Finally, let me turn to some first passage phenomena of random walks
So, as the name suggests first passage phenomena refers to
asking the question: When does the random walker first reach a specified level
and I'd like to illustrate the interesting first passage properties of a
one dimensional random walk in the continuum limit
So, let's ask the following two basic questions about a one dimensional random walk
First one is, what is the probability of
eventually, and notice the word eventually sitting here,
eventually hitting the origin when starting from 'x'
So 'x' is just some arbitrary point on the one dimensional line,
let's take it to be positive, and we're asking the question:
What is the probability that a random walker that starts at 'x'
eventually hits the origin?
And second question is:
What is the time to hit the origin?
Now, we actually know the answers to both these questions from much earlier on
in this tutorial because I showed that in dimensions two and below
a random walk is recurrent and that means that it hits every point infinitely often
and it particular, means that no matter where I start
I'm guaranteed to hit any point
So if I started at 'x' I'm guaranteed to hit the origin
Similarly, and this is the part that will be new
which is we can also compute the time that it takes to hit the origin
and even though we're guaranteed to hit the origin
the amazing feature is that it takes an infinite amount of time
to hit the origin
So again, I want to emphasize this dicotomy between
being sure that you're gonna hit a specific point
but it takes infinitely long to get there
So, I's like to derive these two results
in the simplest possible way and for this I will use
the continuum aproximation
So, namely I'm going to solve the diffusion equation on the positive
half lines
I'm going to solve 'dc' by 'dt' is equal to
'D', 'd' second 'c' by 'dx' squared
So, I'm using the letter 'c' just because it's conventional for concentration
It's the same as 'p' in an earlier slide
With the initial condition at 'c' of 'x' at 't' equals zero
is equal to delta of 'x' minus 'x' naught
So, I'm starting at some position 'x' naught
on the positive half line
And there's also a boundary condition to make this
a well defined problem
Namely, 'c' at 'x' equals zero at any time 't'
is equal to zero
This is known as the absorbing boundary condition
And it basically is stating that when a random walker
reaches the origin
the problem is over