Chapter 7
Brownian Motion
Previous chapters can safely—indeed beneficially—be read on the understanding that the distribution functions are of the familiar non-negative kind. But, except where otherwise stated, those distribution functions are actually permitted to be complex-valued.
In contrast, this chapter deals with a specific, complex-valued distribution function (and hence a specific observable X) based on the Presnel-type integrands discussed in Chapter 6.
7.1 c-Brownian Motion
The subject of this chapter is an observable XT with complex-valued (or c-valued) distribution function 
, the indefinite integral or Stieltjes version of the incremental Fresnel density function 
described in Chapter 6.
Take T = [0, ∞[or any subset or interval of it; for instance, T = ]0, 
] with t0 = x(t0) ≔ 0. The list BM1, …, BM7 below is a summary of properties associated with Brownian motion.
Taking c = −
gives the standard version of Brownian motion. But some of these properties are present in observables or processes other than standard Brownian motion, and, in anticipation of this, it is not stated ...
Get A Modern Theory of Random Variation: With Applications in Stochastic Calculus, Financial Mathematics, and Feynman Integration now with the O’Reilly learning platform.
O’Reilly members experience books, live events, courses curated by job role, and more from O’Reilly and nearly 200 top publishers.
