Courant Events Right: So I'm very happy to introduce my friendly collaborator, Amrit Ahmad, who will speak about environment gives measures and propagation preventedness for non-specific patients.
Courant Events Right: Thank you. Oh, thank you very much, Bjorn. Thank you to the organizers, especially Nina and Roland. And let me first,
Courant Events Right: besides thanking the invitation, I apologize to the… all the people that I'm not going to mention, that I should mention, but it's a… it's a… it's hard to… to… to do justice, to everybody who has,
Courant Events Right: Yo, this.
Courant Events Right: And, also to the physicists, you know, I'm not a physicist, and so probably the thesis asks me a question, I won't know the answer, so…
Courant Events Right: All right, so let me start by actually, talking a little bit… so they asked me to sort of do a little bit of a survey of, like, sort of, like, what is it known for, dispersive feed is, and sort of what I call these measures of…
Courant Events Right: fields, but the… so let me start by saying, okay, so the… the first thing that I want to do is to sort of do a little bit of a review of Morgan's ideas.
Courant Events Right: who started this, say, in the context of the Gibbs measure for 2D periodic, MLS.
Courant Events Right: And then it's kind of like I want to, discuss a little bit, the work that we did more recently, with, Yu Deng and HGN Yue, that somehow allow us to extend, these results for, in 2D for any old.
Courant Events Right: power. And then, sort of, like, try to develop a theory for the dispersive context.
Courant Events Right: That is kind of, like, gives, the local reposeness for any dimension and any power, the linearity says for Schrodinger, in what we call the full subcritical branch, which is kind of, like, sort of this black box here, so local pulseness that, were obtained in the stochastic… in the singular stochastic PDE,
Courant Events Right: By marking structures, and I will talk a little bit about normalizations later, because we don't do that. And then in passing, I'm going to mention… sorry about Bjorn, because I'm very fond of this paper, with Bjorn and, and Haitian and Yuden, in which we did the… what we call the hyperbolic fee for 3, which is essentially the invariance of the 3D Gibbs measures under the flow of the cubic
Courant Events Right: A sort of nonlinear way of grief.
Courant Events Right: And,
Courant Events Right: I won't have time to sort of, like, say much about this, so… but I want to mention a couple of things.
Courant Events Right: So, the first two slides are… I learned them from Bjorn, because I always start very high, and so it's like he tells me to start… he starts very low, so this… the first two slides are from him. So, so everybody knows, let's just…
Courant Events Right: set some ideas that it is sort of, like, basic ODs, if you have some physical systems with some, some forcing acting on it, so you have a potential energy.
Courant Events Right: with some nonlinear interactions, then you can actually, from there, you can construct the Hamiltonian ODE,
Courant Events Right: Or if you have some fluctuations of random forces, you can actually construct a large amount of SDE.
Courant Events Right: And the point is that no matter which of these two ways you go, for both of them, you essentially obtain, essentially, the same, gist measures, which you can prove that
Courant Events Right: is invariant, so in the context of Hamiltonian of these, because of the conservation of Hamiltonian and Leo's theorem, and in the context of, SDEs, you have to use equipot, but essentially, you have to do that.
Courant Events Right: And so, of course, we… having this in mind, we go to infinite dimensions.
Courant Events Right: And so, I don't see anything, so… So, we start with a field. So, I'm going to always to be in the periodic case. P here is going to be always odd, bigger or equal than 3, and we have here, like, an energy functional.
Courant Events Right: Which I call H of phi, which is given by, sort of, the mass, the gradient… oops, sorry. The mass of the gradient, and, our potential energy now is going to be phi to the P, P plus 1. So, P plus 1 is always even,
Courant Events Right: And, and this is kind of like a positive, coercive, a positive definite functional, okay? So formally, to this… to this, taking into account what we just said for ODEs, you can associate
Courant Events Right: you know, formally, sort of like a Gibbs measure. The way that it's written is total nonsense, because it's kind of like, all these three terms don't make any sense. But there is a way of constructing. So my starting point is, I'm not… I'm not… it's not… I'm going to assume that the measure is given to me.
Courant Events Right: And, what we're trying to understand is what are the dynamics
Courant Events Right: under the flows of this measure and under the flows. And though in the picture that we just saw.
Courant Events Right: So when we go into the Hamiltonian ODE side, we can go for, with complex value, fields, which will give you, sort of the non-Nier-Schrodinger equation, and you can go with real value fields, which is a real Hamiltonian, which will give you the nonlinear wave equation, that's that side, and on the other side, you get the stochastic heat equation.
Courant Events Right: And so, it's the same measure, but you can… the dynamics are different depending which flows you take. So, the case of the heat is very well understood, so my talk is about the A and B.
Courant Events Right: Okay, so let me just go on this table, which we have actually in the paper of the wave with Bjorn, which is a little bit the history of what happened with all of these things. So the first column has the measure.
Courant Events Right: In dimensions 1, 2, 3, and above. So it's… that… that is about the existence of the construction of the measure. The second one is about the… the second, the third, so heat wave of Schrodinger is about the dynamics.
Courant Events Right: And so, the story starts, I think with Nelson in the 60s, in which the measure, he constructed, I guess, the measure in dimension 1 and 2.
Courant Events Right: And then, it continues, sort of, in the 73 with Liman Jaffe, who constructed in 3D the measure, but only when P is 3. So in dimensions 1 and 2, you can construct it for any P0, bigger or equal than 3, and in dimension 3, only when P is 3.
Courant Events Right: And then, you have, in the 80s, the results of, Einzmann and, and Frollig.
Courant Events Right: Who proved that you have, sort of, marginal triviality in dimensions bigger or equal than 5, and then in recently, in 2019, this is Einstman and Duminel-Cope who prove also marginal triviality 4 in dimensions 4. This is for the cubic, so it's fee for 4.
Courant Events Right: And so how about the dynamics? So the story starts with the wave, and starts with Freelander in 85, who proved that for any power in dimension 1, the measure remains invariant.
Courant Events Right: So you have a global flow, so you have to have a global… global solution, so you have a global solution, and then, for… for the… for data that is in the… that has… gives you data, that you're in the support of the measure.
Courant Events Right: And, and he proved that you have invariance. Then, it follows Vayuata in 87, who proved the same thing in dimensions 1 for, for the heat.
