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