 Thank you very much. Thank you very much to the organizers for this nice school and conference. I really enjoyed my stay here. So my talk will be related, of course, to the lectures of Sergei Kuxin and the talk of Armin. So we will see how this controllability approach can be applied to prove ergodicity for equations in unbounded domains. So in unbounded domains, usually you have two types of problems. So first of all, the operator, the stocks operator, the spectrum of the operator will not be discreet. And so we will see that this controllability approach applied in a suitable way does not see at all how looks the spectrum. If the equation is controllable in the suitable sense, then no matter how looks the spectrum of the operator. Then there are also a lot of problems coming from the lack of compactness in the system, where you are working on unbounded domains. So the Sobolev inclusions are no more compact. And so, in fact, we are going to replace these compactness properties by other compactness coming directly from the equation. Some compactness induced by the equation will replace compactness coming from Sobolev spaces. So I will start by a short introduction. Study the classical Navier-Stokes system. So the viscosity will not play any role. So I take it equal to 1. So we are in the incompressible setting. And to fix the ideas, we will consider directly boundary conditions, but any reasonable condition will do. We consider the space variable in a domain d, which will be unbounded. So we will not be able to work in any unbounded domain. So in fact, we will assume that d is an unbounded domain of Poincare type, which means exactly the following that the classical Poincare inequality is satisfied on this domain. For any smooth compactly supported field. It is bounded in one direction, so it can. So it is in two dimension. You can take a strip. You can take a strip. So in fact, what I need from the equation is the dissipativity. So I take Poincare inequality. So I assume that on the domain, the Poincare inequality is satisfied in order to have dissipativity in the equation. As soon as the system is dissipative, it will work. So I don't know if for your setting Navier-Stokes is dissipative or not, but it is to be checked. So in any sense, in any case, it is not this condition which is important, but the property of dissipativity. So there is a lambda such that this inequality holds for any v in this space. So we assume this condition since Poincare implies dissipativity Navier-Stokes system, which means exactly the following that the solution satisfies the following inequality minus lambda t u0 squared plus lambda minus 1 eta squared l2 0 t h minus 1 d. So this dissipativity is crucial for us because without this dissipativity, the energy can go to infinity without creating even stationary measure. So even to have a stationary measure, we need the dissipativity property. So in fact, we can work in arbitrary d, domain d, in R2, even for R2, provided that there is a dumping in the equation. For example, dumping term of the form alpha u with alpha, which is strictly positive. So in this situation, we will have dissipativity in the equation no matter how looks the domain. So what are the assumptions about the noise? The form of the noise will be exactly the same as in the lectures of Sergei Kuxin and talk of Armin. So it is of the following form. So you have characteristic function of the interval k minus 1 k. Then you have these random perturbations. And about these perturbations, we assume the following eta k are independent identically distributed random variables in the space L2, 0, 1 with values h. And h is the classical Navier-Stokes space, which is the closure of smooth vector fields that are divergence free. So the main result is the following. So assume that these perturbations are E with respect to L2 norm, with respect to L2 norm. Thanks. The usual L2 norm. So the main result is the following. So assume that eta k are decomposable, which means the following is that they are of the form eta k equals to infinity sum over j of Bj Ejk Ej, where Ej is an orthonormal basis L2, 0, 1, h. So it is a spacetime basis. Bj are positive numbers with square summable. So all of them are known 0. So we are working in the non-degenerate setting. The goal of this work is to understand what happens, what we need to do in unbounded domains. This is the difficulty that I would like to understand here. Of course, a degenerate version of this also can be regarded. So all this is deterministic. All the randomness comes from the coefficients cjk. Cjk are independent, identically distributed random variables in R that are bounded, which means by 1, for example, and whose law is absolutely continuous with respect to the Lebesgue measure real line. This density is c1 smooth and positive at 0. So it is exactly what we saw yesterday and the beginning of the week. So under this decomposability and non-degeneracy assumption, we have exponential mixing. So then the NS system is exponentially mixing, which means exactly the following. There is a unique stationary measure if this is a probability measure over this Navier-Stokes space h, compact h in this space. There are numbers c, c positive such that the law of the solution at integer times converges to this stationary measure with respect to the dual-lipschitz norm that we saw several times. So it is c exponential minus ck for any k and the initial condition u0 will be in this compact. So we have uniform exponential convergence for initial data given in this compact. So this is the main result that we are going to prove using this controllability approach. So of course, this is based on the