Courant Events Right: And then, essentially, nothing happened for about 10 years.
Courant Events Right: until 1994, in which Burguin proved that under the flow of the NLS for any odd power nonlinearity, as long as you are defocusing, you have also the invariance of the measure for NLS.
Courant Events Right: And it's… it's actually a little bit surprising that the… the… the way in which things happen, because they happen first in the dispersive context, and then, in the… in the, sort of, in the… in the heat con… at least in dimensions, too.
Courant Events Right: So the next, the next…
Courant Events Right: Results came, and this is a seminar paper of Bugin.
Courant Events Right: very important paper in 1996, that in 2D, for the cubic NLS defocusing, he proved that, indeed, the measure is also invariant.
Courant Events Right: Okay, and we're going to talk about… this is the paper that I want to stop a little bit about. And then, this is in his Park City Lecture notes. He proved also for the wave equation any power, that you are invited. So here is for any P. For the wave, it's for any P, but in… for… for… for Schrodinger, it's only for P equals to 3.
Courant Events Right: Okay? And then after that, what happened took essentially a few more years until 2003, which the Prato de Bush proved that in 2D, for the heat equation and any power, you also have
Courant Events Right: the invariance of the measure under the flow. And essentially, you know, we will talk about this, there is a sort of a connection between the method, this linear, nonlinear decompositions that Burgain and that Prato de Bush used, but there's some differences.
Courant Events Right: And then nothing happened for about 10 years until, the work of Martin Hare, who was the breakthrough in Dimension 3, that allowed to prove that,
Courant Events Right: okay, you have the local theory, and then it's also the paper of Martin here with Mattesky and Murat with Weber for the global part, in which you essentially have here that you have also the invariance of the Gibbs measure, and then the flow
Courant Events Right: Of the cubic, say, stochastic heat.
Courant Events Right: Okay, in 3D.
Courant Events Right: And then after that, it took a few years, but essentially, with Yudeng and Haiti and Yue, we sort of… we took this question in 2D, in which the measure exists for any P, so why P equals to 3? Why not quintic? And so something like the Quintic NLS is very natural, because it shows up when
Courant Events Right: when you do, sort of, false Einstein condensates, and you take ternary collisions, the question that you see is the quintic, you know, if you do binary collision, you see the cubic, but if you do ternary, you see the quintic. So, it's relevant.
Courant Events Right: And, so we retook that question, and sort of we were able to prove, and this is one of the things I'm going to talk about, that, actually for any P odd, you have also the invariance of the measure, okay?
Courant Events Right: And then after that, in 2022, with Bjorn, is we proved that in dimension 3,
Courant Events Right: and P equals to 3 for the cubic nonlinear wave equation, the measure is also invariant, and this is what we call… I don't know who calls it the hyperbolic P43. Maybe… is that yours, I don't know who put this name. Okay, all right.
Courant Events Right: So, and that came in 2022. So, this is essentially the picture that took… take us essentially from 1960 all the way to 2022, and this is what is known, this is, like, the global picture of at least what I know.
Courant Events Right: about, this, TP plus 1D models in the periodic case, and there is, three different.
Courant Events Right: flows.
Courant Events Right: Okay, so this doesn't mean that you cannot build Gibbs measures for other flow, for other equations, right? I mean, if you have a Hamiltonian equation, you can always build one, but let's just stick to these ones, okay? All right, so let me actually set up some ideas. So we take Schrodinger, and I take it… yes?
Courant Events Right: Yes, of course.
Courant Events Right: Here, you… the bottom two lines, you call red. Oh, because then here you have marginal triviality, there is no… there's only the Gaussian measure that is… you have, so you can ask dynamical questions, but…
Courant Events Right: I'm just sticking to the gifts measure. So, yes, you could, you could ask some dynamic questions, but I'm not going there. Yes, thank you. Thank you.
Courant Events Right: You know, for businesses, it would be very complicated to prove that it converges when you start from some special… some initial condition that it converges with the measure. And when you have the measure, the…
Courant Events Right: For academic version.
Courant Events Right: Not involved this measure soil.
Courant Events Right: What is the problem with Z2?
Courant Events Right: Why is it a problem for you? Well, the problem is that the first thing that you need to prove is that for data that isn't support of these measures, and in dimensions 2 and larger, the data are distribution that are very raw.
Courant Events Right: That you have a solution at all. The local solution, I'm sure the probability that if I take, say, data in the support of this measure, that you have a solution for some finite time.
Courant Events Right: Then it's a question of paying for that final emission system, and you make it low, and then how do you go from there? The infinite emission solution, and then after that.
Courant Events Right: you actually prove that the final dimensional measures that are associated to the final dimensional Hamiltonian converge quickly during measurement.
Courant Events Right: So can I ask, what about in input volume and default content? We're getting some work on that, but I'm not talking about clinical.
Courant Events Right: there is some work done, and I guess, Bjorn with Gilbert Fatilani, they did it for the wave, the analog of what Burgain did for Schrodinger, correct? Schrodinger and Eagle of 5.
Courant Events Right: Oh, so you stay in Schrodinger. So Bourgain did it for P equals to 3 on the line, and Bjorn and Staffilani… Brigman and Staffilani did it for P bigger or equal than 5. Okay, still Schrodinger. Yeah, okay.
Courant Events Right: I don't know anymore about this.
Courant Events Right: But, I mean, for heat, of course, but waiver-Schrodinger is very different, vinyl, yeah.
Courant Events Right: All right, so… so here we have the Schrodinger equation, and this is the defocusing equation, and and this is important. So the Hamiltonian here is, so the mass… that means the L2 norm and the Hamiltonian are conserved quantities of time, so these are constants of time.
Courant Events Right: And the Hamiltonian is positive definite, so you have a sign.
Courant Events Right: And, and so, this equation, you can actually do it by hand, by rescaling your variables, your X and D, when you are on RD.
Courant Events Right: But if not, you can… heuristically, you can also do it by understanding scaling is about, how the high frequencies of your solution interact
Courant Events Right: When you have two high frequencies that interact and give you a high frequency, then from those heuristics, you can also guess what your deterministic scaling is, which for the equation, the deterministic scaling is the dimension over 2 minus 2 over P minus 1.
