The Wiener Stochastic Convolution Integral – as a scalar Random Variable with a single parameter – is a good start to getting robust tools for studying turbulence.  But the results must be expanded to vector quantities of 4 parameters (3 space and time).  This is a complex task made more difficult by the fact that the Wiener Process (Brownian Motion) is continuous but has unbounded variation.

The first step is to insure that the Stieltjes-form integral with the Wiener Process as integrator actually exists, can be extended to infinite limits, has a derivative, and has appropriate stationary and ergodic properties.  “PrettyGoodIntegrationTheory” (today added to Other Topics) is designed to answer some of the basic existence questions posed by these integrals.

From these results, various chapters in The Revised Thesis (to be added to this blog by and by as revisions are completed) will expand the basic integral to vector form with multiple parameters.  And from this we shall form Wiener’s “Homogeneous Polynomial Functionals” (HPF) and “Orthogonal Polynomial Functionals” (OPF).

These Polynomial Functionals are then used to express velocities (et alia) which are then inserted into the equations of motion.  We then solve these equations of motions either analytically, or by a Galerkin best fit process, of by direct numerical computation.