WEBVTT

1
00:00:00.040 --> 00:00:07.860
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.

2
00:00:08.560 --> 00:00:18.660
Courant Events Right: Thank you. Oh, thank you very much, Bjorn. Thank you to the organizers, especially Nina and Roland. And let me first,

3
00:00:18.780 --> 00:00:33.579
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,

4
00:00:34.020 --> 00:00:35.610
Courant Events Right: Yo, this.

5
00:00:35.740 --> 00:00:42.870
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…

6
00:00:42.930 --> 00:00:56.700
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…

7
00:00:57.090 --> 00:01:07.470
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.

8
00:01:07.780 --> 00:01:14.680
Courant Events Right: who started this, say, in the context of the Gibbs measure for 2D periodic, MLS.

9
00:01:14.850 --> 00:01:30.319
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.

10
00:01:30.320 --> 00:01:38.129
Courant Events Right: power. And then, sort of, like, try to develop a theory for the dispersive context.

11
00:01:38.160 --> 00:01:58.579
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,

12
00:01:58.580 --> 00:02:21.939
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

13
00:02:21.940 --> 00:02:24.080
Courant Events Right: A sort of nonlinear way of grief.

14
00:02:24.320 --> 00:02:25.740
Courant Events Right: And,

15
00:02:26.000 --> 00:02:32.120
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.

16
00:02:32.140 --> 00:02:44.379
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…

17
00:02:44.380 --> 00:02:53.940
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.

18
00:02:54.160 --> 00:03:00.580
Courant Events Right: with some nonlinear interactions, then you can actually, from there, you can construct the Hamiltonian ODE,

19
00:03:00.990 --> 00:03:06.540
Courant Events Right: Or if you have some fluctuations of random forces, you can actually construct a large amount of SDE.

20
00:03:06.720 --> 00:03:18.429
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

21
00:03:18.430 --> 00:03:32.579
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.

22
00:03:32.580 --> 00:03:37.659
Courant Events Right: And so, of course, we… having this in mind, we go to infinite dimensions.

23
00:03:37.860 --> 00:03:53.019
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.

24
00:03:53.020 --> 00:04:08.360
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,

25
00:04:08.390 --> 00:04:20.260
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

26
00:04:20.269 --> 00:04:36.769
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.

27
00:04:37.010 --> 00:04:40.899
Courant Events Right: And, what we're trying to understand is what are the dynamics

28
00:04:40.920 --> 00:04:46.940
Courant Events Right: under the flows of this measure and under the flows. And though in the picture that we just saw.

29
00:04:46.940 --> 00:05:07.129
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.

30
00:05:07.540 --> 00:05:21.090
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.

31
00:05:21.540 --> 00:05:34.700
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.

32
00:05:35.040 --> 00:05:46.700
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.

33
00:05:47.080 --> 00:05:56.819
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.

34
00:05:57.220 --> 00:06:14.040
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.

35
00:06:14.040 --> 00:06:20.170
Courant Events Right: And then, you have, in the 80s, the results of, Einzmann and, and Frollig.

36
00:06:20.170 --> 00:06:35.939
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.

37
00:06:36.500 --> 00:06:51.090
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.

38
00:06:51.090 --> 00:07:05.750
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.

39
00:07:05.750 --> 00:07:15.100
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.

40
00:07:15.420 --> 00:07:18.650
Courant Events Right: And then, essentially, nothing happened for about 10 years.

41
00:07:18.800 --> 00:07:35.329
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.

42
00:07:35.580 --> 00:07:50.879
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.

43
00:07:51.070 --> 00:07:53.280
Courant Events Right: So the next, the next…

44
00:07:53.430 --> 00:07:57.089
Courant Events Right: Results came, and this is a seminar paper of Bugin.

45
00:07:57.460 --> 00:08:06.800
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.

46
00:08:06.910 --> 00:08:26.190
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.

47
00:08:26.750 --> 00:08:41.529
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

48
00:08:41.530 --> 00:08:55.619
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.

49
00:08:55.830 --> 00:09:08.320
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,

50
00:09:08.660 --> 00:09:23.440
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

51
00:09:23.560 --> 00:09:26.100
Courant Events Right: Of the cubic, say, stochastic heat.

52
00:09:26.500 --> 00:09:28.580
Courant Events Right: Okay, in 3D.

