We now examine the diffusion phenomenon in light of the probabilistic interpretation of the Gaussian kernel. First, we write out the convolution (2.10) explicitly:
Now, consider a randomly moving particle that ``carries'' an intrinsic value
u. Suppose the final position (at time t) of this particle is 
,
and the total displacement of this particle from time 0 to time t is 
; then the value at 
at time 0 appears at
position at time t. In other words, given , 
if, for the
moment, we consider fixed. However, we consider this displacement
to be random. Then we can compute the average or expected value
of u at position and time t:
where 
denotes computing the expected
value with respect to the random displacement . Let 
denote the probability density for the displacement
vector; the parameter t denotes the duration of the time interval of the
particle's motion. Then (2.19) becomes:
But recall that, for all t, 
is a valid probability density function of a random n-dimensional vector with mean value 
, independent coordinates and
with each coordinate having variance 2ct. In particular, as 
, this function approaches an impulse, that is, the density of a random
quantity which is with probability 1. This suggests that can indeed describe the
random motion of a particle, that is, it can serve the role of used above. Thus, (2.18) has the form (2.20). The motion described by the Gaussian kernel is called Brownian motion, and has tremendous significance in statistical
thermodynamics. We summarize the result:
The solution to the diffusion equation evolves from an initial distribution to the distribution at some later time according to the ensemble average of particles undergoing Brownian motion.
Some important properties of Brownian motion are that the trajectory of a
particle moving in this fashion is continuous with probability 1, that the
motion over a given time interval is independent of the initial position and
is independent of the motion over any other non-overlapping time
interval, that the motion along the coordinate directions are independent of
each other, and that there is no directional preference or orientation to
the motion. In fact, it can be shown that the only statistical
distribution with these properties is Brownian motion (except for degenerate
cases, such as a particle that does not move at all- corresponding to c=0- or a particle that instantaneously runs out to infinity- corresponding
to 
).
