Stochastic Covariant Derivatives in a (Curved) Space-Time: a Glimpse into the Fractoid Spaces

Abstract

A study on the notion of covariant derivatives in flat and curved space-time via It\^o-Wiener processes, when subjected to stochastic processes, is presented. Going into details, there is an analysis of the following topics: (i) Besov space, (ii) Schr\"odinger operators, (iii) Klein-Gordon and Dirac equations, (iv) Dirac operator via Clifford connection, (v) semi-martingale and Stratonovich integral, (vi) stochastic geodesics, (vii) white noise on a (4+)D space-time $\mathfrak{H}$-geometry (with the Paley-Wiener integral), and (viii) torsion of the covariant derivative. In the background stands the scale relativity theory, together with a sketch of the concept of fractoid spaces.