INTRODUCTION

Purpose. The purpose of this report is to provide a multi-type extension, allowing weak interaction in both space and type, of the well-known results of Dawson and Gärtner (1987) [DG] regarding large deviations from the McKean-Vlasov limit for weakly interacting diffusions. This is achieved byintegrating more systematically the previous work Djehiche and Kaj (1994) [DK], in which a large deviation result is derived for a class of measure-valued jump processes, with the setting of the Dawson-Gärtner large deviation principle. Necessarilly, some aspects of such an extension will be mere notational rather than substantial. We will try to focus on those parts that are less evident and to point out some techniques from [DK] which can be used as an alternative to those of [DG].

INTRODUCTION

The definition of several types of stochastic integrals for anticipating integrands put the basis for the development, in recent years, of an anticipating stochastic calculus. It is natural to consider, as an application, some problems that can be stated formally as stochastic differential equations, but that cannot have a sense within the theory of non-anticipating stochastic integrals. For example, this is the case if we impose to an s.d.e. an initial condition which is not independent of the driving process, or if we prescribe boundary conditions for the solution.

In this paper, we will try to survey the work already done concerning s.d.e. with boundary conditions, and to explain in some detail a method based in transformations and change of measure in Wiener space. An alternative approach is sketched in the last Section.

Transformations on Wiener space provide a natural method, among others, to prove existence and uniqueness results for nonlinear equations. At the same time, a Girsanov type theorem for not necessarily adapted transformations allows to study properties of the laws of the solutions from properties of the solution to an associated linear equation. A natural first question about these laws is to decide if they satisfy some kind of Markov (or conditional independence) property.

In Section 2, we introduce stochastic differential equations with boundary conditions, the particular instances that have been studied, and the kind of results obtained concerning conditional independence properties of the solutions. The short Section 3 outlines the idea of the method of transformations and change of measure. In Section 4, we recall the necessary elements of Wiener space analysis in order to enounce the Girsanov type theorem we want to apply.

In March 1994 a two week workshop on Stochastic Partial Differential Equations was held at the University of Edinburgh under the auspices of the International Centre for Mathematical Sciences. The meeting attracted an international audience including researchers from all aspects of the field aswell as participants with backgrounds in biology, physics, engineering and finance. This volume is a collection of articles contributed by participants at that meeting. Some report work presented at the meeting and some are surveys, but all are accessible to a wide audience. Although the subject matter reflects the diversity of the field, all the contributions are closely related to the central theme of stochastic partial differential equations.

There are many people who contributed to the meeting. Only a few of them are mentioned here, but I am deeply indebted to them all. The original proposal for the workshop would never have been completed without the invaluable input of Terry Lyons. I should like to thank him and the other members of the Scientific Committee, Peter Donnelly and Steve Evans. A great deal of the administration was handled by the International Centre for Mathematical Sciences and I should especially like to thank Lucy Young, not only for her hard work but also for her patience with my inefficiency. Thanks also to Sandra Bashford for her sustained cheerfulness and limitless supplies of coffee. The Science and Engineering Research Council provided funding under grant number GR/H94092.

The production of the present volume could not have taken place without the help of Roger Astley at CUP and of course the contributors themselves. The referees must remain anonymous, but they have my heartfelt thanks.

Introduction

It is well-known that the leading term in Varadhan's and Wentzell-Freidlin's large deviation theories (Varadhan (1967), Wentzell and Freidlin (1970)) and Maslov's quasi-classical asymptotics of quantum mechanics (Maslov (1972)) involves the solution of a variational problem which gives a Lipschitz continuous solution of the Hamilton Jacobi equation if it exists (Fleming (1969, 1986)). It was proved by Truman (1977) and Elworthy and Truman (1981, 1982) that before the caustic time the Hamilton Jacobi function which is C1,2 gives the exact solutions of the diffusion equations geared to small time asymptotics. The main tools in the theory are classical mechanics and the Maruyama-Girsanov-Cameron-Martin formula. The philosophy of the theory is to choose a suitable drift for a Brownian motion on the configuration space manifold and to employ the MGCM theorem to simplify the Feynman-Kac representation of the solutions for the heat equations. An extended version of this theory to degenerate diffusion equations was obtained in Watling (1992). The Brownian Riemannian bridge process was obtained by this means in Elworthy and Truman (1982). The extension to more general Riemannian manifolds was obtained in Elworthy (1988), Ndumu (1986,1991). The same methods have been applied to travelling waves for nonlinear reaction diffusion equations in Elworthy, Truman and Zhao (1994).

Introduction

For the last thirty years, there has been interest in numerical simulation of solutions of stochastic differential equations (SDE's). More recently, along with the growing general interest in stochastic partial differential equations (SPDE's), there has been a desire to produce numerical solutions to SPDE's. There are very few SDE's for which analytical solutions can be obtained and this is naturally also true for SPDE's. One hope is that using numerical methods to generate solutions to such equations will lead to better understanding of the equations. Theory and numerical work often go hand in hand: while theory can provide useful numerical methods, pictures obtained numerically can lead to conjectures that the analyst may then succeed in proving. Another hope is that numerical methods will be of use to people outside mathematics. There are already SPDE's used in practical applications, such as in finance.