Courant Events Right: So, let's stick with the, say, dimension is 2 and P is 3. In the cubic case, in two dimensions, the deterministic scaling is L2.
Courant Events Right: Okay?
Courant Events Right: Now, what is amazing is that the biggest theorem there is, deterministically, about this Schrodinger equation on the torus
Courant Events Right: Is that, provided this number is non-negative.
Courant Events Right: And provided that you take data that is strictly bigger than that, so that your data has to be strictly bigger than the… than this, so if you are in cubic 2D, the data has to be strictly bigger than zero. You cannot take that in L2.
Courant Events Right: Then, the equation is locally well-posed.
Courant Events Right: But nobody knows.
Courant Events Right: It's amazing, but nobody knows. If you take data in L2,
Courant Events Right: whether this equation has deterministically any local solutions, okay? But…
Courant Events Right: Can you speak up? Sorry, I think there is something recent.
Courant Events Right: Yeah, that is… but it's not quite this. It's a little different. The setting of Stratler, yes, but it's not quite the full result. I heard that, though, not talk later. It's not quite… it's not quite the full result, yeah. I mean, they have something, but it's not… it's…
Courant Events Right: It's not the full result.
Courant Events Right: So.
Courant Events Right: As far as I understand. And you have some impulseness when S is strictly, less than the critical one, and this is,
Courant Events Right: Okay, some work.
Courant Events Right: So, anyway, so, we are going to be interested in random data, because, we are going to be studying the Gibbs measure, so we start with, sort of, like, a random variable, which is formally given, it's the law of which is formally given by a Gaussian measure. And so here, these are, Gaussian… these are Gaussians.
Courant Events Right: And here I take a power alpha, say alpha. This is important that you remember, because it's going to appear, later. And so, so, so, so alpha is, say, S,
Courant Events Right: What is it?
Courant Events Right: with the D. So, as dimension over 2,
Courant Events Right: Okay, or X is the same as alpha minus dimension for 2. So,
Courant Events Right: So you take, random data in this fashion, so these are IID, you can take, let's say you take Gaussian, random, IID random variables.
Courant Events Right: And so when alpha is 1, this is very special because this corresponds to, to, to sort of the statistical ensemble of a Gibbs measure.
Courant Events Right: Okay, at least in dimensions 1 and 2, we can talk about dimensions 3 later. And what I want to point out now here is that, you see, if you are in dimensions 1, and alpha is 1, then your measure is going to be… so the regularity of the measure, which is given by the Gaussian one.
Courant Events Right: It's actually supported on functions, which lie in HF minus.
Courant Events Right: But if you're in dimensions 2 or 3, where the measure exists, then your, your Gaussian part, your, your, your,
Courant Events Right: Reduction measure is going to be supported on distributions, which in two dimensions are below L2, and in dimension 3, they are below H minus a half. So they are… these are solid spaces, so they are very, very rough.
Courant Events Right: And so, this is one serious problem, okay? Remember that I said that deterministically, the… in L2 for the cubic equation is L2, is the critical deterministically, so here already we are in the supercritical regime.
Courant Events Right: for, solutions of the cubic equation in 2D.
Courant Events Right: So… In principle, we don't expect… I said, you expect, deterministically, you expect ill-poseness.
Courant Events Right: Okay, and… and… okay.
Courant Events Right: So what is it that we were interested in? So, the questions that we were interested in is in the study, you start with this random initial data that is distributed according some canonical law, like the Gaussian, and suppose that you have first independent Fourier coefficients. This happens at least in dimensions 2.
Courant Events Right: And we want to understand how this random structure gets transported under the flow of the dispersed PDE. So what happens? You feed in, you have an equation that is deterministic, you have data that is random, you feed in this into the nonlinear flow, and you want to see what happens.
Courant Events Right: In other words, how is this data transported under this flow? What is the optimal regime where solutions exist and are unique?
Courant Events Right: Or surely, at least locally in time.
Courant Events Right: Can you describe the solution in terms of the random structure? In other words, not just, do they exist, but can you give me a description of how do they look like? I mean, the theory of relative structure, that's what relationship structures does. They give you a description, a very precise description of the solution. We are asking the same question.
Courant Events Right: At least for short times.
Courant Events Right: And if there is formally, at least formally defined, if there is a Gibbs measure associated to this, can we justify the invariants? Can we prove that they are invariant? Which will come after we prove that we have global solutions.
Courant Events Right: Okay, so let me actually talk a little bit about what is for us the threshold. So this goes to the first question. What is the optimal regime in which solutions are actually expected to exist when you take random data for local time?
Courant Events Right: And so the first thing that we want to… that we understood is that, you see, the deterministic scaling has nothing to do with this problem, because the deterministic scaling, when you do it.
Courant Events Right: does not see the central limit theorem, right? So it doesn't see anything probabilistically, so it's not the… it's not the true… it shouldn't be the true threshold.
Courant Events Right: And so to find the true threshold, we actually essentially will do the same calculation that you will do in the deterministic case, but instead of putting a 1 here, we put the Gaussian. So, suppose that you take data, so you take a random variable, so they are all supported at the same frequency.
Courant Events Right: they are all sort of like n to the minus… this… I had here before a K to the alpha, now they're all… K is like n, so that comes out here as n to the minus alpha. Then you have your random variables, and that's the Fourier series. So you take data like this.
Courant Events Right: And then you take the linear evolution of that, so the semi-group for Schrodinger is e to the iT laplacian. On the Fourier side, you will have e to the iT mod k squared.
Courant Events Right: And… and essentially, this is going to have, unusual unit size in HS, the same S. S and Alpha related in the same way.
Courant Events Right: And so, what is it that you want? If you're going to try to prove local well-poseness, okay, the linear evolution is going to be well-defined locally in time, obviously, but what you want is that if you take, say, the second iterate, so in other words, you take the… you put on the right-hand side the linear evolution, say, the cubic one.
Courant Events Right: No, I put anything, any power. So this is the linear revolution.
Courant Events Right: So, if this is going to be locally webpost, what it should happen is that for fixed time.
Courant Events Right: and say fix any fixed here, then the second iterate should remain in the same space you started with. So if you start in HS, you expect that the second iterate, this, should stay in the same space.