53
00:09:28.870 --> 00:09:47.550
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

54
00:09:47.550 --> 00:09:58.410
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.

55
00:09:58.550 --> 00:10:12.690
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?

56
00:10:13.010 --> 00:10:19.939
Courant Events Right: And then after that, in 2022, with Bjorn, is we proved that in dimension 3,

57
00:10:19.940 --> 00:10:34.709
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.

58
00:10:34.860 --> 00:10:49.440
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.

59
00:10:49.440 --> 00:10:58.589
Courant Events Right: about, this, TP plus 1D models in the periodic case, and there is, three different.

60
00:10:58.590 --> 00:11:00.020
Courant Events Right: flows.

61
00:11:01.420 --> 00:11:19.450
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?

62
00:11:19.920 --> 00:11:21.300
Courant Events Right: Yes, of course.

63
00:11:21.640 --> 00:11:34.170
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…

64
00:11:34.170 --> 00:11:43.220
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.

65
00:11:43.220 --> 00:11:54.809
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…

66
00:11:54.820 --> 00:11:56.929
Courant Events Right: For academic version.

67
00:11:57.810 --> 00:11:59.949
Courant Events Right: Not involved this measure soil.

68
00:12:00.350 --> 00:12:02.460
Courant Events Right: What is the problem with Z2?

69
00:12:02.470 --> 00:12:14.580
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.

70
00:12:14.580 --> 00:12:25.370
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.

71
00:12:25.370 --> 00:12:34.009
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.

72
00:12:34.200 --> 00:12:42.129
Courant Events Right: you actually prove that the final dimensional measures that are associated to the final dimensional Hamiltonian converge quickly during measurement.

73
00:12:44.900 --> 00:12:54.320
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.

74
00:12:54.520 --> 00:13:05.710
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.

75
00:13:05.710 --> 00:13:18.529
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.

76
00:13:18.760 --> 00:13:21.189
Courant Events Right: I don't know anymore about this.

77
00:13:21.310 --> 00:13:26.820
Courant Events Right: But, I mean, for heat, of course, but waiver-Schrodinger is very different, vinyl, yeah.

78
00:13:28.220 --> 00:13:45.760
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.

79
00:13:45.760 --> 00:13:50.099
Courant Events Right: And the Hamiltonian is positive definite, so you have a sign.

80
00:13:50.440 --> 00:13:58.820
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.

81
00:13:58.850 --> 00:14:10.249
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

82
00:14:10.250 --> 00:14:25.389
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.

83
00:14:25.450 --> 00:14:35.719
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.

84
00:14:35.950 --> 00:14:36.980
Courant Events Right: Okay?

85
00:14:37.080 --> 00:14:46.289
Courant Events Right: Now, what is amazing is that the biggest theorem there is, deterministically, about this Schrodinger equation on the torus

86
00:14:46.400 --> 00:14:49.799
Courant Events Right: Is that, provided this number is non-negative.

87
00:14:50.060 --> 00:15:04.940
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.

88
00:15:05.110 --> 00:15:08.930
Courant Events Right: Then, the equation is locally well-posed.

89
00:15:09.250 --> 00:15:11.070
Courant Events Right: But nobody knows.

90
00:15:11.330 --> 00:15:15.310
Courant Events Right: It's amazing, but nobody knows. If you take data in L2,

91
00:15:15.490 --> 00:15:21.720
Courant Events Right: whether this equation has deterministically any local solutions, okay? But…

92
00:15:22.370 --> 00:15:25.920
Courant Events Right: Can you speak up? Sorry, I think there is something recent.

93
00:15:25.920 --> 00:15:44.480
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…

94
00:15:44.480 --> 00:15:45.789
Courant Events Right: It's not the full result.

95
00:15:47.070 --> 00:15:48.280
Courant Events Right: So.

96
00:15:48.420 --> 00:15:56.329
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,

97
00:15:56.560 --> 00:15:58.619
Courant Events Right: Okay, some work.

98
00:15:58.620 --> 00:16:17.069
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.

99
00:16:17.140 --> 00:16:29.209
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,

100
00:16:29.310 --> 00:16:30.149
Courant Events Right: What is it?

101
00:16:30.510 --> 00:16:33.770
Courant Events Right: with the D. So, as dimension over 2,

102
00:16:34.430 --> 00:16:40.910
Courant Events Right: Okay, or X is the same as alpha minus dimension for 2. So,

103
00:16:44.130 --> 00:16:53.420
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.

