Within the context of a harmonic bath, the term ``random force'' is something of a misnomer, since R(t) is completely deterministic and not random at all!!! We will return to this point momentarily, however, let us examine particular features of R(t) from its explicit expression from the harmonic bath dynamics. Note, first of all, that it does not depend on the dynamics of the system coordinate q (except for the appearance of q(0)). In this sense, it is independent or ``orthogonal'' to q within a phase space picture. From the explicit form of R(t), it is straightforward to see that the correlation function
i.e., the correlation function of the system velocity 
with the random force
is 0. This can be seen by substituting in the expression for R(t) and
integrating over initial conditions with a canonical distribution weighting.
For certain potentials 
that are even in q (such as a harmonic oscillator),
one can also show that
Thus, R(t) is completely uncorrelated from both q and , which is a property
we might expect from a truly random process. In fact, R(t) is determined
by the detailed dynamics of the bath. However, we are not particularly interested
or able to follow these detailed dynamics for a large number of bath degrees of
freedom. Thus, we could just as well model R(t) by a completely random process
(satisfying certain desirable features that are characteristic of a more general bath),
and, in fact, this is often done. One could, for example, postulate that
R(t) act over a maximum time 
at discrete points in time 
,
giving 
values of 
, and assume that
takes the form of a gaussian random process:
where the coefficients 
and 
are chosen at random from
a gaussian distribution function. This might be expected to be suitable for
a bath of high density, where strong collisions between the system and a bath
particle are essentially nonexistent, but where the system only sees feels the
relatively ``soft'' fluctuations of the less mobile bath. For a low density bath,
one might try modeling R(t) as a Poisson process of very strong collisions.
Whatever model is chosen for R(t), if it is a truly random process that
can only act at discrete points in time, then the GLE takes the form of
a stochastic (based on random numbers) integro-differential equation. There
is a whole body of mathematics devoted to the properties of such equations,
where heavy use of an It 
calculus is made.