Courant Events Right: So let's try to find out when does it stay in the same space.
Courant Events Right: Okay, so when does it stay in the same space? So, okay, we are working on the Fourier side, because this is the best thing for NLS and this per CPD. You take the Fourier transform, you have a cubic P power, so you're taking the Fourier transform of a product, you get a convolution, so you write down what that is.
Courant Events Right: And essentially, okay, so in this… in these pictures, so the N to the minus p alpha is because you take p powers of this n, so you get n to the minus p alpha.
Courant Events Right: And then you get here, which I call it a base tensor, this is deterministic, you get here the characteristic function of this set. So what is this set? This set is deterministic, it's just telling you that you took the Fourier transform of a multilinear expression, of a p power of u. So the first one, the first equation tells you that it's a convolution.
Courant Events Right: And the second one comes, this is what we call the resonant equation, comes because, you know, you have the linear evolution of that, and you took a Fourier transform, so then you have this quadratic relationship among the frequencies of each of the U's. You have that K1 squared minus… so the plus minus is because in Schrodinger, you have solutions at complex value, so every time that you see
Courant Events Right: So P-1 is even, so this is a U times a UR, right?
Courant Events Right: So, divided by 2. So, so you have a quadratic equation. Now, because you see a quadratic equation here, the first thing that you, you, you, you understand is that
Courant Events Right: something that is non-trivial, that has to do with counting integer lattice points enters. Something that is beyond just a geometric problem or a dimension counting. And so let me explain that. So if I want to understand what's the size of this theory coefficient.
Courant Events Right: Then, for… by the large deviation estimate, you know, materialized deviation estimate, you know that then this… this in absolute value is controlled by the little l2 norm of this… of this, coefficient.
Courant Events Right: So, but a little nuances coefficient is simply the cardinality of this set to the power 1 half. That's the one half from the central limit period, okay? So, if you have here 1, 1, which is the deterministic case, you wouldn't have this 1 half
Courant Events Right: And this same calculation will recover the deterministic scaling. Okay, so now we're doing it
Courant Events Right: probabilistically, right? So let me tell you where this number… why is it that the cardinality of this set
Courant Events Right: is this number in blue, which is ND times t minus 2 plus D minus 2 plus epsilon. Well, the reason is because you need to count how many integers at these points you have, say, in the quadratic equation, so what you do… okay, K is fixed, omega is some constant that is fixed, and so what you do is you count to full dimension.
Courant Events Right: say, K2 through KT. So that gives you
Courant Events Right: to full dimension, that gives you n to the dimension times P minus 2. So that's this number. That's just dimension counting.
Courant Events Right: And then you have to… now you have an equation that essentially reads K1 squared equals to a constant.
Courant Events Right: So what do you do? You count the last D-2 coordinates, so you are in ZD,
Courant Events Right: Right? K is an integer in ZD. You count the last D-2 coordinates to full dimension. That gives you n to the D minus 2,
Courant Events Right: And then, essentially, what you have left is the equation of a circle. You have the first two coordinates of K1, so you have K1 in X and K1 in Y,
Courant Events Right: That are satisfying the equation of a circle.
Courant Events Right: And this is where the number theory comes in, because now you have to use the sphere counting estimate that tells you, essentially, that the number of integer lattice points that lie on this circle is, like, n to the epsilon. It's smaller than n to the epsilon, and that's where this plus comes in.
Courant Events Right: Okay? And that, that, that sphere counting, which you can do with higher dimensions, and it's, like, n to the epsilon times n to the D minus 2, comes, you know, it's related to this,
Courant Events Right: device or estimates, and so on. So the omega is just a constant, okay? So, you can slowly show me. There is a little bit of… I… I'm ignoring the time variable, so there is a little bit of facilitance here that has to do with time, and I'm ignoring it. So, just think of this, ignore it.
Courant Events Right: So think of omega… think of omega for a second to be, like, zero.
Courant Events Right: We're, like, one, okay? So…
Courant Events Right: I mean, there is an integration that has to do with time in omega that I'm not… I'm not doing. Yeah, yeah, there is a… there is… I mean, there is a… there is a log there loss that I'm not talking about. But, you know, there's already another log lost, on my part, another end to be expert here, so it's definitely… there is an integral of 1 over omega.
Courant Events Right: like, visual, another log of n, which I'm…
Courant Events Right: Not dealing with it, so I'm not taking time into account. You know, time is short, we can ignore it.
Courant Events Right: So, now you ask the question, okay, now you have this bound.
Courant Events Right: And as I said, if instead of Gaussians you have one, then that would be essentially the deterministic scaling. And so now you say, when is this bonded in the sublet space? And the answer is, if and only if
Courant Events Right: The S that you have, that S, is strictly bigger than minus 1 over P minus 1.
Courant Events Right: So, independent of the dimension.
Courant Events Right: Right? For any power P, in order to expect… I mean, the best that you… oops, what happened?
Courant Events Right: The best that you can expect in local webpulseness is to reach the threshold of minus 1 over P minus 1, and that's the best… that doesn't mean that you will reach it, but that's the best that you can expect for, Schrodinger.
Courant Events Right: And our result, actually, we actually prove it. We prove that you can reach it. I mean, that's actually some… one of the things I'm going to talk about. And so you can do the similar type of calculations for other equations, so you can actually repeat this for the stochastic heat equation, and the number that you get in this probabilistic scaling notion for the heat is exactly the same as the one that you obtained Martin computed, which, when P
Courant Events Right: Phase 3 for the cubic equation gives you minus 1.
Courant Events Right: Okay? Which is what…
Courant Events Right: you expect. And in the case of the wave, it gives you… when P is… when P is 3, it gives you minus 3 quarters, so the wave is kind of, like, in between,
Courant Events Right: Schrodinger and… and heat. So, in heat, you have minus 1. In Schrodinger, you have minus and half.
Courant Events Right: And, in the weight, you would have minus 3 quarters, say, for P equals to 3, okay? And for other piece, okay, so somehow, this is in between. And that's actually, consistent with the fact that for Schrodinger, you would never have any regularization
Courant Events Right: So that your Wuhanan, so the provider doesn't give you any regularization.
Courant Events Right: The weight gives you one derivative, essentially, and heat gives you two, but Schrodinger gives you zero.
