A great deal of research into the theory of random processes is concerned with the problem of constructing a process that has certain properties of regularity of the trajectories and has the same finite-dimensional probability distribution as a given stochastic process xt. It is a complicated theory and one that is difficult to apply to those properties that we most need for the study of Markov processes (the strong Markov property, quasi-left-continuity, and the like.)

The problem can be usefully reformulated. In an actual experiment we do not observe the state xt at a fixed instant t, but rather events that occupy certain time intervals. This is the motivation behind the Gel'fand-Itô theory of generalized random processes. Kolmogorov, in 1972, proposed an even more general concept of a stochastic process as a system of σ-algebras ℱ(I) labelled by time intervals I. Developing this approach, we introduce the concept of a Markov representation xt of the stochastic system ℱ (I) and prove the existence of regular representations. We construct two dual regular representations (the right and the left), which we then combine into a single Markov process by two methods, the “vertical” and the “horizontal” method. We arrive at a general duality theory, which provides a natural framework for the fundamental results on entrance and exit spaces, excessive measures and functions, additive functionals, and others. The initial steps in the construction of this theory were taken in [6]. The note [5] deals with applications to additive functionals (detailed proofs are in preparation). We consider random processes defined in measurable spaces without any topology: the introduction of a reasonable topology allows of a certain arbitrariness.

McSHANE'S INTEGRAL

We shall generally be concerned with the following objects: A subset T ⊂ R with an interval [a, b] ⊂ T;

A probabili.ty space (Ω, F, μ) with a family, or filtration, σ of a-algebras {Fτ:τ ∈ T} such that Fσ ⊂ Fτ ⊂ F whenever σ < τ;

Banach spaces G1, …, Gq and a Hilbert space H, all separable;

Maps zρ:[a, b] → £o(Ω, F;Gp) ρ = 1, …, q

and a map B:T x Ω → ╙ (Gl, …, Gq;H).

To save space, and brackets, we will sometimes use the notation for zρ (t) and Bτ, or B(τ), for B(τ, -), with the corresponding convention for other processes.

We shall usually require our processes to be adapted to the filtration: for zρ and similar processes this means that each zρ(t) is Ft-measurable, and for B it means that B(τ) is Fτ-measurable for each τ in T. The σ-algebra Fτ can be thought of as consisting of those events which are known about at time τ, or the ‘past’ at time τ. The term non-anticipating is sometimes used instead of ‘adapted’, but it also has a more technical definition; a process adapted to {Fτ:τ ∈ T} is also called an F*-process.

From time to time our maps will be subjected to some of the following conditions (for positive integers rand p):

Condition A(r)

Each zρ is adapted and there exist constants K > 0, δ > 0 such that if a < s < t < b and t-s < δ then, almost everywhere

ABOUT THESE NOTES. ACKNOWLEDGEMENTS

These are a much expanded and revised version of notes of seminars given at the University of Warwick during the year 1973/74. These were given and written up jointly with J. Eells. The audience consisted of non-probabilists with a reasonably good background in manifold theory. The aim was to go through, from the beginning, the basic properties of stochastic differential equations, extend the theory to manifolds and in particular use this in the proof that by polygonal approximation the Cartan development or ‘rolling’ has a ‘stochastic extension’ which gives a geometric construction of Brownian motion on Riemannian manifolds (see §§11 A, B Chapter VII). Another aim was to describe, from this point of view, the stochastic parallel transport discussed in Itô's Stockholm article (1963).

The previous year we had been on leave separately: myself at the University of California at Santa Cruz, and Aarhus University, beginning to learn some Stratonovich calculus after earlier suggestions from R. Curtain; and Eells at I.A.S. Princeton, and I.H.E.S. Bures-sur-Yvette, where he collaborated with P. Malliavin (1972/3) in an examination of diffusions on vector bundles, horizontal lifts, etc. using a different approach. We met up at I.H.E.S., and all these institutions deserve thanks for their hospitality.

We will define stochastic differential equations on manifolds in a way which will make them look as simi.lar to ordinary differential equations as possible. It may be helpful to keep in mind the fact that ordinary differential equations will be a special case of our stochastic differential equations.

The notation in this chapter i.s the same as before, but now we assume that we have a filtration {Ft:a < t < b} with b allowed to be infinite. In particular the σ-algebras Ft each contain all sets of measure zero in F. Throughout G and E will denote separable Banach spaces and H a separable Hilbert space, while

z:[a, b] → £o(Ω;E)

is sample continuous and satisfies condition A(q), q > 4, on each compact subset of [a, b). Some of the terminology from manifold theory and differential geometry is explained in Appendices A and B.

STOCHASTIC DYNAMICAL SYSTH1S

(A) Let M be a separable metrizable C3 manifold modelled on the Hilbert space H. Let Ḙ denote the trivial E-bundle, E x M → M, over M, and let TM be the tangent bundle of M. A stochastic dynamical system, (S.D.S.), (X, z) on M consists of a section X of ╙ (Ḙ, TM) for some E, together with a stochastic process z with values in E. In particular X(m) ∈ ╙ (E, TmM) for each m ∈ M. We will write X ∈ L (Ḙ, TM).

We shall only consider stochastic dynamical systems with X of class C1 and having locally Lipschitz first derivatives (in coordinate charts) e.g. X of class C2, and where z satisfies our standing hypotheses.