministic flows can be considered classical, the stochastic counterpart has only been developed in the past decade. Much of this work was done in close connection with the theory of diffusion processes, where dynamical systems implicitly enter probability theory by means of stochastic differential equations. The aim of this book is to give a broadly based presentation of Stochastic flows to the scientific public, both to serve as a guide to the field and to illustrate its rich structure, unifying capability and broad domain of applications. The book contains 15 carefully reviewed papers from a conference hosted by the University of North Carolina at Charlotte during the weekend of March 16-18, 1990. The articles discuss stochastic flows generated by stochastic differential systems and iterated function systems. They explore in particular, the relation between flows and the intrinsic geometry of the Riemmanian manifold, stochastic versus control flows, heredi- tary systems, Lyapunov spectra, invariant measures, stability and statistical equilibrium. Many previously unpublished research results, often embedded in illuminating expositions, reflect the various directions into which the field is evolving. CONTENTS Preface Conference Participants Part I : Diffusion Processes and General Stochastic Flows on Manifolds Stability and equilibrium properties of stochastic flows of diffeomorphisms P. BAXENDALE . . . . . . . . . . . . . . . . . . . . . . 3 Stochastic flows on Riemannian manifolds K.D. ELWORTHY . . . . . . . . . . . . . . . . . . . . 37 Part II: Special Flows and Multipoint Motions Isotropic stochastic flows R.W.R. DARLING . . . . . . 75 The existence of isometric stochastic flows for Riemannian Brownian motions M.LIAO . . . . . . . . . . . . . . . . . . . . 95 Time-reversal of solutions of equations driven by Levy processes P. SUNDAR . . , . . . . . . , . . . . , . . . . . . . . 111 Birth and death on a flow E. QINLAR and J .S. KAO . . . . . . . , , . . . . . . . 121 Part III: Infinite Dimensional Systems Lyapunov exponents and stochastic flows of linear and affine hereditary systems S.E.A. MOHAMMED . . . . . . . . . . . . . . .. .. 141 Convergence in distribution of Markov processes generated by i.i.d. random matrices A. MUKHERJEA .................... 171 Part IV: Invariant Measures in Real and White Noise-Driven Systems Remarks on ergodic theory of stochastic flows and control flows F. COLONIUS and W. KLIEMANN .......... . 203 Stochastic bifurcation: instructive examples in dimension one L. ARNOLD and P. BOXLER . . . . . . . . . . . . . . .241 Lyapunov exponent and rotation number of the linear harmonic oscillator M. PINSKY . . . . . . . . . . . . . . . . .. ... 257 The growth of energy of a free particle of small mass with multiplicative real noise V. WIHSTUTZ ..................... 269 Part V: Iterated Function Systems Iterated function systems and multiplicative ergodic theory L. ARNOLD and H. CRAUEL . . . . . . . . . . . .283 Weak convergence and generalized stability for solutions to random dynamical systems J. ELTON and J. EZZINE .307 Random affine iterated function systems: mixing and encoding M. BERGER ...................... 315