Courant Events Right: Okay, so that's actually a little bit coherent. So, okay, so as I said, this is just the guiding principle. That doesn't mean that you will always reach it.
Courant Events Right: And in many cases, you cannot… the wave is one of them, because you have problems with, not just high, high into high interactions, you have also problems when two waves that are at a high frequency interact and give you a low frequency, in which case you cannot even, you know, you have problems.
Courant Events Right: And of course, these calculations depend on the global geometry and on the particular dispersive relation, or whatever you have it, okay? So, so, okay. Any questions?
Courant Events Right: All right.
Courant Events Right: So, okay, so let me talk about Burgain.
Courant Events Right: And so, of course, this is what I was telling you before. Before, you start with some finite-dimensional approximation to your equation, and because we are doing things on Fourier, n here is the dyadic number, so N is something… is a frequency, so it's kind of like something that is
Courant Events Right: In here, okay? And it seems, if you are used to the parabolic case, what you should think is that n is like your 1 over epsilon. When you do the modifications.
Courant Events Right: of your equation, with, sort of approximations to the identity, the… the 1 over epsilon that you see in the parabolic case is the same as the n. So… so, okay.
Courant Events Right: So then you… you take… so what is it that… what is this? You take your data, you truncate it, up to frequencies no larger than n.
Courant Events Right: And you can take a sharp cutoff, or a smooth cutoff, it doesn't matter, here. And then UN is just the solution of this equation with this data, okay? And we take the weak order of this.
Courant Events Right: Which is the usual renormalization that you do.
Courant Events Right: where, you know, you have infinite mass here, so you need to somehow subtract this term from here. Otherwise, you cannot make sense of the nonlinearity, right? The nonlinearity doesn't make any sense as a distribution unless you renormalize it properly.
Courant Events Right: So now, so local well-poseness means whatever you think it means. It means that somehow outside some set, there is a small time, and some small set, probability very small, such that if omega is outside that set.
Courant Events Right: then the sequence of solutions here, UN, will converge to a solution U in the continuous functions.
Courant Events Right: With values in HS minus, which is where the random data is, where F omega… so you see the solution and the data is in the same space.
Courant Events Right: H is minus.
Courant Events Right: Sorry, here, here you're going to be confused, because I didn't put the alpha.
Courant Events Right: If you put alpha, then the S is the alpha. If you don't put the alpha, because I'm going to talk about Burgain, I mean, 2D for the cubic, so for the measure, the alpha is 1, so this S should be 0, 0 minus, okay? But if you put alpha, then the S is like that.
Courant Events Right: Okay, and so this is well-defined, the nonlinearity is well-defined, in the limit, in the sense of distributions, and the U solves the equation in the sense of distribution. So, this is what you, what you expect this should be. So what is that work again did?
Courant Events Right: So, he considered the cubic equation in 2D,
Courant Events Right: And he considered the geese measure.
Courant Events Right: And the Gibbs measure in 2D, as I told you, is supported just below L2. So it's supported… the Gibbs measure is supported in zero mind.
Courant Events Right: The probabilistic… the probabilistic scaling… the deterministic scaling is L2, so this is super critical with respect to the deterministic scaling, but it's actually subcritical with respect to the probabilistic scaling, which in the cubic case is minus a half.
Courant Events Right: Okay? So, what happens is that this is not a perturbation problem. If you want to solve this equation by a fixed-point argument, you have to do a fixed-point argument in a space which is super critical, where you cannot prove it. It's not a… you cannot do a fixed-point argument in 0 minus, below and 2, okay? It's not going… you're never going to be able to close your S.
Courant Events Right: So what he does is he does this linear, nonlinear decomposition, which some of you know also as what the Prato de Bush did, in which you sort of, like, separate the linear evolution of the random, which is the roughest term.
Courant Events Right: But it's random. You know precisely the random structure of that term, and it's the worst, you separate it, and then you solve for… you find the difference equation. You solve V solves the equation for U minus that, you find the equation for that, and then you try to solve a fixed point argument, you try to find V in a smoother space.
Courant Events Right: than L2. So, in other words, you do a fixed point argument for V,
Courant Events Right: In some sublet space of regularity strictly bigger than zero.
Courant Events Right: And so when you do that, this is the difference equation.
Courant Events Right: In order to prove that, this is like the cubic one, you have to understand the interactions of things that are linear evolutions against themselves. So, say you have three randoms against, you know, random, random, random.
Courant Events Right: you have three, deterministic ones, you treat V as a deterministic… it's not deterministic, but you treat it as deterministic, so you have three cubics, and then you have interactions of both.
Courant Events Right: And so what happens is that when everything is random.
Courant Events Right: you can actually… what saves you is exactly that you can use large deviation estimates, like we did in the scaling. When everything is deterministic, because you are trying to put it above… above L2,
Courant Events Right: you have the… you use the local… the deterministic local theory that you already know, so you have the estimates for when everything is big. And the problem that you have is when you have a combination of these things.
Courant Events Right: Because when you have a combination of these things.
Courant Events Right: then, you're in trouble. You cannot do the large variation estimates, and so the three main ingredients here
Courant Events Right: is, okay, you have multilinear large deviations, we talked about that. You have to use a little bit of analytic number theory, this integer-lattice counting instruments, we talk about that in the context of the scaling. But there's actually one of the most important ideas in Bourguin. It's not just this linear and nonlinear decomposition, it's this idea that when you have no
Courant Events Right: gain of regularity from your duvelle.
Courant Events Right: How are you going to exploit randomness?
Courant Events Right: And the answer to that is via these random matrix estimates, or these… these TP star arguments that he does, which is just a random matrix estimate. And that's actually… for me, that's the most important idea in this… in this work, is that
Courant Events Right: You cannot prove anything for Schrodinger unless somehow you exploit randomness using these random matrix estimates.
Courant Events Right: Okay, so let me give you an example.
Courant Events Right: Of what that means. So if you have something so… okay, so as I said, you have to do estimates with random, random, random, random, deterministic, deterministic, all mixtures, and the frequencies, you have to take large frequency, or small frequency, all possible frequencies everywhere. So suppose that you have one case, this one, which is DRR.
