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