previous works of Armin and our joint work with Sergei, Kuxin, and Armin. Here you come. So the references here would be, so his works of 2011 and 2017, where he used controllability approach to study ergodicity of degenerate noise but degenerate in the space, which means that the noise was supported in his works in a given subdomain of the main domain or on a piece of the boundary. And then with Sergei Kuxin, we generalize this approach in two papers to study degenerate noise but in Fourier spectrum. So in the literature, there are only a few works in unbounded domains for related equations. So it is the works of Bartin and Hanyin, where they consider the Burgers equation on the real line. So they prove exponential mixing for this equation, but their proof does not work for this more generic type of equations like Navier-Stokes Ginsburg-Landau. They use in a very critical way the special features coming from Burgers equation. But I need to mention that what they prove is stronger in the following sense that they are able to consider space homogeneous noises. So the stationary, the unique stationary measure they get is translation invariant, which is physically the natural situation. This is a situation which for the moment is open for the Navier-Stokes system. OK? Yeah. So we will proceed. So I will give a formulation of an abstract, sufficient condition a little bit as Sergei Kuxin and Armin Shiri can, but with less compactness properties. And then we will check one by one the properties of the conditions in this theorem for our concrete example. So this was the first part. Now the second part, abstract result. So this is a version of a result that we saw in the previous two talks by Sergei Kuxin and Armin Shiri. So as in their case, in fact, so we can look at the restriction of the system, we can consider it as a random dynamical system, UK equals to s UK minus 1 eta k. The restriction to integer times of the system can be written in this form, where s is a mapping from this product space to this. So h is exactly the Navier-Stokes space here. And e will be the space of controls here. And s will be the resolving operator of the Navier-Stokes system, which sends initial condition and control to the solution at time 1. But in this section, we will forget about Navier-Stokes. We will formulate conditions in terms of s and eta k, which imply exponential mixing. And then in the last third section, we will check the condition for Navier-Stokes. So what are the conditions? So there are five conditions in this theorem. So assume that the following five conditions are satisfied. The first one is the following. So s is s from h times e to h is c to smooth. And the derivative d us u eta satisfies the ease of the form d us u eta is of the form c1 plus e to u eta. So for any u in h and eta in the support of the noise, I will define it in the last condition. This is going to be exactly the same set as in the talks of Sergei Kuxen and Armeshi again. So there is a difference already at this level with respect to the previous talks, because s is not regularizing into a space v that is compactly injected. And this is important because we do not have that property. So already here, we lost some compactness. But we recover, in fact, some compactness here, because we will assume the following. We will assume the following c to u eta from h to h is compact. The norm of c1 is less than 1. So this decomposition means that the linear system can be decomposed into a sum of two operators. One is dissipative, the second one is compact. So second condition. OK, let me call this random dynamical system 1. OK, there are two. OK, s. So the system s is asymptotically compact. So this means the following. So for any sequence of initial conditions, or any bounded conditions, 0n in h, for any controls, so for any times n integer times ln, n controls eta 1, n, eta ln, n in the support of the noise, we have that the sequence sln, 0n, eta 1, n, eta ln, n is relatively compact. So this is a condition that is often used in the theory of attractors to construct attractors for PDC in unbounded domains. We use this property here in the stochastic setting. Then there are three conditions that we already met before. The first of them is this approximate controllability. So the system 1, system s, is approximately controllable to some point. So this means the following. So for any compact h, for any epsilon, there is an integer m and controls eta 1, eta m in the support of the noise. We are never allowed to choose a control outside the support of the noise. So we can choose controls times such that the solution of my system at time 1m. I need starting from u0 and taking controls eta 1, eta m is close to this final point, distinguished point you had for any initial data in h. So we are doing this uniformly with respect to the initial condition in compacts. So for any compact of initial condition, we are able to find a uniform time. Of course, I need to put this before, right? The controls depend on you, of course. So I need to put it after epsilon, no, after m. And then fourth condition is approximate controllability of the linearization, which means the following. So the image of the linear operator, you take the derivative with respect to the noise of our nonlinear resolving operator and then consider it as a mapping from control space to fast space. The image of this linear operator is this. It's this for any u in our space and any eta in here. Again, so exactly, exactly. For the Navier stocks, this follows from the dissipativity. But of course, we may have some system with multiple equilibrium states where this is not obvious and can be derived using the geometric