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