Mise en Scène: A General View

In Chapter X §1, we started with three assumptions – three perceptions – about a Brownian particle's trajectory: (i) its direction at any instant cannot be determined; (ii) displacements over disjoint time intervals are unrelated; (iii) ‘statistics’ of displacements over time intervals of equal length are the same. In a framework of probability theory, the strongest interpretation of these perceptions implies that a Brownian particle's position X(t) at time t ∈ [0,1] is Gaussian with mean 0 and variance ct. Specifically, we argued in Chapter X §1 that if Brownian displacements are statistically independent, symmetrically distributed random variables with distributions homogeneous in time, then {X(t) : t ∈ [0,1]} is necessarily a Wiener process (Definition X.1). A Wiener process, however, conveys an idealized view: while haphazard and difficult to predict, Brownian displacements are not, in reality, independent of one another. At the end of Chapter X, imagining Brownian motion to be a random walk, we departed from the classical model, and viewed statistical independence as the first and indeed simplest instance on a scale of stochastic complexity. This view – under assumptions of time-homogeneity, finite variance, and prescribed ‘randomness’ – led us to α-chaos processes. The case α = 1, exemplified by a Wiener process, is a continuous-time model for the simple random walk, and the case α > 1, exemplified for integer α by the Wiener homogeneous chaos, is a continuous-time model for walks that manifest greater levels of ‘randomness’.