control tools of Agra, Chevon, and Sarichev. But in our setting, this will be immediately following from the dissipativity. And this is one of the points where we use this Poincare hypothesis. Of course, for our concrete example of Navier stocks, this will not depend. We are going just to switch off the control, take 0, and wait. But in abstract level, it may depend. And the last one is that eta k is decomposable. Decomposability is defined before. Although in this abstract setting, we can relax it a little bit. But so I stop here. So for this abstract result, as soon as you have these five conditions for your random dynamical system, then the system is exponentially mixing, again, in the sense that I gave before. So where do we use this dissipativity? It is used, for example, here to check this condition. And it is also used to check this asymptotic compactness property. We will see that. So why is this condition of asymptotic compactness important for this approach? This is because this asymptotic compactness implies that our trajectories live in compact. So this is important. In fact, so if we define the set of attainability for our system, so the set of attainability, so at time k first, so I take initial conditions from some set h. And I define the set of attainability at time k as follows. So I take sk initial condition u0 and controls eta 1, eta n, this for any initial condition in h, and any controls from the support. And then I consider the set a h, which is the union of these sets of attainability at time k. And then I take the union in the space h. And then I take the closure in the space h. So this is compact, provided that the set of initial conditions is compact. So this allows to have compact for space, which is very important, because when trajectory lives in compact, you apply the usual Bagalubov-Krylov argument to show the existence of session measure. And if you work in a compact, you have many estimates that are uniform in space. And it is really convenient to work with. Of course, there is this open problem mentioned yesterday by Arme Chirician. So it is a nice problem to understand what should be done when the controls, the noise, are unbounded so that the fast paces is no more compact. So now what we are going to do, we are going to check the conditions. So I will concentrate myself only on the first two conditions, because the rest we already saw two times discussed. So I will only discuss what is new with respect to the previous works. So in the third part, we are checking the conditions of this theorem. So I will keep these two conditions. So we are checking this first condition. So the first regularity part is obvious. This is a usual smoothness property, dependence property from the right-hand side of the equation and initial condition. So we will check this decomposition. So the linearization of the equation is given. So the linear equation is the following. d minus delta u. OK, u will stand for the solution of Navier-Stokes. Here I will use w plus b w u equals to 0. There is an initial condition, u0, that is equal to w0. So more exactly what we are doing. As usual, Navier-Stokes system is projected to the space of divergence-free vector fields. So that the system is written in the pressure-less form. It becomes in this divergence-free space, an evolution system. And then you linearize it. You get something like this, where u is exactly the trajectory around which you linearize. So in fact, this operator ds, d-u-s, is nothing else, but the value of this over some point w0 is exactly the solution of this at time 1. We can write wt as a sum of w1t plus w2t. w1 is the solution of the following equation. w1 minus delta w1 equals to 0. And I have an initial condition. And then in the second, I will have the remaining part of the equation. This is w2 delta w2 plus bw u equals to 0. And now initial condition will be 0, of course. So here, we do not have any dependence on the trajectory or control around which we linearize. So this is completely uniform. And using this Poincare hypothesis and the dissipativity, we get immediately that w1t squared is less than exponential minus t lambda w0 squared. Yes, yes, it is not important the exact that it is. But it is the same lambda exactly in the Poincare. It is positive constant. So if I take t equal to 1, t equal to 1, I get exactly this. Because this norm will be the exponential of time 1, yes? Yes, yes, because in fact, this u is the trajectory corresponding to initial condition u0, u, and control eta. We denoted the same symbol. It is usual stuff. I'm sorry. So u and eta are both encoded here. So here, we have this uniform estimate. And then here, we have compactness, in fact. Why do we have compactness here? So here, this is a polynomial non-linearity. w here is multiplied by a function that decays at infinity. So in fact, this resolving operator that sends w0, because here, there is w0 in the definition of w. So the mapping that sends w0 to w2 at time 1, this mapping is a composition of several mappings, which are all continuous. And one of them is compact, which one? In the middle of this composition, there is here a multiplication with something that decays at infinity. And it brings compactness into this application. So here, where we use the structure of the non-linearity. And in fact, here, it is quite general property, which still remains true for many other polynomial non-linearity. Navier's talks here will not play an essential role. This argument remains essentially true for many other non-linearity. So this is the verification of the first property. Then what about this second property? Checking condition 2. Here, we use some formulas. So