104
00:16:53.520 --> 00:17:02.430
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.

105
00:17:02.480 --> 00:17:21.719
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.

106
00:17:21.720 --> 00:17:27.130
Courant Events Right: It's actually supported on functions, which lie in HF minus.

107
00:17:27.440 --> 00:17:35.370
Courant Events Right: But if you're in dimensions 2 or 3, where the measure exists, then your, your Gaussian part, your, your, your,

108
00:17:35.390 --> 00:17:49.619
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.

109
00:17:49.710 --> 00:18:06.849
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.

110
00:18:06.960 --> 00:18:11.539
Courant Events Right: for, solutions of the cubic equation in 2D.

111
00:18:11.720 --> 00:18:20.810
Courant Events Right: So… In principle, we don't expect… I said, you expect, deterministically, you expect ill-poseness.

112
00:18:21.170 --> 00:18:23.430
Courant Events Right: Okay, and… and… okay.

113
00:18:23.540 --> 00:18:39.759
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.

114
00:18:39.770 --> 00:18:57.829
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.

115
00:18:57.830 --> 00:19:07.079
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?

116
00:19:07.270 --> 00:19:10.710
Courant Events Right: Or surely, at least locally in time.

117
00:19:11.140 --> 00:19:27.280
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.

118
00:19:27.480 --> 00:19:29.150
Courant Events Right: At least for short times.

119
00:19:29.590 --> 00:19:42.469
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.

120
00:19:43.270 --> 00:19:58.629
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?

121
00:19:58.740 --> 00:20:08.339
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.

122
00:20:08.340 --> 00:20:19.399
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.

123
00:20:19.400 --> 00:20:36.209
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.

124
00:20:36.440 --> 00:20:51.249
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.

125
00:20:51.550 --> 00:21:04.570
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.

126
00:21:04.980 --> 00:21:13.769
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.

127
00:21:14.030 --> 00:21:32.050
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.

128
00:21:32.050 --> 00:21:36.119
Courant Events Right: No, I put anything, any power. So this is the linear revolution.

129
00:21:36.180 --> 00:21:41.319
Courant Events Right: So, if this is going to be locally webpost, what it should happen is that for fixed time.

130
00:21:41.360 --> 00:21:56.760
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.

131
00:21:57.060 --> 00:22:00.359
Courant Events Right: So let's try to find out when does it stay in the same space.

132
00:22:00.440 --> 00:22:19.170
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.

133
00:22:19.530 --> 00:22:29.340
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.

134
00:22:29.370 --> 00:22:50.969
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.

135
00:22:50.970 --> 00:23:14.910
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

136
00:23:14.910 --> 00:23:19.849
Courant Events Right: So P-1 is even, so this is a U times a UR, right?

137
00:23:19.850 --> 00:23:30.680
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

138
00:23:31.130 --> 00:23:50.930
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.

139
00:23:51.230 --> 00:24:06.040
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.

140
00:24:06.080 --> 00:24:22.129
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

141
00:24:22.720 --> 00:24:28.369
Courant Events Right: And this same calculation will recover the deterministic scaling. Okay, so now we're doing it

142
00:24:28.520 --> 00:24:34.390
Courant Events Right: probabilistically, right? So let me tell you where this number… why is it that the cardinality of this set

143
00:24:34.550 --> 00:24:54.400
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.

144
00:24:54.510 --> 00:24:59.370
Courant Events Right: say, K2 through KT. So that gives you

145
00:24:59.770 --> 00:25:06.949
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.

146
00:25:07.720 --> 00:25:13.600
Courant Events Right: And then you have to… now you have an equation that essentially reads K1 squared equals to a constant.

147
00:25:13.800 --> 00:25:20.930
Courant Events Right: So what do you do? You count the last D-2 coordinates, so you are in ZD,

148
00:25:21.180 --> 00:25:31.969
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,

149
00:25:32.210 --> 00:25:40.369
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,

150
00:25:40.800 --> 00:25:44.699
Courant Events Right: That are satisfying the equation of a circle.

151
00:25:44.910 --> 00:26:03.670
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.

152
00:26:04.280 --> 00:26:16.309
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,

153
00:26:16.990 --> 00:26:36.070
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.

