Mise en Scène: A Historical Backdrop and Heuristics

The Wiener process – a stochastic process with independent Gaussian increments – was originally conceived as a probabilistic model for Brownian movement, and has been, ever since, among the most influential mathematical constructs in the twentieth century. For our purposes, we used it in Chapter VI §2 to produce a canonical example of an F2-measure that cannot be extended to an F1-measure. In this chapter and the next, we examine and develop ideas underlying this example.

We begin here with some of the history and heuristics behind Brownian motion and the Wiener process. (In this book, ‘Brownian motion’ or ‘Brownian movement’ will refer always to a physical phenomenon, and the ‘Wiener process’ to Norbert Wiener's mathematical model of it.)

From Brown to Wiener

In the sciences at large, Brownian movement generically refers to haphazard, erratic, difficult-to-predict trajectories of particles. Such movements exhibited by tiny particles suspended in liquid first became known to naturalists in the seventeenth century, soon after the invention of the microscope, and for a long time were thought to be vital – always manifesting life. Refuting that ‘vitality’ was the cause, the botanist Robert Brown recorded in 1827 that erratic movements, such as those observed by his colleagues and predecessors, were in fact performed by inorganic as well as organic particles. He guessed these particles to be nature's most basic constituents, and referred to them as ‘active molecules’ [Br]. Brown almost got it right.