Courant Events Right: So you have deterministic random non-random. Suppose the deterministic function, the V, is in a high frequency, and the other two are in a low frequency. So what does it mean that they are in a low frequency? It means that the decay that you have here is useless, because this could be one.
Courant Events Right: So that means that this doesn't help you.
Courant Events Right: And you need to control this in L2. Now, you know nothing about the independence between this and that, so you cannot use any large deviation estimates
Courant Events Right: to do this, to compute this, Helder doesn't work, because as I said, you don't have any decay, so in other words, you have no way to sort of compensate or to offset the growth that you get from the counting estimates of this set. So, if these were large.
Courant Events Right: You could have said
Courant Events Right: the number that you get here, when the Ts are low… There is nothing you can do.
Courant Events Right: So, the correct way to exploit this independence somehow, because you do have some independence, and you have no smoothing whatsoever, is to use this. What he does is he considers this matrix, which I call G, or sigma KK1,
Courant Events Right: Which is defined as, okay, N2 and M3 are the size of the frequencies of K2, K3, and then you have the sum as a bar with the two Gaussians, okay? Forget the V. And you think of this expression as being this matrix being applied to the vector V in K1.
Courant Events Right: And now what you want to do is you want to understand, the operator norm, the L2 operator norm of this matrix.
Courant Events Right: And the way to do so is by what we call a TT star argument. You look at the operator, the matrix, and you look at the adjoint of this matrix.
Courant Events Right: And then the L2 operator norm of G is the same as the square root of the operator norm of GG star. Now, when you do GG star, you are going to double the number of Gaussians, you are going to have to be very careful with what is,
Courant Events Right: independent, what is not, so the proof of this is very long, several, several pages, there's a lot of analysis you have to do, box localization, so it's quite complicated, but in the end.
Courant Events Right: he gets the right decay for this object to close the SD.
Courant Events Right: Without this, you're dead in the water, you cannot do anything. In the wave case, this doesn't happen. You can get away with streetcars, essentially, and sort of, like, you know, counting estimates that come from geometry.
Courant Events Right: and multilinear, large deviation estimates. You don't really need to bring this TT star argument into play, but for Schrodinger, you absolutely need to, okay? And this is actually something that is very important, and I want to emphasize this because it's not just about the ANZAT,
Courant Events Right: That is linear and nonlinear. It's not just about that. Bourgain's…
Courant Events Right: Sending an idea is to bring in this, random matrix estimate to bear to compensate for the fact that Schrodinger has no smoothing whatsoever.
Courant Events Right: All right, so, after, okay, so after you have proved the local well pauses with this, there is something that is called Burgain's globalization argument that I'm not going to go through that, but I put the slide so that if you want to read it afterwards, because you're going to have the slides, you can read it on your own.
Courant Events Right: And essentially, this is what the question that you asked me, is how do you do the whole procedure, and so it's explained here, step by step.
Courant Events Right: And, let me just not, not, not say anything, because I'm already, like, super behind.
Courant Events Right: But, you know, this is pretty well understood how to do it. Once you have the local well… the probabilistic local well pauses, there is
Courant Events Right: this argument that tells you, okay, how do you get the global well-poseness for the final dimensional approximation, and also for the solution itself. And it's not obvious, okay? To go for the finite dimensional approximation.
Courant Events Right: from local to global, you need to exploit the fact that the final dimensional approximation is also Hamiltonian, and so if you look at the Gibbs measure for that, then that's invariant, and you use that invariance, or the final dimensional one, as a conservation law.
Courant Events Right: To extend the local solution of the finer dimensional approximation to a global one.
Courant Events Right: And then, you cannot compare the infinite-dimensional solution with the finite-dimensional solution globally, because you know nothing about you.
Courant Events Right: So there is a very intricate and delicate argument with an intermediate system, which is the infinite equation with data, and you have to do this game, which is kind of like this commutator and stability arguments and their small perturbations, that allow you to walk
Courant Events Right: the infinite solution next to the finite solution, one time step at a time, okay? So it's not trivial, but it's… it's understood.
Courant Events Right: So, in the end, you get that the solutions of the infinite-dimensional system are almost surely global.
Courant Events Right: And once you have that, you prove the invariance of the infinite-dimensional measure. I say, that's the icing in the cake. You've proved it at the very end.
Courant Events Right: Okay, so then let me ask, let me ask this question. I mean, okay, so…
Courant Events Right: What if you are not interested in the Gibbs measure? So, Burgain, this is what happened. This is what was non-deterministically, this is what Borgain did. He went below L2 with the… with this, this method. He proved the local repositis here.
Courant Events Right: But we know that even for the cubic equation, the parallelistic standing is minus a half. So what happens if you go below the support of the Gibbs measure, say, in 2P.
Courant Events Right: What happens is that this method doesn't work anymore. It breaks down, okay? In other words, Bourgain's method fails as soon as you leave the support of the measure. As soon as you become rougher, you're dead, okay? And the idea, the reason is because
Courant Events Right: if you do this nonlinear decomposition, say, you need to put this V, or, like, in this case, right, you need to put this V in a space where you know your deterministic theory. So, but the problem is that no matter what, if you start below this.
Courant Events Right: So, Burgain, what he did with the TT star is gain
Courant Events Right: Enough regularity to go from below level 2 to a level 2.
Courant Events Right: But if I stop here.
Courant Events Right: I need to gain all of this regularity. I need to somehow put this V above L2 when I start below… much lower than this.
Courant Events Right: And so… and what happens is that when you do this, there is no way
Courant Events Right: This V is always below L2. You can never put it above L2, there's no way of closing, okay? And the same problem happens when you take any power bigger than 3, when you take powers bigger than… bigger or equal than 5, and you stay with the two-dimension, and you say, why did Burgett only prove it
Courant Events Right: for the cubic, when on the wave, he put proof for any power. It's the same problem. It's because the cube… if you take power of 5, let's suppose that power is 5, so the quintic equation in 2D has deterministic scaling at H,
Courant Events Right: One half.
Courant Events Right: So, if I start.
Courant Events Right: sort of, like, with random data, say, in the support of the Gibbs measure, corresponding to power of 5, I will be just below L2.
Courant Events Right: And what that means is that now.
Courant Events Right: instead of going from below L2 to a little bit above L2, which is the cubic case, you have to go from below L2, which is from 0 minor, all the way above H. So you have to gain not a little bit of regularity, you have to gain a lot of regularity.
