 Networks of oscillators. So already from the title, I think you can see a link to all three of the many courses that we had so far, right? On the first line, you have controllability. On the second line, I'm talking about mixing for stochastic differential equations. And on the third line, I'm hopefully trying to, I'm going to talk about some models that make sense physically and hopefully are interesting. And so I'm just going to write some equations, the basic equations on the board so that we can keep them there and you'll have them for all of the talks. So we are in Rn, Rd, I think, if I want to be consistent. And we have an SDE, stochastic differential equation, which is of the form d of xt is f of xt dt. So far, it's just a plain old ODE. And now I add a term, which is sigma times dZt, where sigma is a linear map from Rn to Rd. And Zt is a random process which lives in Rn. So the word degenerate in the title means that I'm not going to assume that n is at least d, right? So n will typically be smaller than d. So the noise is degenerate. It does not act on all degrees of freedoms. That's the analog and sort of infinite dimension of the m, strictly smaller than infinity that Professor Cookson had in his PD context. And so I'm interested in mixing. So what that means is I want to know if there is an invariant measure for this equation, if it is unique, and if some generic initial condition when I let it evolve according to this equation converges to this invariant measure. So why am I interested in systems of this type, which are essentially finite dimensional analogs of what Professor Cookson was talking about? Well, there are two reasons. The first one is that whenever you do, if you do PDs and you're trying to do numerics on them, most likely what you'll do is you'll project onto a finite dimensional thing, right? You'll do Gediakin projections. So that becomes a finite dimensional system or pretty interesting systems from Hamiltonian physics. So if you have a finite number of masses and springs that are held together, then this will be a finite dimensional system. And there are some interesting physical questions there. By now, you should be convinced after the two talks by Professor Ekman that there are some interesting questions there. So those are good systems. So I'll make a short recap of what I mean when I talk about the networks of oscillators. Then I'll move to the abstract setup closer to this. And I'll talk about the method and the result and the method. So the result that I'll present is a result that was obtained in collaboration with Vahegh and Narcissian. But I should also give some credit to Armand Chirichian because I learned all of this stuff from him during the time that he was in Montreal during our master's thesis. So let us start. This is the simplest thing you can think about. You have L masses. They're all connected to each other by spring, so by linear forces. This is this K times the difference between the position of the two masses, right? Q is the position of each mass. And they're also pinned to the ground. So there's no translation invariant that allows the system to just go away to infinity. This model is pretty simple. In fact, a bit too simple. There are many ways you can deal with this. So what I'm going to do to make things just slightly harder is just add a small non-linear rt to the potential. So I'm not in this... I'm in the case which is harder than linear. But I'm not in the hard, hard case which is where, for example, as we see, you get problems when you have different behaviors for the pinning and the coupling. If they behave like different powers and the power for the coupling is higher, then you can't run into problems. I won't have this here because both are of the same order, right? They're quadratic potentials, so linear force, plus something that grows sub-linearly. And what I want to do is study... Right, so now the system is conservative. So if I want to do talk about mixing and things like that, what I have to do two things. I have to add some noise. But if I add some noise and don't add any dissipation, everything will just break loose. So I'm looking at this equation where on the first oscillator, I put some dissipation and some noise, typically in the literature, what you'll see is white noise. And I do the same thing on the last oscillator. So the noise is degenerate. The small n here would be two, right? There are only two degrees of freedom on which the noise acts directly. But nevertheless, I'm not right. If I make the assumption that no spring is broken or anything like that, then somehow we expect the noise to still propagate fairly well through the system to have something like some notions of controllability for this system. Okay, so from the meaning course by Professor Eichmann, what we know is that there's a series of papers which prove existence, uniqueness, and convergence to some invariant measures. Invariant measure. And one thing that's to be noted is that as of fairly recently, they are able to deal with geometries which are more complicated than the chain. Right here, each mass just has two nearest neighbor at most except for this one. But they can tackle more complex geometries where you have a complicated networks of masses and springs. Of course, they have assumptions there. So, I mean, this is an oversimplification, but what they assume is that the network is sufficiently connected for the noise to propagate. So they have this, it's a controllability condition. It's formulated in terms of the graph, of the connections between the different masses. And they have some non-degeneracy assumptions on the interaction, so the coupling potential. Their method is really, those are beautiful results. They allow for the treatment of strong linearities, so which is more than I'm hoping for here. But on the other hand, the proofs rely on the Gaussian structure of the noise. So you don't expect this exact same method to carry through if you change the nature of the noise. And so what I wanna do is tell you about the coupling and controllability approach for such system, which will