Signalled Diffusion Book II: Exponentiating Diffusion

Book II of my "Signalled Diffusion" project can be downloaded here. Book I can be downloaded from a link in this post. Below I reproduce the introduction to  book II. 

Introduction

This is Book II in my current series about diffusion. It investigates the possible meanings of equations of the form

…where  the "house" symbol represents the multidimensional equivalent of the "Del" operator.

In interpreting the meaning of equations of this kind my concern is to investigate the possible sub-microscopic mechanisms which will return equations like the above, at least as an approximation. But this logic cannot be reversed: This equation doesn’t logically entail the submicroscopic approximations from which it is derived. That is, we have to hypothesise these mechanisms and then derive the equation; the mechanisms themselves don’t necessarily follow from the equation. Therefore this equation  only constitutes evidence that these submicroscopic mechanisms are in operation and not proof.

As I said in book I my sights are on the case where  the equation becomes a quantum equation: that is when the diffusion constant D, and also V,  incorporate the imaginary number “i”.