154
00:26:36.070 --> 00:26:40.680
Courant Events Right: So think of omega… think of omega for a second to be, like, zero.

155
00:26:43.560 --> 00:26:46.509
Courant Events Right: We're, like, one, okay? So…

156
00:26:46.940 --> 00:27:05.919
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.

157
00:27:06.090 --> 00:27:09.180
Courant Events Right: like, visual, another log of n, which I'm…

158
00:27:09.370 --> 00:27:15.819
Courant Events Right: Not dealing with it, so I'm not taking time into account. You know, time is short, we can ignore it.

159
00:27:16.490 --> 00:27:20.019
Courant Events Right: So, now you ask the question, okay, now you have this bound.

160
00:27:20.670 --> 00:27:34.889
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

161
00:27:34.890 --> 00:27:40.879
Courant Events Right: The S that you have, that S, is strictly bigger than minus 1 over P minus 1.

162
00:27:40.960 --> 00:27:44.620
Courant Events Right: So, independent of the dimension.

163
00:27:44.830 --> 00:27:51.489
Courant Events Right: Right? For any power P, in order to expect… I mean, the best that you… oops, what happened?

164
00:27:57.230 --> 00:28:09.500
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.

165
00:28:10.680 --> 00:28:35.659
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

166
00:28:35.660 --> 00:28:39.280
Courant Events Right: Phase 3 for the cubic equation gives you minus 1.

167
00:28:39.280 --> 00:28:41.229
Courant Events Right: Okay? Which is what…

168
00:28:41.400 --> 00:28:51.659
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,

169
00:28:52.160 --> 00:28:59.499
Courant Events Right: Schrodinger and… and heat. So, in heat, you have minus 1. In Schrodinger, you have minus and half.

170
00:28:59.590 --> 00:29:17.020
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

171
00:29:17.020 --> 00:29:22.620
Courant Events Right: So that your Wuhanan, so the provider doesn't give you any regularization.

172
00:29:22.660 --> 00:29:31.819
Courant Events Right: The weight gives you one derivative, essentially, and heat gives you two, but Schrodinger gives you zero.

173
00:29:31.850 --> 00:29:40.450
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.

174
00:29:40.450 --> 00:29:56.929
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.

175
00:29:57.020 --> 00:30:08.209
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?

176
00:30:10.040 --> 00:30:10.830
Courant Events Right: All right.

177
00:30:10.970 --> 00:30:13.679
Courant Events Right: So, okay, so let me talk about Burgain.

178
00:30:13.880 --> 00:30:29.579
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

179
00:30:29.580 --> 00:30:39.649
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.

180
00:30:39.700 --> 00:30:52.260
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.

181
00:30:52.460 --> 00:31:00.809
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.

182
00:31:01.160 --> 00:31:13.010
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.

183
00:31:13.160 --> 00:31:16.279
Courant Events Right: Which is the usual renormalization that you do.

184
00:31:16.760 --> 00:31:33.910
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.

185
00:31:34.230 --> 00:31:48.310
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.

186
00:31:48.310 --> 00:31:54.499
Courant Events Right: then the sequence of solutions here, UN, will converge to a solution U in the continuous functions.

187
00:31:54.690 --> 00:32:03.220
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.

188
00:32:03.700 --> 00:32:05.020
Courant Events Right: H is minus.

189
00:32:05.410 --> 00:32:10.920
Courant Events Right: Sorry, here, here you're going to be confused, because I didn't put the alpha.

190
00:32:11.150 --> 00:32:28.380
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.

191
00:32:30.490 --> 00:32:46.019
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?

192
00:32:46.170 --> 00:32:49.690
Courant Events Right: So, he considered the cubic equation in 2D,

193
00:32:49.850 --> 00:32:51.980
Courant Events Right: And he considered the geese measure.

194
00:32:52.470 --> 00:33:00.800
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.

195
00:33:01.110 --> 00:33:16.120
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.

196
00:33:16.230 --> 00:33:35.690
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.

197
00:33:35.970 --> 00:33:48.700
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.

198
00:33:48.700 --> 00:34:08.220
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.

199
00:34:08.230 --> 00:34:13.830
Courant Events Right: than L2. So, in other words, you do a fixed point argument for V,

200
00:34:14.040 --> 00:34:18.119
Courant Events Right: In some sublet space of regularity strictly bigger than zero.