work even for noises that are very far from being white noise. So different types of noises, which are not usually treated in literature. So that's, as I said, maybe you want, right, what the white noise does in an equation like this if you think about, I mean, one of the early works of Einstein and Brown and Langevin and whatever, right, it's mimicking the fact that there's this particle in the medium that's getting kicked. And in some approximation, right, you can model that by white noise. But white noise is an approximation. And I'm gonna be interested in actually considering kicks. So I'll consider some sort of compound Poisson process type noise. So where there's no noise for a while and then the particle gets kicked and then there's no noise for a while, maybe not the same duration of time and then it gets kicked around again. And so this is a perfectly fine SDE as far as we're concerned. And I'm interested in dealing with that. L is fixed. Yeah, yeah, L or in the abstract set of my dimension here. Yeah, yeah, I'm not claiming that I can control things well as the dimension goes to infinity. Okay, and not only are they not Gaussian, these random kicks at random times, they're not, or at least not obviously, decomposable in the sense of the lecture by Professor Cookson, right, this writing in the basis where you have independent coefficients in the basis. It's not clear that you have that here. So you have to adapt a bit, even the approach from there. And the kind of assumptions that I want to make, right, the sort of qualitative features of this network, as I said, is that there is some dissipativity. If you let the system evolve without noise, it will sort of relax to near the center, right? Maybe not quite to zero because there's this nonlinear term here, which is small, but still the overall behavior is that if you try to go far, you'll be brought back near the center. What? Think bounded. I can do a bit more than bounded, something that grows that one over, I don't know if it's twice or four times the dimension, but what? No, I just mean that its growth is not comparable to the linear, think bounded, but not small, okay? And there's this sort of very naive notion of controllability saying that everyone's connected to everyone through the spring, so there should be some good propagation because I'm close to the harmonic case, which is, but I'll be more rigorous on what that means here. I'll make precise conditions. So as I said, the real abstract setup is this. I have an SDE, so the noise is injected from a small dimension to a big dimension by some linear map. I'm gonna make dissipative assumptions, which are behavior about F at infinity. I want it to point inwards and I'll make assumptions on the properties of the images of the solution map and its derivatives in the noise, right? So that's really the same notion of controllability that we add in the mini course. Okay, so that's the SDE. Just to introduce a bit of notation that comes in handy when you do these things. The SDE, we say that it generates, I mean it induces a semi-group on the space of bugle probability measures, right? You can think of, I mean, I can write down the definition, it's simply, so if I have P, T star of, say the simplest example is a Dirac mass, so if I have an initial condition which is concentrated in one point and I'll make it evolve, then this, it's the probability that if you start it in X and wait time T, you're somewhere, right? This is just what it means and then in the general case, you just integrate with respect to your initial condition. So yeah, if I apply this to a set gamma here, then the gamma goes there and then lambda dx. I'm assuming that my noise is nice enough that this all makes sense. Okay, and my question is, oh, that one. Is there a measure which does not change under this evolution? And if I take an initial condition that's not too bad, then does it converge to my invariant measure? I say not too bad because I think if lambda is such that it does not have a first moment or things like that, then our result does not really carry through, but for a large class of lambda, this will work. For example, anything that's supported on a compact or has, I think, first or second moment. Anyway, okay? And when I talk about convergence here, I will not use the dual Lipschitz norm but I'll use total variation, which was also introduced in the mini course. And I just have one slide of comments on this notion of convergence and it's the following. So there are different ways you can define it. One that I like is defining it as the, if you look at the average, it gives to some bounded measurable functions and you take the worst one. You divide by two so that it stays between zero and one. That's my notion of distance between two probability measure. And now there's this nice lemma, which was also referred to in, it was stated in terms of random variable here. I stated it in terms of measure, but it's really the same. So what it's saying is if you wanna know what's the total variation distance between two measures. So say you have, I'm gonna, right, so this says you go to twice the space. So if I have a measure on R, that looks like this, is this big enough? Okay, and another one, right, which is on another copy of R, apriari, which looks like, I don't know, something like this. Then to know what's the total variation distance between this thing and this thing, you ask, now I work in the square, right? The product of the two spaces. And I ask, can I define a measure on the square, which has the right projections and as much weight as possible on the diagonal. So if the two measures are the same, right, it's like this and it's the same here, then you can put all the weight on the diagonal. It will have the right projections and you'll have a lot of weight on the diagonal. So this, you will find a P, right, something on the square, which gives zero weight to the complement of the diagonal. So the total variation distance is zero, right, they're the same measure. If the measures have disjoint support then there's no way you can put anything on the diagonal