Analysis of the exit problem for randomly perturbed dynamical systems in applications

In the preface of his book entitled 'Theory and applications of stochastic differential equations', Z. Schuss (1980) noticed a gap between the theory of stochastic differential equations and its applications. In addition to the work of Schuss and many others in the field, the present work aims at narrowing this gap.This thesis deals with randomly perturbed dynamical systems. Such systems frequently arise in the modelling of phenomena in biology, mechanics, chemistry, and physics. In some cases random perturbations form a minor aspect of the problem under study. Then a deterministic description can be used. In the present work the behaviour of the dynamical systems depends essentially on the random perturbations. We encounter systems with so-called 'diffusion across the flow' (Chapters 1,2) and systems with 'diffusion against the flow' (Chapters 1,3-5). The stability of equilibria of these systems (and thus, the lifetime, reliability of these systems) is affected by random perturbations.In the study of so-called 'exit problems' we consider a domain in the state space of the dynamical system and try to compute statistical quantities related to escape from this domain, such as the probability density function of the exit-time, the probability density function of exit points on the boundary of the domain (or, less ambitiously, the first few statistical moments of these densities: mean, variance, etc.). The expectation value of the exit-time can be used to express the stochastic stability of the system.We speak of randomly perturbed dynamical systems, so we assume that the stochastic fluctuations are small. This is often a realistic assumption. To derive expressions for the statistical quantities mentioned above, we employ asymptotics where the small parameter is related to the intensity of the random perturbations. The asymptotic method used in Chapter 2 is well-established. The asymptotics in Chapters 3-6 are of a formal character. The asymptotic analysis is performed to the lowest order necessary to incorporate the essential effects. In view of the complexity of this simplest approach, we did not carry out higher order calculations.The first chapter forms an introduction to some important topics in the theory of exit problems. We discuss the relevant (initial-) boundary-value problems, the classification of boundaries of domains of stochastic dynamical systems, and we give elementary examples of systems with 'diffusion across the flow' and 'diffusion against the flow' and their asymptotic solution. This chapter facilitates access to literature on exit problems and to the remaining chapters of this thesis. A more detailed treatment of the topics touched upon in this chapter is found in the cited literature.Chapter 2 is concerned with the dynamics of a loaded stiff rod. The load consists of a deterministic part and a small stochastic part. An accumulation of stochastic load fluctuations may drive the energy of the rod across some critical level. The expectation value of the time to reach this critical energy level is a measure for the reliability of the system that contains the rod. According to the directions in which the loads act, various cases are distinguished. We derive expressions for the expectation value of the exit-time and (for some of the cases) of a number of other statistical quantities, as the exit-time density, its moments and cumulants and the probability density function of the square root of the energy (as a function of time). We use an asymptotic method known as the averaging technique. As a matter of fact, the model is a randomly loaded slightly damped oscillator. Since many practical systems near equilibrium behave essentially like a slightly damped oscillator, the. results obtained may be expected to have a wide range of application.In Chapter 3 we study the exit problem for a stochastic dynamical system of interacting biological populations. Exit from the domain (the positive orthant) corresponds with extinction of a population. We start with a birth and death process, having a discrete state space, and subsequently formulate an 'approximate' Fokker- Planck (or forward Kolmogorov) equation in a continuous state space. It is assumed that the deterministic system associated with the stochastic dynamical system has a point attractor in the positive orthant. The biological system will remain for some (probably long) time in a neighbourhood of the attracting point, but after a (rare) succession of random fluctuations, one of the populations will get extinct. Determining the expected time of exit (of whichever of the populations), and of which population will probably get extinct first, requires the numerical solution of a system of so-called 'ray equations' (obtained from the Fokker-Planck equation by the WKB-method). In literature these differential equations are provided with initial conditions, which entails difficulties in the numerical construction of contours in the state space on which the eikonal function attains a constant value (confidence contours). We define boundary conditions instead of initial conditions and thereby resolve these difficulties. The ideas are illustrated by a two-dimensional generalized Lotka-Volterra model. This model allows a nice demonstration of the concepts of deterministic stability and stochastic stability. Numerically constructed confidence contours are shown for predator-prey, mutualism and competition variants of the model. We carry out numerical simulations of birth- death processes to check the results.A discussion of various ways of numerical solution of the system of ray equations is found in Chapter 4. In particular we explain the boundary-value method referred to above. Moreover we give some details on the numerical construction of rays and confidence contours. At the end we present an example with intersecting rays. This phenomenon is investigated analytically in Chapter 6.In Chapter 5 we are concerned again with a stochastic version of the two- dimensional generalized Lotka-Volterra model. The approach differs from that in Chapter 3 in that now we pay attention to what happens near the boundaries of the domain (the positive coordinate axes). The main difficulty is caused by the fact that the normal components of both the drift and the diffusion coefficients vanish near the boundaries, as linear functions of the distance to the boundaries. To obtain expressions for the statistical quantities of interest, we generalize a method of other authors in the study of a similar one-dimensional problem. The asymptotic expressions contain some unknown constants, that can be obtained numerically. Explicit calculations are carried out for a predator-prey system as an example,Applying the WKB-method to the forward Kolmogorov equation, we obtain the ray equations. In the solution of the ray equations one sometimes observes intersecting rays forming caustic surfaces. This phenomenon is studied in Chapter 6. Near locations of intersecting rays, the WKB-approximation does not hold. We derive a uniform asymptotic expansion in terms of new canonical integrals whose validity extends over regions containing caustics. We start with the simple case of a cusp arising in a diffusion problem for which explicit results can be obtained. Subsequently, we generalize it to a formal approach to singularities arising in the forward Kolmogorov equation.The text of each of the chapters has appeared as a report or a publication in a scientific journal.