201
00:34:18.350 --> 00:34:21.519
Courant Events Right: And so when you do that, this is the difference equation.

202
00:34:21.780 --> 00:34:33.319
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.

203
00:34:33.320 --> 00:34:43.369
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.

204
00:34:43.770 --> 00:34:46.799
Courant Events Right: And so what happens is that when everything is random.

205
00:34:47.030 --> 00:34:58.679
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,

206
00:34:58.680 --> 00:35:11.670
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.

207
00:35:11.720 --> 00:35:14.360
Courant Events Right: Because when you have a combination of these things.

208
00:35:15.190 --> 00:35:23.400
Courant Events Right: then, you're in trouble. You cannot do the large variation estimates, and so the three main ingredients here

209
00:35:23.400 --> 00:35:42.839
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

210
00:35:42.960 --> 00:35:46.380
Courant Events Right: gain of regularity from your duvelle.

211
00:35:46.640 --> 00:35:49.020
Courant Events Right: How are you going to exploit randomness?

212
00:35:49.500 --> 00:36:05.829
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

213
00:36:05.910 --> 00:36:14.950
Courant Events Right: You cannot prove anything for Schrodinger unless somehow you exploit randomness using these random matrix estimates.

214
00:36:15.070 --> 00:36:17.750
Courant Events Right: Okay, so let me give you an example.

215
00:36:18.070 --> 00:36:37.120
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.

216
00:36:38.240 --> 00:36:52.630
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.

217
00:36:52.790 --> 00:36:55.100
Courant Events Right: So that means that this doesn't help you.

218
00:36:55.240 --> 00:37:05.069
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

219
00:37:05.170 --> 00:37:21.789
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.

220
00:37:22.190 --> 00:37:23.719
Courant Events Right: You could have said

221
00:37:23.950 --> 00:37:28.830
Courant Events Right: the number that you get here, when the Ts are low… There is nothing you can do.

222
00:37:29.320 --> 00:37:45.970
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,

223
00:37:46.040 --> 00:38:02.339
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.

224
00:38:02.860 --> 00:38:09.629
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.

225
00:38:09.760 --> 00:38:20.460
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.

226
00:38:20.550 --> 00:38:35.799
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,

227
00:38:35.800 --> 00:38:48.510
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.

228
00:38:48.710 --> 00:38:53.399
Courant Events Right: he gets the right decay for this object to close the SD.

229
00:38:53.690 --> 00:39:07.219
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.

230
00:39:07.320 --> 00:39:26.330
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,

231
00:39:26.330 --> 00:39:30.690
Courant Events Right: That is linear and nonlinear. It's not just about that. Bourgain's…

232
00:39:30.690 --> 00:39:41.249
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.

233
00:39:43.150 --> 00:39:58.429
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.

234
00:39:58.430 --> 00:40:05.180
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.

235
00:40:05.180 --> 00:40:10.690
Courant Events Right: And, let me just not, not, not say anything, because I'm already, like, super behind.

236
00:40:10.790 --> 00:40:18.480
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

237
00:40:18.480 --> 00:40:33.260
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.

238
00:40:33.380 --> 00:40:48.659
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.

239
00:40:48.660 --> 00:40:53.930
Courant Events Right: To extend the local solution of the finer dimensional approximation to a global one.

240
00:40:54.240 --> 00:41:03.189
Courant Events Right: And then, you cannot compare the infinite-dimensional solution with the finite-dimensional solution globally, because you know nothing about you.

241
00:41:03.200 --> 00:41:19.959
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

242
00:41:19.960 --> 00:41:29.579
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.

243
00:41:29.760 --> 00:41:37.409
Courant Events Right: So, in the end, you get that the solutions of the infinite-dimensional system are almost surely global.

244
00:41:37.680 --> 00:41:45.700
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.

245
00:41:46.440 --> 00:41:50.709
Courant Events Right: Okay, so then let me ask, let me ask this question. I mean, okay, so…

246
00:41:51.340 --> 00:42:05.410
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.

247
00:42:05.520 --> 00:42:17.149
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.

248
00:42:17.290 --> 00:42:31.570
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

249
00:42:31.780 --> 00:42:48.310
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.

250
00:42:48.420 --> 00:42:52.469
Courant Events Right: So, Burgain, what he did with the TT star is gain

251
00:42:52.950 --> 00:42:57.789
Courant Events Right: Enough regularity to go from below level 2 to a level 2.