because it would have to project this thing. So it's a way of thinking about distance between measures as how much weight you can put on the diagonal when you try to couple them, okay? And this minimum is achieved, right? We saw that in the lecture too. There is such a maximal coupling, so one that puts as much weight as possible on the diagonal. Okay, so now the question is, why am I talking about couplings? What does it have to do with mixing? And the answer is seen through a basic lemma or maybe a proposition about convergence of Markov chains on compact facepaces, which is, I mean, I don't know who I should attribute this to maybe. I mean, it goes back to ideas of Derblin and people like that. So it says that if you have a discrete time Markov family on a compact face space, if there's a small number, so something between zero and one such that, when you look at the evolution of Zirak masses in two different points in one time step, then this is always bounded by some number which is strictly less than one. Then you have a unique invariant measure and you have exponential convergence. So really when I say the total variation distance between these things, it's really about, it's a total variation between the transition functions in time step one. So how do you prove this? Well, through couplings, that's why I'm talking about this. Well, there are two proofs of this actually. I think in Armin, in Professor Schirrikin's and Cookson's book on 2D turbulence, there are these two proofs. And the first one is just an argument sort of abstract about contractions. The second one uses coupling and that's where you get the insight that I wanna tell you about. So it's an existence and uniqueness theorem. Well, the existence part is not so bad because you're in a compact space, the space of probability measures on a compact space with a different topology, right? It's compact, so it's sequentially compact. So to sort of engineer an invariant measure is a classical argument that goes by the name of Bogolyubov Krilov. So I'm not gonna spell it out, but because of compactness, you have existence. Now what about uniqueness? Well, this is maybe a big block of text, but it's, we've seen this idea before. If I wanna, I know a priori, right, that transition functions get close to each other. So I can construct a random variable for any initial condition. Sorry, I should have mentioned. I work on twice the space as we did before. And I construct a random variable there, which depends on initial conditions. And the properties that it has is that if I project onto one of the components, why write pi projects into this one and pi prime projects into this one, if I look at the distribution of this random variable, only its first coordinates, then it looks like the measure, one of the measures I'm interested in. And if I look at the other projections, it looks like the other measure I wanna compare. And it has this nice property that the probability that the thing does not hit the diagonal is at most gamma. So I can do that because the total variation distance is the n for overall coupling. I can find a coupling, which does that. And I proceed inductively. I take many copies of this randomness and I define a process inductively. So I first take my initial conditions and then to go from step K to step K plus one, I sort of use this coupling. Sorry, there should be a pi here. Sorry. Z K plus one you take is the first component of this map and Z K prime you take is the second component of this map. And then you can check that this new, what you've created is you have a process which is like this. And if you look at the distribution of the first component, it has the first distributions as the original process and the same for the second. And the probability that the two components never meet each other is exponentially small because at each step, I'm sort of optimizing for them to meet. So the probability that they never meet is pretty small. And so, right, the probability that Z K is not equal to Z prime K is small, but that forms a coupling for, right. If you have this, then because this has the distribution of, sorry, this is a K starting in X and this as the distribution like this, right, those X and X prime are the initial conditions for that process, then you know that these things in total variation is small because you've managed to create a coupling which often hits, which has a lot of weight on the diagonal. Okay, so that's the idea. The idea is you want to prove mixing. So you're trying to double the space, create a process there which often hits the diagonal and that's sufficient to prove that the evolutions of Dirac masses get close together. But then if you, right, if you integrate this equation in X with respect to the invariant measure which you know exists and this one with respect to some other measure, you get that the distance between the target invariant measure and the evolution of your initial conditions is also small. Does that make sense? I think so. So this is the first lemma that I talked about. If you have a uniform upper bound on the variational distance between evolutions of Dirac masses, then you can prove exponential mixing. But this is a bit ambitious, right? Getting a bound like this, which is uniform in all X and X prime is not necessarily reasonable, especially if X is not compact, right? So this worked in compact spaces but you can always hope to apply this theorem to any situation you're interested in. So there's a second lemma which is sort of slightly more sophisticated version of this idea, so it still goes through a coupling and it's that if you have this estimate only in some set, but the time you need to wait to get into this set does not have too big of a tail, then you're good to go. You have exponential mixing. And the idea is the following. You wanna create something on twice the space. So at first if they're far from B and they're not equal, then you just let them be independently. And the time you have to wait for them to enter just B is not too large. Well maybe at first only one of them will enter in B but if, and then