Let's now compute the probability distribution of
a 1-dimensional random walk in the continuum limit.
So let's start with our master equation from the
previous slide for the probability distribution.
I'm going to write it in a slightly different form that
facilitates taking the continuum limit.
So let's look at p(x), but instead of looking at it
time t, let me look at time t+1. So instead of having
1, we're going to write the time increment as dt which
will allow us to take the limit as dt goes to 0.
So the master equation from the previous
slide will be p(x,t+dt) is one half for a probability
of hopping from x-1 to x. But instead of writing
x-1 here, we'll write x-dx comma t, plus one half
p(x+dx, t). Let's now take a Taylor series expansion
of this equation. So, I'll write p(x,t) plus dt
to partial derivative dp by dt to higher-ordered
terms. And on the right-hand side I have one
half and the (x-dx) term when we expand it
in a Taylor series will be p(x,t-dx