 Thanks. Hi everyone. It's a pleasure to be here to present some of my work and I'd like to thank the organizers for giving me this opportunity. My name is Raphael Fourien. I'm a PhD student with Amandine Weber at Polytechnique and Alison Etheridge at Oxford University. And today I'm going to talk to you about gene flow through geographical barriers. What do I mean by geographical barriers is anything that prevents people from moving across. So you can think about mountain ranges, but also if you think of say deer or frogs, highways, political borders, also like geographic features like straits and seas and rivers. So I'm not going to talk about any particular species, so you can think of humans, but birds, anything. And why are we interested in this? Well, we're interested in the consequences of these kinds of features on the genetic composition of populations that live on both sides of a barrier. And I'm going to tell you about a very simple model to represent these kinds of situations. So each square here is a population with n individuals living in it. And my mountain range is here. So this is towns in France and towns in Spain. This is the Pyrrnes. And at each generation I replace every individual in each colony by children of the previous individuals. And a proportion 1 minus m of children in each colony come from other children of parents from the same colony. And a proportion m of them come from one of the neighbouring colonies, except of course for these two populations around the barrier, where a proportion half of c m, half of m comes from this population and half of c m comes from the population that's on the other side of the barrier. So you have to think of c as a very small parameter, something between 0 and 1 and potentially very small. So there's a reduced exchange of migrants between these two populations. And I want to know how the genetic composition of this system evolves in time, generation after generation. And this is what it looks like. So suppose I have two types in my population, types 0 and 1. And initially all the types 1 are on the right and all the types 0 on the left. And I let it evolve for some generations. And the proportion of type 1 in each colony is going to follow this dark line here. So you can see that there's a jump in a proportion of type 1 at the interface. And there's a very nice way to think about this model here is by instead of thinking of individuals reproducing forwards in time, I think about sampling individuals and looking at where the ancestors lived back in time. So suppose I sample an individual in x and I look at its ancestor t generations ago and I ask the question, what's the probability that the ancestor of this individual lived to the right of the barrier? Well this probability is going to be exactly the proportion of type 1 in my population. So this is the equation that's written here in xi is around the mark on z. That's tracing the position of the ancestor of a uniformly sampled individual in my model. So it's very easy to write the transition probabilities of xi. It's exactly the migration rates in this model. So whenever it's in one of the populations here or here, it's doing a simple random walk with jump probability m. And when it's here, it jumps to the right with probability 1 and to the left with probability 1 over 1 plus c and to the left with probability c over 1 plus c. And the goal of biologists studying these kinds of models is to try to use genetic data to reconstruct the age of ancestors like say sample 2 individuals. I ask what's the age of their most recent common ancestor. And that's giving me some information about the geography of my system. And so for this we need some formulas to describe what xi t is doing. But since it's a random walk, it's very discrete and we don't have any nice formulas for the law of xi t. So the situation here is roughly like this. We have this simple model for how gene type frequencies evolve in time. And we know that we can approximate it with this continuous line here that's solving a partial differential equation that I haven't written down. And we have a way of describing this model in terms of a random walk and we want to describe the continuous version with a continuous process here. And that's what I'm going to be telling you about. So take a sequence of random walks, each with a different parameter cn and cn is going to zero. So the bar is getting stronger and stronger as you rescale space and time. I look at the position of my ancestor t generations ago and I rescale it just as I would rescale a simple random walk to get stunt up brown and motion. And I suppose that cn is of order something over square root of n and this something could be zero, infinite or something finite in the middle. And in all these cases I have a convergence of my sequence of processes xn to some continuous process x. Which is essentially a Markov process but not quite on R because it doesn't have the same behavior if it comes to zero from the left or if it comes to zero from the right. Which is fairly logical given that the random walk doesn't have the same behavior when it's sitting here or when it's sitting here. And so I call this limiting process here partially reflected brown and motion because we're going to see that it looks like a brown and motion but sometimes it's allowed to cross the origin. So I'll give you two constructions of this process and then I'll give you an argument for why the random walk converges to this process. The first construction is the following. Start with standout brown and motion. Draw two lines here and essentially we want to keep only the excursions outside of this set and we want to forget about all this. So I want to forget about all the red bits. I take them out and I join all the excursions like this. That gives me a process and it's obviously not Markov because the behavior at zero depends is a bit weird but you can make it Markov by splitting zero in two states. One that's zero plus and one that's zero minus and it becomes Markovian. So that's the first construction and you can see that for a while if you start from here for a while it looks like reflected brown and motion. And after some