I need to take an arbitrary sequence of initial data that is bounded. Then I am taking a sequence of times, controls in the compact. And I need to show that these solutions at time ln are relatively compact. This is linear. This is linear. And the other condition is non-linear. So it doesn't follow. So at least what I'm going to do, it is not based on this. It uses really the non-linear equation. So how we proceed? So we denote. Let us discuss it after. So because the argument here is already not complicated. So we are going to just multiply the equation and integrate and pass to the limit. So let us do this. So vk is this sequence. So passing to a subsequence, if necessary, I can assume a lot of convergences. So what type of convergence? I can assume that the initial conditions converge weekly in h, because this is bounded. Then I can assume that eta e n converges strongly to eta e. Strongly in e. And moreover, I can assume that vk converges weekly because of the dissipativity. This will be a bounded sequence. So I can extract a weakly convergent subsequence, w. I have these three limits. So as a consequence of this weak convergence, we immediately have that w, the norm of w, is less or equal to lim int n goes to infinity of y I denote it by k, I don't know. It is n. I pass to the limit, and I have this inequality, right? What I need to do is to prove, so go, lim sup as n goes to infinity of vn is less than w, the norm of w. If you are able to do this, you have convergence of the norm so that you have convergence strongly. And all these modulo subsequence, OK? So I took some subsequence. I denoted the same symbol, all the sequences. So now I'm going to check this. That lim sup satisfies this inequality. So we consider a sequence with delay, s ln minus m, 0n eta 1 eta n ln minus n. I take any integer m. So then I use a diagonal argument to show that, again, passing to subsequence, but denoting with same symbols, everything, we can converge weakly to some limit, wm, as n goes to infinity. Of course, this comes, again, from the boundedness. I can extract a subsequence so that we have this limit. So what is the consequence of this? So if we denote this sequence, wnm, then sm of wnm with controls eta 1, more precisely, it is, let us write it like this, but it depends on m. So this equals to w. What is this? In fact, we know that these initial conditions converge weakly. Then I use continuous dependence in weak topology from the, with respect to the initial condition, and strong topology with respect to the controls. I can pass to the limit and get here the same w as here that can be written as follows for any m. Of course, I'm taking the limit here, so there is no n. Wn is this, because this is the initial conditions that converge to this limit. The m remaining controls, m remaining controls that are not included there, converge to some limit that I call this, this, this. And I put sm here, so this is equal to w, equal. It is m, of course. OK, thank you. OK, so great. So what we do next is just applying some energy equality for the Navier-Stokes equation. We apply the following equality. So w squared equals to exponential minus lambda m wm, which is the w here, squared, plus 2 integral 0m exponential minus lambda m minus s, ms, ms. I will define these quantities in a minute. So then there is this norm. So what is this? So in fact, you just take this color product of equation. Equation with initial condition wm, and eta m is the function formed by sequence of controls eta 1m, eta mn. On the 0, 1, it is this. On the 1, 2, it is eta 2m, et cetera. On the last interval, it is eta m. So this is the step function, simply. And this is the associated solution. So the value of the solution at the initial time is this. At the final time is this. So this is just what you get by taking this color product and using a Duhamel formula. Then we do the same, exactly the same for, so let me write here, so that I'm close. So I do the same for the other trajectory. And then I pass to the limit. So I need to specify what is this. This is exactly the square of h1 norm minus lambda over 2, the l2 norm. So because of the Poincare inequality, this is a square of a norm, which is equivalent to the norm of h1. And this is important, we will see in a minute. So I rewrite a similar equality for the other trajectory. So I'm almost done. So what is this? This is the same energy equality written for the solution starting from this point and corresponding to our controls here, eta nm, which starts from this. And at time m, it is here. Next, we pass to the limit as n goes to infinity. What happens? What happens? This term goes to this term. So here, you use the weak convergence in the h1 space. You use the equivalence of this norm to the usual norm of h1. And you notice that here, there is a minus. When you pass to lim inf, because of this minus, it becomes lim soup. And so when you pass to the limit, and you notice that m is arbitrary, and this is bounded. So it goes away and you get exactly what you were looking for. This is lim soup as n goes to infinity of vn, which is less than the w. So you pass to the limit. You use these two equalities. And it is important that the sign here changed. And you get this property. Again, we used Navier-Stokes. Only one property, in some sense, two properties from the Navier-Stokes. The non-linearity is conservative, because when you consider energy equality, there is no non-linearity. So it is convenient. And then the dependence of the solution weakly in the initial condition, and strongly in the control. And this is, again, a very usual properties for PDs. So this is what I wanted to tell you. Thank you very much.