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"Randomly" is ill-defined. Brownian motion is defined from four axioms. Two that I recall are "everywhere continuous" and "nowhere differentiable".

Which brings us back to the difficulties with different definitions of terms as used by different disciplines...
Something about "parsimony" way earlier in the thread. :-)

But seriously, the only thing I remember about Brownian motion was reading Einstein in translation, ages ago; and a writeup of a molecular dynamics study using "simulated" Brownian motion to attempt to incorporate its effects on solvent caging of a substrate at an enzyme's active site. So your "everywhere continuous" and "nowhere differentiable" fail to ring a bell...Although it does present room for speculation about the size or scale of the system (number of particles and detail of interaction potential) during which a bunch of discrete particles can begin to be successfully modeled as a continuum. Could you please post a reference to a link or two? Enlightenment gratefully accepted. :-)

My considered opinion is that Einstein and Feynmann are two of the most elegant and economical writers in English I have come across (except J.R.R. Tolkien).