## On the Unification of Geometric and Random Structures Through Torsion Fields: Brownian Motions, Viscous and Magneto-Fluid-Dynamics

**Authors:** Diego L. Rapoport

We present the unification of Riemann-Cartan-Weyl (RCW) space-time geometries
and random generalized Brownian motions. These are metric compatible
connections (albeit the metric can be trivially euclidean) which have a
propagating trace-torsion 1-form, whose metric conjugate describes the average
motion interaction term. Thus, the universality of torsion fields is proved
through the universality of Brownian motions. We extend this approach to give
a random symplectic theory on phase-space. We present as a case study of this
approach, the invariant Navier-Stokes equations for viscous fluids, and the kinematic
dynamo equation of magnetohydrodynamics. We give analytical random
representations for these equations. We discuss briefly the relation between them
and the Reynolds approach to turbulence. We discuss the role of the Cartan
classical development method and the random extension of it as the method to
generate these generalized Brownian motions, as well as the key to construct
finite-dimensional almost everywhere smooth approximations of the random representations
of these equations, the random symplectic theory, and the random
Poincare-Cartan invariants associated to it. We discuss the role of autoparallels
of the RCW connections as providing polygonal smooth almost everywhere
realizations of the random representations.

**Comments:** recovered from sciprint.org

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### Submission history

[v1] 30 Nov 2007

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