it'll go out and then go in but if you wait a little more and you have good estimates on how long you need to wait, then both of them will be there. And once they're both there, you sort of do a version of the previous lemma and then you can show that there's a good probability that it hits the diagonal. The two components are equal and then you have estimates like this and then you can prove exponential mixing. Now, so I've introduced there were two main words in the first line of my title. It was a coupling and controllability approach. So that was the coupling part. Now why controllability? Why controllability is important in this business? Well it's because in those two lemmas, you need an upper bound on the difference. And to get an upper bound on the difference, it's sufficient to get a common lower bound, right? Cause if this measure is lower bound bounded by something and this thing also lower bounds this one, then you can at least put that on the diagonal. It will have the right projections and it won't be a problem. So the idea is that if you wanna prove a bound on the difference, well you can just prove a common lower bounds which hold for all x near some point or in some small set. And how do you do that? Well, that's where controllability comes in. The measure which you obtained by evolving a delta, what it is, it's just the push forward of the law of the noise through the solution map, right? We've seen this in the other mini course. So you have a space for the noise, which is very big which is very big and you have the solution map which I call S say TX dot. So it takes an initial condition and a control here and it gives you a point in your, well I'm working in RN, so maybe I shouldn't draw anything compact but I know the plane. And that's the image of this map is once you chose that control, right? You solve the ODE and then you get a point at time T and that's what it maps to. So you have a law here on the controls, right? The law of the noise, call it L. I think I called it L, we'll see. Then the law of the solution is just the push forward of this measure through the map. So if you think back to basic calculus and integral calculus, right? If you want to push a measure through a nice map, you can do it via a change of variable if you have a nice Jacobian. If the Jacobian is nice, right? If the Jacobian of this map has full rank, then you can do essentially change of variable and you can get a handle on the push forward of the measure and that's the whole idea, right? The controllability conditions were conditions on this, having full rank or without the D just hit covering a ball and so if it covers a ball then by SAS theorem you can find a point where the derivative as full rank and things like that, so that's essentially the idea. And now in the limiter was this other condition that the time to hit that ball where you have good estimates on that should not be too large, it should be exponentially. Well, this is closely related to the notion of approximate controllability to the center of B, right? If the system is controllable to B then there exists a lot of control that take you there and then if the law here is nice then you get a lower bound on the probability of going into the set B, so that's the idea. So now I'm ready to present the main result which was obtained with a Hagen-Nercesian. So it is if you have an SD in RD of this form, that I talked about and this process here is a n-dimensional Poisson process with some distribution for the jumps that has a positive continuous density with respect to the Lebesgue measure and you have a dissipativity assumption and two controllability conditions. So there exists one point, one point in the phase space to which the system is approximately controllable and from which the system is solidly controllable then you have exponential mixing. So a few remarks, this noise is unbounded. It's not Gaussian but because of the Poisson structure there's still a way to reduce it to discrete time problem and still get good estimates. So that's what we did in the paper. So I just maybe if people are not sure what I'm referring to with that type of noise it's really I have a random function here which is of this form. So there's a sequence of waiting times which are IID and distributed according to an exponential law and I have a sequence of jumps which are also IID with some distribution which is nice, right? I'm assuming it has a positive continuous density with respect to the Lebesgue measure. And then my nose is of this form. It's nothing then at time tau k which is just tau one which is just t one it jumps by eight to one. Then at time tau two which is t one plus t two the time you have to wait for the second kick it jumps again but now by eight to two and this eight to two is also random. So I read them times you're kicked in random directions. So because of that I can rewrite the random dynamical system in this way. If you wanna know what time t small what happens then you just there's no kick yet so you just evolve with zero control. And when you hit the time of a kick then you just add it. So this has nice enough structure that we can estimate things properly and I wanna just recall the definition of solid controllability and approximate controllability. So there's this distinguished point x hat in the phase space. Yeah and this is my assumption on distribution on jumps. So there's this distinguished point x hat. There's a number small. There's a time. There's a compact set in this space of continuous functions and a non degenerate ball g in R n. Such that if I have a function which well approximates my solution map on that compact set then this function covers a ball. Okay so why this condition? It goes back to this picture here. This picture here what you wanna do is you wanna do a change of variable so you want this map to have a full rank Jacobian. So the way to obtain a full rank Jacobian is usually applying some version of South's theorem but turns out that because of the arguments a lot of times you don't just need this map to cover a ball but also some small perturbations of that