252
00:42:58.110 --> 00:42:59.920
Courant Events Right: But if I stop here.

253
00:43:00.060 --> 00:43:07.700
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.

254
00:43:07.960 --> 00:43:11.350
Courant Events Right: And so… and what happens is that when you do this, there is no way

255
00:43:11.520 --> 00:43:30.520
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

256
00:43:30.520 --> 00:43:46.269
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,

257
00:43:46.700 --> 00:43:47.710
Courant Events Right: One half.

258
00:43:48.040 --> 00:43:49.519
Courant Events Right: So, if I start.

259
00:43:49.950 --> 00:43:58.729
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.

260
00:43:58.880 --> 00:44:00.699
Courant Events Right: And what that means is that now.

261
00:44:01.300 --> 00:44:15.159
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.

262
00:44:15.320 --> 00:44:21.580
Courant Events Right: And if P is bigger than 5, then you have to actually go from 0 minus all the way here.

263
00:44:21.990 --> 00:44:36.259
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.

264
00:44:36.600 --> 00:44:50.819
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?

265
00:44:50.900 --> 00:45:05.499
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.

266
00:45:05.500 --> 00:45:10.870
Courant Events Right: And low frequencies in the other terms, in the… here, I just wrote something cubic.

267
00:45:10.870 --> 00:45:26.229
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.

268
00:45:26.920 --> 00:45:43.719
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.

269
00:45:43.720 --> 00:45:51.509
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.

270
00:45:51.660 --> 00:45:54.020
Courant Events Right: In Schrodinger, this never happens.

271
00:45:54.360 --> 00:45:57.729
Courant Events Right: It doesn't matter, I removed this, I stay…

272
00:45:57.850 --> 00:46:06.230
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.

273
00:46:06.240 --> 00:46:25.160
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.

274
00:46:25.370 --> 00:46:40.510
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.

275
00:46:40.700 --> 00:46:43.850
Courant Events Right: And here, I tried to explain why this doesn't work.

276
00:46:44.080 --> 00:46:53.310
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

277
00:46:53.330 --> 00:47:09.560
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.

278
00:47:09.590 --> 00:47:23.029
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.

279
00:47:24.460 --> 00:47:34.440
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.

280
00:47:34.790 --> 00:47:53.330
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.

281
00:47:53.660 --> 00:47:59.029
Courant Events Right: then you change your answer to something that looks like this, which is that now your answer is

282
00:47:59.300 --> 00:48:01.379
Courant Events Right: the solution, plus

283
00:48:01.510 --> 00:48:10.330
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.

284
00:48:10.330 --> 00:48:22.279
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.

285
00:48:22.360 --> 00:48:28.349
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.

286
00:48:28.920 --> 00:48:48.510
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.

287
00:48:48.750 --> 00:48:52.060
Courant Events Right: And so here is some pictures of what's going on.

288
00:48:52.420 --> 00:48:54.199
Courant Events Right: This is the Duchamel.

289
00:48:54.320 --> 00:49:08.319
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.

290
00:49:08.360 --> 00:49:21.519
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.

291
00:49:22.100 --> 00:49:46.190
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…

292
00:49:46.190 --> 00:49:51.530
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.

293
00:49:51.760 --> 00:50:04.049
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.

294
00:50:04.650 --> 00:50:21.619
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.

295
00:50:21.640 --> 00:50:24.859
Courant Events Right: Where this is essentially given to you is data.

296
00:50:25.160 --> 00:50:38.909
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.

297
00:50:38.940 --> 00:50:46.090
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.

298
00:50:46.310 --> 00:50:49.420
Courant Events Right: What this operator is, is a matrix

299
00:50:49.640 --> 00:51:03.439
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

300
00:51:03.650 --> 00:51:06.120
Courant Events Right: And these two objects are independent.

301
00:51:06.280 --> 00:51:07.430
Courant Events Right: See here?

302
00:51:07.700 --> 00:51:10.739
Courant Events Right: I'm going to ignore it, it can be taken as a parameter.

303
00:51:10.830 --> 00:51:28.810
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.

304
00:51:29.140 --> 00:51:31.900
Courant Events Right: Alright, so in the end, this is your answer.

305
00:51:32.330 --> 00:51:50.469
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.

306
00:51:50.500 --> 00:51:57.520
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.

