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