map to still cover a ball so that you can carry all of the analysis through. So this is the notion of solid controllability. I think Professor Akratschev talked about this notion in the course. And the other assumption is that the system is approximately controllable to x hat. So in a compact, if you're in a baller regions R and you allow yourself some time enough time then you can approximately reach x hat from any point in the big ball. So that again should. This is the kind of thing that will ensure that you hit near this points x hat where you have the squeezing estimates. There are preceding three steps for the theorem. As I said you first wanna reduce to a discrete time problem. Then you wanna construct a coupling for the discrete time problem and then you wanna come back to the continuous time problem. Steps one and two are, I mean you have to write things down but I don't think there's, I mean it's fairly straightforward and the trick is how you construct that coupling and it has a similar flavor to what was done for those PDEs that we had this morning. So this construction has to depend on some integer m which is sort of the number of kicks you have to wait for one time step because the dimension where the kicks happen is much smaller, you can't expect in one kick to get wherever you want. But if you sort of allow for many different kicks intertwined with free evolutions then you sort of, you can hit more places. So m is large enough that if you have different kicks and free evolution in between them then you have good attainability properties. This m has to be chosen in a nice way but anyway. So if your coupled process, the two coordinates are different and not both in B then you just let them evolve independently until they hit B, we have good control on the time you have to wait for that and once they're both in B you do a very similar construction to what was done in this lemma one and then you can guarantee that they will coincide after some time and then they will coincide forever. And so where each of the controllability conditions come in is in the following way. So there are three steps, right? This is my whole face space. I have a big compact which is the ball of radius r. Then I have a little neighborhood of x hat. So if both z and z prime are far and different then I wait and because of the dissipativity, right? The fact that f points inwards they'll eventually both be inside this compact. Then inside this compact because of approximate controllability I can estimate the time they need for both to enter this ball near x hat. And then from x hat I use solid controllability to make sure that they eventually coincide. And that gives me total variation estimates for the difference between evolved measures. Yeah, so that's what's written here. That's the potential of the proof. Then of course there are some estimates you have to carry out that I hope I sort of gave the main ideas for the abstract result. And now I have about, do I have 10 or 20, 30? 20 minutes. I wanna go back to the more concrete model and maybe explain how you would go about proving these conditions for the concrete model. So the first thing you do is you wanna rewrite the system in this form. And so what you do is you write, do I want, I might want p here, then q dot. It's gonna be a sketch of the form that it has because it has a lot of ones and zeros but just to give you an idea. There's a gamma one here, then there are a bunch of zeros and then there's a minus gamma L. All right, that's the dissipation terms. p dot, the first one is minus p, the second one. Here you have, wait, how do you do this? You do q dot should be equal to p. So that's, all right, I put dot, q dot, sorry. Anyway, I think if I put this here then I know how it goes, it goes like this and then there's, no, there's omega star and then omega and then you probably have a zero here. Right, then I have plus a small, no, not a small but a sublinear term and then I have plus a matrix which looks like a bunch of zeros and sigma n and it's an L and then lots of zeros applied to some noise. Okay, where omega star and omega, right, or should I say omega star, omega is a matrix which encodes the all the kappas in the case and it's a matrix which has a bunch of the square which is like the kappas in the case. And so how do you ensure that such a system has good controllability properties? Well, maybe I shouldn't, well, you first focus on the linear part and you have this so-called Kalman condition which we, I don't think we talked about it in the mini course but if you, this is one of the case that we saw in the mini courses actually, right? It's a fixed vector field plus a linear combination of some controls and what you do if you wanna check controllability properties, one way to do it is to compute Lie brackets of this constant vector fields with these constant, what? Yeah, sorry, this linear vector field with this constant one but computing Lie brackets for something linear and something constant is pretty easy. It's just product of matrices, you can compute that and there's a rank condition on the product of matrices so you can check that without the sublinear terms it has good controllability properties and then because this is really sublinear and if you ensure, if you put the right conditions on the sublinear term then these controllability properties will be preserved. So at the heart of the study of this model is the study of the linear case and that's why with this model I cannot do a Q to the four or things like that that the people in Geneva were about to do because at the heart of the controllability arguments there's the study of the linear case which is very well known. I mean people have worked on the linear case for ages and a lot of is known about it so it's the basis of the analysis. And again this so-called Kalman condition on this matrix with this matrix is related to how the springs are connected and things like that. So physically it makes sense but I cannot do the other order of potentials in quadratic. So that's essentially all I wanted to say so thank you for your attention.