time it flips on the other side and it looks like reflected brown and motion on the other side etc etc. And the second construction makes this more explicit. This time start with reflected brown and motion and I'm going to flip some excursions up and down. And when do I do this? Well I look at its local time at the origin that's the dark line here. And I wait until the local time reaches an independent exponential random viable. That's the red line here. And when it reaches this level and the local time reaches this level I change the sign of the process. And then I draw another exponential random viable. I wait until the local time reaches this level and I flip it again etc etc. And the first question that we want to ask is are these two constructions the same thing? Is this guy the same as this guy? Well the first thing to note is that the local time to the right of zero that this process accumulates is the same as the local time that the original brown and motion accumulates at this level. And there's a very well known theorem that's the Ray-Knight theorem that tells you that the local time that brown and motion accumulates here before first reaching this point is an exponential random viable with some parameter that depends on the size of this space here. And that allows you to say that the time at which you have to flip the process is the time when the local time of this process reaches an independent exponential viable. Right so why now given you construction of the process and I haven't told you why the random wall converges to this process I'm only going to sketch the main states. The first thing that's fairly clear to see is that the absolute value of the random walk is going to behave like the absolute value of a brown and motion because the only weird behavior is jumping from minus one to plus one. But outside of this set is behaving as a simple random walk. So the absolute value is basically the absolute value of a simple random walk. So the first point is clear. The second point, well think about how many times the random walk has to visit this point before first going across the barrier and exiting from this side. This is a geometric random viable because of the Markov property each time you visit this point you have some probability of exiting from this side and some probability of exiting from this side. And as we all know geometric viable with a very small parameter looks like an exponential random viable. And that's what's written here and then you use the Markov property to get some independence. So the local time accumulated by the random walk between different crossings of the barrier is a scale geometric random viable and so converges to an exponential random viable. And then you have to give some argument for why they're essentially independent in a limit. I won't go into more details. And this is an illustration of this result. So the blue dots are the transition probabilities of the original random walk just computed by iterating the transition matrix. And the red line is the transition density of the partially reflected Brownian motions of the limiting process. We happen to have an explicit formula for this. So it's easy to draw and you can see that the law of XIT is very close to the law of XT. Yes. So to sum up I started with this microscopic model character to illustrate how genes evolve in time in the presence of a barrier. And I could approximate the evolution of types with this partial differential equation. And this random walk gave me a description of this of the evolution of types in this model and I can approximate it. I've just shown you this red arrow here. I can approximate it with partially reflected Brownian motion. And the actual question is can I draw the arrow here? And the answer is yes. If I start this process from the point X, I let it evolve from time and I ask for some time, I ask what's the probability that's sitting to the right here. Well, this probability gives me exactly a solution to the partial differential equation that's driving this dark line here. Thank you. So thanks Rafael for this nice talk. Is there any questions? Are there any experiments that kind of confirm the prediction that we should have a behavior like this partially reflected Brownian motion? So no one has traced down the ancestry of any individuals in populations and see so that this isn't a proper behavior, but this jump in genetic composition of populations has been used and is being used to study different situations for natural populations. So this description is very theoretical in this one. We know we can use it to caricature what's going on in space. The main purpose of this is to use these models to do some inference of some demographic parameters in the populations. So we don't really care if what's happening in this model isn't really what's happening in real life. What we want is that essentially the processes look the same, the resulting processes look the same, so that we can sum up what's happening in the real population with some effective parameters in this model. So is there any other questions? So you have a fairly explicit construction of your process using Ray-Knight's formula and certain properties of Brownian motion. Can you generalize it to general one-dimensional diffusions? I haven't thought about that. I guess you could still do this one for general diffusions, for the construction, but for the convergence you ask? Well, so maybe you wouldn't look at the process that you obtain by erasing the excursions in a certain region, but still the barrier process, can you define it kind of directly for a general diffusion? I don't know about the existence. So this correspondence gives you a characterization of this process as a solution to a martingale problem, and you can still write down the martingale problem for a more general process, and the proof of this thing uses properties of Brownian motion, but doesn't really use what's happening above and below this set. So as long as the process behaves as Brownian motion in this set, then I guess you could say something about it. That's an interesting question, yeah. So is there any other questions? If not, let's thank Florian again.