The note concerns the structure of the Brownian excursion filtration (C ~, xeR). This filtration, indexed by the space variable, has infinite martingale dimension. We show how it can be characterised by the martingale properties of the reflecting Brownian local time.
The problem examined here has its origins in the Markovian properties of diffusion local time discovered independently by Ray [8] and Knight [-3]. Essentially they proved that if the local time is observed at a diffusion killing time then it is itself a diffusion in the other (space) variable. An account of these ideas can be found in [4] or [9].
In his attempts to understand this phenomenon Williams [11] discovered the existence of an entire filtration indexed by the space variable. This called the excursion filtration and is written here as gx. The point is that the various local time processes are not just Markovian in their own filtrations, but are d ~ Markovian as well. Now the usual way of constructing a filtration is to start with the process which generates it. Walsh's basic structural question, posed implicitly in [9], is whether there exists some well-behaved process which generates gx. We claim to have found such a process.
To explain more fully it is helpful to recall briefly the basic terminology and notation of [6], to which we refer the reader for further details. Suppose that B~ is a Brownian motion which is started at zero. We denote its filtration by ~, assumed right continuous and complete. Then write t A(x, t) = j I~B~ < ~ ds 0 and denote by z(x,.) the right continuous inverse of the increasing process A(x,.). Because A(x,t) is ~t adapted it follows that z(x,t) is a ~ stopping time. Define B(x, t)=B(z(x, t)) noting that this process satisfies the inequality