Our aim is to show how simple techniques, already in use for numerical solution of SDE's, can be extended to provide numerical solutions to SPDE's. We will concentrate on strong solutions, that is solutions obtained for one particular realisation of Brownian motion or of a Brownian sheet. We describe how Brownian paths and sheets can be generated and stored in tree-like structures, allowing for subsequent refinement and therefore comparison of solutions using different meshes.

Motivation

Suppose that we have a dynamical system whose (non-linear) equations depend on a parameter λ, and that there is a bifurcation to periodicity or chaos as we increase λ up to a critical value λ0. It is then interesting to add noise to the system, and to ask whether this induces the system to bifurcate earlier, at some λ < λ0, or whether it delays the onset until λ > λ0. It might on the other hand change the nature of the criticality, of destroy it altogether.

Here we report on a study (made with L. Rondoni) of the chemical system known as the Brussellator modified to follow the activity-led law of mass action. For certain values of the rate constants the system exhibits a limit-cycle, which is converted into a global attractor when the system is coupled to thermal noise in a particular way.

Introduction

In 1990, Pardoux & Peng initiated the study of backward stochastic differential equations motivated by optimal stochastic control (see Bensoussan, Bismut, Haussmann and Kushner). It is even more important that Pardoux & Peng, Peng recently gave the probabilistic representation for the given solution of a quasilinear parabolic partial differential equation in term of the solution of the corresponding backward stochastic differential equation. In other words, they obtained a generalization of the well-known Feynman-Kac formula (cf. Freidlin or Gikhman & Skorokhod). In view of the powerfulness of the Feynman-Kac formula in the study of partial differential equations e.g. K.P.P. equation (cf. Freidlin), one may expect that the Pardoux-Peng generalized formula will play an important role in the study of quasilinear parabolic partial differential equations. Hence from both viewpoints of the optimal stochastic control and partial differential equations, we see clearly how worthy to study the backward stochastic differential equations in more detail.

Introduction.

Multilevel branching particle systems were introduced recently by Dawson and Hochberg [DH] as models for a class of hierarchically structured populations. In such a model individual particles migrate and branch, and in addition collections of particles are subject to independent branching mechanisms at the different levels of the hierarchical organization. The analysis of these models is complicated by the fact that the higher level branching leads to the absence of independence in the particle dynamics. Several aspects of multilevel branching systems have been investigated by Dawson, Hochberg and Wu [DHW], Dawson, Hochberg and Vinogradov [DHV1, DHV2], Dawson and Wu [DW], Etheridge [E], Gorostiza, Hochberg and Wakolbinger [GHW], Hochberg [H], and Wu [W1,W2, W3,W4]. Some of these references, specially [DHV2], include examples of areas of application where hierarchical branching structures are present.

Wu [W1,W2] studied the long-time behaviour of a critical two–level superprocess constructed from a system of branching Brownian motions in Rd. This process is obtained in a similar way as the usual (one-level) superprocesses, i.e., as a limit under a high-density/short-life/small-mass rescaling, which in the two-level case is applied simultaneously at the two levels. He proved that this critical two-level superprocess goes to extinction, as time tends to infinity, in dimensions d ≤ 4, and from his results in [Wl] it follows that it does not become extinct in dimensions d > 4. Gorostiza, Hochberg and Wakolbinger [GHW] proved the stronger result that persistence holds in dimensions d > 4.

Introduction

In this paper we give a detailed exposition of the way in which the recent work of S. Fajardo and H. J. Keisler can be used to establish existence of solutions to stochastic Navier-Stokes equations. Fajardo & Keisler develop general methods for proving existence theorems in analysis, with the aim of embracing the many particular existence theorems that can be proved rather easily using nonstandard analysis. The machinery developed centres round the notion of a neocompact set – which is a weakening of the notion of a compact set of random variables with values in a metric space M - and the notion of a rich adapted probability space, in which any countable chain of nonempty neocompact sets has a nonempty intersection.

In the papers Capiński & Cutland used nonstandard methods to greatly simplify some known existence proofs for the deterministic Navier- Stokes equations and (using similar methods) solved a longstanding problem concerning existence of solutions to general stochastic Navier-Stokes equations. The aim here is to show how the main results of these papers can be obtained using the neocompactness methods developed in. In addition, these methods yield additional information concerning the nature of the set of solutions and existence of optimal solutions.

Introduction

Hierarchically structured multilevel branching particle systems were first introduced by Dawson, Hochberg and Wu and have since been the object of much investigation. Such a two-level system consists of individuals or “firstlevel particles” undergoing some spatial motion and branching, which are, on the second level, grouped into clusters. Each of these clusters constitutes a “superparticle” which is simultaneously affected by some other, independent branching mechanism. The idea is to describe not only reproduction or death of individuals, but also replication or catastrophic elimination of whole families or clans. See for several examples arising in various fields of applications.

In this paper, we consider a system of Brownian particles in IRd with particularly simple reproduction mechanisms – namely, critical binary branching at each level – and we show that in dimensions d ≤ 4, these systems do not persist, in the sense that no nontrivial equilibria with finite first moments exist. (In light of a recent result of Bramson, Cox and Greven on nonexistence of equilibria of branching Brownian particle systems in dimensions one and two, the restriction “with finite first moments” might be superfluous here; the determination of this, however, is beyond the scope of the techniques used in this paper.)