307
00:51:57.720 --> 00:51:59.510
Courant Events Right: So, no questions.

308
00:52:00.280 --> 00:52:02.440
Courant Events Right: All right, so,

309
00:52:05.000 --> 00:52:24.480
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.

310
00:52:25.650 --> 00:52:28.850
Courant Events Right: And what is actually remarkable, I think.

311
00:52:29.480 --> 00:52:34.010
Courant Events Right: Is that the decomposition that we get here

312
00:52:34.210 --> 00:52:38.209
Courant Events Right: I mean, this structure of the solution is global.

313
00:52:38.870 --> 00:52:44.959
Courant Events Right: It remains for all times. It's not destroyed. We prove it locally, but then you can actually check

314
00:52:45.200 --> 00:52:52.169
Courant Events Right: That, it remains like this for all time. The same as the linea-linear is also global. This is global.

315
00:52:52.640 --> 00:52:56.409
Courant Events Right: Okay? So… so that's actually quite remarkable.

316
00:52:56.960 --> 00:52:59.329
Courant Events Right: Okay, so more generally.

317
00:52:59.650 --> 00:53:23.709
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…

318
00:53:24.090 --> 00:53:39.099
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.

319
00:53:39.260 --> 00:53:40.899
Courant Events Right: But,

320
00:53:41.150 --> 00:53:49.990
Courant Events Right: this theory gives you, what it does is it keeps on peeling off. It keeps on peeling off from the remainder.

321
00:53:50.970 --> 00:53:56.860
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.

322
00:53:57.450 --> 00:54:16.539
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.

323
00:54:16.610 --> 00:54:22.629
Courant Events Right: Okay, and this is actually done in full generality. So I'm going to skip this because

324
00:54:22.690 --> 00:54:25.930
Courant Events Right: I'm sure that you're tired. I need a lot of pictures.

325
00:54:25.950 --> 00:54:37.580
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…

326
00:54:37.580 --> 00:54:55.369
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.

327
00:54:55.370 --> 00:54:56.499
Courant Events Right: Take the gate.

328
00:54:57.130 --> 00:55:07.299
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.

329
00:55:07.550 --> 00:55:12.490
Courant Events Right: But again, you view this as a mapping from K1, K2 into K,

330
00:55:12.590 --> 00:55:18.419
Courant Events Right: And again, you can… you have to… you can prove that this carries the information of the low frequencies.

331
00:55:18.670 --> 00:55:21.020
Courant Events Right: And that this is independent of this.

332
00:55:21.180 --> 00:55:27.000
Courant Events Right: Okay, and okay, so we developed a theory here. Okay, this is another example.

333
00:55:27.160 --> 00:55:31.420
Courant Events Right: And so we developed a theory that essentially tells us that on the Fourier modes.

334
00:55:31.570 --> 00:55:38.109
Courant Events Right: All the fully modes of the solutions can be written, can be expanded in terms of multilinear Gaussians.

335
00:55:38.120 --> 00:55:53.160
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.

336
00:55:53.160 --> 00:56:01.660
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

337
00:56:01.920 --> 00:56:20.859
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.

338
00:56:20.860 --> 00:56:25.849
Courant Events Right: The random average operator is going to gain 1 half, which is 1 half minus, which is what we proved.

339
00:56:25.850 --> 00:56:37.840
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.

340
00:56:38.160 --> 00:56:49.970
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.

341
00:56:49.970 --> 00:57:00.010
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.

342
00:57:00.520 --> 00:57:10.600
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.

343
00:57:10.750 --> 00:57:21.589
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

344
00:57:21.730 --> 00:57:32.230
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.

345
00:57:33.440 --> 00:57:47.989
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.

346
00:57:48.120 --> 00:57:51.300
Courant Events Right: You are supported in a very rough space.

347
00:57:51.590 --> 00:58:00.679
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.

348
00:58:01.040 --> 00:58:06.949
Courant Events Right: is singular, which means that you have probabilistic dependent Fourier modes, which is a problem.

349
00:58:07.560 --> 00:58:19.350
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?

350
00:58:19.520 --> 00:58:34.779
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?

351
00:58:34.930 --> 00:58:40.149
Courant Events Right: Now, I can explain the proof of this, which has a lot of beautiful ingredients.

352
00:58:40.560 --> 00:58:42.580
Courant Events Right: But let me just mention, to finish.