This chapter deals with a specific, complex-valued distribution function (and hence a specific observable X) based on the Fresnel-type integrands. The observable has continuous sample paths with continuous modification of the integral. The chapter presents some standard results which are used to formulate geometric Brownian motion. It introduces versions of the observables, distribution functions, and their expectations in which the integration in one of the dimensions is omitted. For that reason, the chapter talks about marginal distribution densities and marginal expectation densities. It provides a verification of a partial differential equation satisfied by the marginal density of the expectation of the random variable. The path integral theory gives rise to a kind of graphical calculus—the Feynman diagrams—for analyzing quantum mechanical interactions. The chapter also discusses U-Observables in c-Brownian Motion, regularized partitions, and also step functions in R^T.

Controlled Vocabulary Terms

distribution function; Feynman diagrams; Fresnel integrals; integration; marginal distribution; reaction-diffusion equations