Courant Events Right: And if P is bigger than 5, then you have to actually go from 0 minus all the way here.
Courant Events Right: all the way to above 1 minus 2 over P minus 1. So, how do you do that? I mean, you know, how do you do that? So, this doesn't work, okay? And so, you cannot actually do this using just the regularity.
Courant Events Right: called the V, you have to really understand the random structure. You have to stop pretending that the V is deterministic, and you really have to get your hands dirty and understand the random structure of this remainder, okay?
Courant Events Right: And so, the first question that you do is where this bad regularity comes from, and this is similar to what happened in the heat, that the bad regularity comes from the interactions when you have high frequencies in the linear evolution of random.
Courant Events Right: And low frequencies in the other terms, in the… here, I just wrote something cubic.
Courant Events Right: So it's the same kind of problem that you have in the heat, except, okay, in the heat you have problems when you have a quadratic term on the noise, and then the low. So you have, like, Z squared, and so that here, okay, here you have it linear in here, and it's, again, high, low into high interactions.
Courant Events Right: The problem is that in the paracontrol theory, for example, of Govinelli, it's enough to remove this term once, which is essentially, this term is like a paraproduct. And once you have changed your answer, in the sense that you do, okay, you do linear evolution of random, you separate this term.
Courant Events Right: then immediately what remains becomes regular, and you can close. Thanks to the fact that the heat operator gains you two derivatives.
Courant Events Right: In Schrodinger, this never happens.
Courant Events Right: It doesn't matter, I removed this, I stay…
Courant Events Right: I stay below L2. I remove this in the second, you know, du Hamel, whatever. I keep expanding, I remove it in… I always stay below L2.
Courant Events Right: So, what you need to do to solve this problem is you need to remove them, all of them, in the whole Du-Homel expansion. Like, in the whole expansion, you have to remove them in every single term. So here, I tried to explain a little bit why the paracontrol doesn't work. So let me skip it, because it's too detailed.
Courant Events Right: But the bottom line of this is that you… you really need to understand the random structure of the remainder, and it's not enough to remove… this is, like, this is the paraproduct in which you have high frequencies here and low frequencies there.
Courant Events Right: And here, I tried to explain why this doesn't work.
Courant Events Right: And, and the whole point of solving it for Schrodinger is that, you need to change your point of view and stop thinking
Courant Events Right: that you want to prove a multilinear estimate for this, because you will never be able to prove any multilinear estimate for this that will allow you to close your estimate. It's impossible. So, as I said, because it's not that you have to remove it once or twice, you have to remove it all the time.
Courant Events Right: So, you have to somehow shift a little bit the point of view. This is the Duchamel of, say, high frequencies in the linear evolution of random, and low frequencies in the rest of the powers.
Courant Events Right: And what you have to do is you need to think of this term as an operator that sends, say, W here, that sends this into this term.
Courant Events Right: And the whole point of trying to understand the randomness of the remainder is encoded in proving the right estimates for these operators, which are marking back the proper… the correct L2 operator norm estimates and Hilbert-Schmidt norm estimates for that. And if you can prove that.
Courant Events Right: then you change your answer to something that looks like this, which is that now your answer is
Courant Events Right: the solution, plus
Courant Events Right: This, which is what we call… this is… this is an operator. When you sum in all frequencies, the low frequencies at N, and then overall frequency is N. You sum all of these.
Courant Events Right: This is what we call the random averaging operator, applied to the linear evolution plus remainder. And this is kind of like removing from all the iterations, from all of them, removing the bad terms.
Courant Events Right: And then after you have done that, then this one can be placed almost close to each one, so then you're fine.
Courant Events Right: Okay? And the way that we prove this is, actually, we have to induction… we have to do an induction on frequencies. In other words, we have to prove that, induction in frequencies, you have to set it up so that if you can prove estimates for the low frequencies, then you can prove estimates for the higher frequencies. So the whole thing goes through an argument, which is an induction on frequencies.
Courant Events Right: And so here is some pictures of what's going on.
Courant Events Right: This is the Duchamel.
Courant Events Right: this is the Duchamel, of the… so, by this picture, I mean the DuCamel of the cubic term, where this is the linear evolution of random, and these two are the low frequencies of the nonlinear one, and then you expand again.
Courant Events Right: So you have this picture, which is to Hamel, you expand on the first entry here, times this, so you have 5, and then so on. And the point is that you have to remove all of them. And to remove all of them… Yes.
Courant Events Right: No, we're doing this for any power. I'm doing the picture for the cubic, because the picture is hard enough. I don't know how to draw this picture. The picture is the cubic, but we do it for any P, bigger or equal than 5, okay? I just don't know how to… I mean, I cannot take any responsibility of claim for any pictures, because I didn't do them. They were done by Haitian, or by…
Courant Events Right: Do you want to know by my student, or… I don't know, but not by me. So, QB is… yes, go ahead.
Courant Events Right: Okay, so no, this is actually done for quintic and septic, but, you know, but it's always the same. You only have high on one, and low on all the other ones.
Courant Events Right: And and essentially what you're doing here is, because you're doing an induction on frequencies, you are supposed to, by the frequency argument, you're supposed to have information on the low frequencies on this. So when you solve, you're trying to solve a parallelinear problem.
Courant Events Right: Where this is essentially given to you is data.
Courant Events Right: Okay, and you're suddenly trying to solve for this. So, essentially, your solution, your random marching operator is nothing but, essentially, summing all of this. It's this thing that I call, like a cube.
Courant Events Right: which is simply the inverse is this, is the inverse of this operator applied to the linear. And if you go to the Fourier transform.
Courant Events Right: What this operator is, is a matrix
Courant Events Right: being applied to the Gaussian that corresponds to the linear evolution in high frequencies, this… this is what we call a 1-1 tensor. It's a matrix that carries all the information of the low frequencies
Courant Events Right: And these two objects are independent.
Courant Events Right: See here?
Courant Events Right: I'm going to ignore it, it can be taken as a parameter.
Courant Events Right: Okay, and and the first person who actually made this observation that you have to be very careful about keeping independence of this and this was Bjorn in a previous work for the… for the wave… for the derivative quadratic wave equation, which is a very important idea.
Courant Events Right: Alright, so in the end, this is your answer.
Courant Events Right: And the way that it works is you start, the Gibbs measure is in 2D, it's in 0-. This operator, you prove that it's in H.5 minus, which is still a supercritical space for the deterministic run, but the remainder you can make, put it almost in H1. And this is the… in the Fourier modes, this is how it looks like.
Courant Events Right: Okay, all right, so I'm super late, so I'm going to… nobody's going to ask any questions, because I'm going to take the time.
Courant Events Right: So, no questions.
Courant Events Right: All right, so,
Courant Events Right: Anyway, so this is how we prove… this is what gives us the local well-poseness, probabilistic local well-poseness, and then, essentially, we use Burgett's globalization to prove that the Gibbs measure type… we have global solutions for data that is in the support of the Gibbs measures in 2D for any power.
Courant Events Right: And what is actually remarkable, I think.
Courant Events Right: Is that the decomposition that we get here
Courant Events Right: I mean, this structure of the solution is global.
Courant Events Right: It remains for all times. It's not destroyed. We prove it locally, but then you can actually check
Courant Events Right: That, it remains like this for all time. The same as the linea-linear is also global. This is global.
Courant Events Right: Okay? So… so that's actually quite remarkable.
Courant Events Right: Okay, so more generally.
Courant Events Right: How do you reach the full range? Because I'm still, in this proof of the random managing operators, we still started with data here, and we took any P. Now what I want is to take any P, any dimension, I don't care anymore about the dimension, I want to take any dimension, any P, and see, say, okay, any P. I wrote it here for P equals to 3. If P is larger than 3, this number moves to the right, but okay, let's say…
Courant Events Right: How do you… how do you prove it? How do you cover, say, the local repositis, the probability local reposit in the full subtractical range? You cannot do it just with these random averaging operators. This is what this theory of random tensors that we developed enters.
Courant Events Right: But,
Courant Events Right: this theory gives you, what it does is it keeps on peeling off. It keeps on peeling off from the remainder.
Courant Events Right: the random structures of this. So we peel it to the first order, but now you… we can keep on peeling them off.
Courant Events Right: To any high order we want, and we know exactly, with these random tensors, what is the… we know we can reach the full subtractical, where we also know what is the exact detailed information or representation of the solution, so we get a very detailed information on the propagation of randomness.
Courant Events Right: Okay, and this is actually done in full generality. So I'm going to skip this because
Courant Events Right: I'm sure that you're tired. I need a lot of pictures.
Courant Events Right: But this is where it comes for, this is, this is what we call the, the, the, the, the Q1 tensors. Q measures how many… so, so these are some examples. So, so Q measures how many…
Courant Events Right: linear evolutions you have in the high frequencies, and one is because you're taking… one corresponds to the output frequency, to the K. So this is a 2-1 tensor, so you have two linears going into… there's always a… that one corresponds to the output frequency.
Courant Events Right: Take the gate.
Courant Events Right: And so, you see, when you have something like this for a 2-1 tensor, then the 2-1 tensor is more complicated, right? Because now the 2-1 tensor has a different structure.
Courant Events Right: But again, you view this as a mapping from K1, K2 into K,
Courant Events Right: And again, you can… you have to… you can prove that this carries the information of the low frequencies.
Courant Events Right: And that this is independent of this.
Courant Events Right: Okay, and okay, so we developed a theory here. Okay, this is another example.
Courant Events Right: And so we developed a theory that essentially tells us that on the Fourier modes.
Courant Events Right: All the fully modes of the solutions can be written, can be expanded in terms of multilinear Gaussians.
Courant Events Right: with coefficients, which are our tensors, which are localized in frequency, T is the parameter, and that they remain independent of the Gaussians that you have outside, and we can make the remainder as smooth as you want, and as small little as you want.
Courant Events Right: And this is the theory that allows us to cover the full range. And one thing that is very important here is that
Courant Events Right: You see, in each iteration, you gain something of the order. How much familiarity you gain from here to here, from here to here, you gain something that is comparable to how far you are from the probabilistic scaling. So if you are far from the probabilistic scaling, so if my scaling was minus a half and I was in 0 minus.
Courant Events Right: The random average operator is going to gain 1 half, which is 1 half minus, which is what we proved.
Courant Events Right: But if I get closer and closer to the probabilistic scaling, then you're getting less and less regularity. And so what happens is that, you see, as you reach the probabilistic scaling, which is your critical problem.
Courant Events Right: you gain nothing, and you cannot stop. I can never stop. We can never stop. Alright, so this is what happens with the theory. Let me skip all of this. Let me just say something very important, that is in this theory.
Courant Events Right: So we have an algebraic part and an analytic part, and in the… in the… and again, the proof goes by an induction on frequency and by induction on the length of the expansion.
Courant Events Right: And the ingredients are the same, are large variation estimates, integer-lantic antibodies, and now here we have to put Bourgain's method in steroids. We have to do TT star.
Courant Events Right: to very high order to be able to reach the probabilistic scaling. So we have to take the star many, many, many, many times, to… to prove estimates that allow us to reach
Courant Events Right: this, and there is a selection algorithm that, exploits the… the… optimizes. You have a choice of estimates, and you have to optimize that. So let me skip all of that.
Courant Events Right: And, okay, let me just, to finish, yes, of 2 minutes to finish. So, talk about this. What happened with the wave? So, so, I discussed all of this. What happened on the wave? On the wave, of course, in 3D, you have the problem that, first.
Courant Events Right: You are supported in a very rough space.
Courant Events Right: which is a minus, a half minus. And, now the measure in 3D is not absolutely continuous anymore, respect of your Gaussian.
Courant Events Right: is singular, which means that you have probabilistic dependent Fourier modes, which is a problem.
Courant Events Right: But on the other hand, for the wave, this is still a subcritical problem, because minus and half minus on the wave is still to the… to the… to that side. What is that side, the right?
Courant Events Right: to that side of minus 3 quarters. So, it's still subtractical. And this was the only reason why, when we started to work on this problem, we had any faith that we should be able to solve it, because it was still subtractical, okay?
Courant Events Right: Now, I can explain the proof of this, which has a lot of beautiful ingredients.
Courant Events Right: But let me just mention, to finish.