19 - SN Partial Differential Equations and Applications Webinars
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Join Diogo Gomes of KAUST as he investigates a priori bounds for a first-order planning problem with a non-vanishing potential and establish a displacement-convexity property. Don’t want to miss another webinar? Register here: https://go.sn.pub/lXgEaxHave an article on hydrodynamic swarming? Submit Your Article: https://go.sn.pub/GT525VLearn more about publishing with SN Partial Differential Equations and Applications: https://go.sn.pub/UFulhi
00:05
Great to be to be here an it's it's a
00:08
great pleasure to talk in this.
00:10
In this seminar Ann and today I'll be talking about
00:14
some work that I did with former student of mind
00:18
Tommaso Sonakshi and then like 30 grand back.
00:21
But I can and creative crater which has to do
00:24
with displacement,
00:26
convexity, inequality is in in meaningful games and because I
00:30
understand that not all of you are experts in Mitchell
00:35
Games or spent.
00:36
A fair amount of time also explaining a little bit
00:40
where mean field games come from and what is the
00:44
relevance of this of these models so.
00:47
The way I I like to explain mean field games
00:51
is something that I called the measure potential framework and
00:56
the idea is that you will get models with large
01:00
populations and this these models quite often feeds the following
01:05
framework.
01:06
You described the population by probability density.
01:11
And this probability density determines the population distribution out of
01:18
this.
01:18
In many of these models there is value function or
01:22
a potential or a pressure that encodes how the population
01:27
affects the environment,
01:29
and this these two variables you and them are coupled
01:34
together with the nonlinear elliptic or parabolic equation.
01:39
And so so the effects on the population on the
01:43
pressure and the pressure drives the evolution of M and.
01:47
And this is exactly the if you think it's exactly
01:50
the framework of many camel taxes models.
01:53
Certainly meaningful games. The yooks model.
01:57
That's another. Another model that fits this.
02:00
This kind of framework. What's special about mean field games
02:04
is that these are models where these agents are rational,
02:09
so they are active. And they seek to minimize the
02:12
cost,
02:13
functional and dysfunctional depends only on statistical information on the
02:19
on the population.
02:20
So these models were introduced by two groups more or
02:24
less simultaneously.
02:26
By Keynes Wagamama, in the engineering community and by lost
02:30
again we ask the more mathematical community.
02:34
And what they comprise is a system of two equations,
02:38
the Hamilton Jacobi equation and the transport or Fokker Planck
02:44
equation.
02:44
And the difficulty on on these models is that these
02:48
equations are coupled and I I sometimes jokingly say that
02:52
we know everything about Hamilton Jacobi equations if we know
02:56
the Hamilton Jacobi equation, and we know everything about transport
03:01
and Fokker Planck equations if you actually know the equations.
03:05
But the the issue is that when this this equations
03:08
are coupled well,
03:09
then it's it's a little bit of a different offer,
03:13
a different game. For example,
03:15
if you look at the typical model,
03:18
so we have the transport equation in this mutual game,
03:22
and we have Hamilton Jacobi question.
03:25
So you can think of M as the density of
03:28
the population and U as a value function for a
03:31
typical agent.
03:33
And if you look at this,
03:34
these two equations and say,
03:37
well, what can we say about this?
03:39
Well, actually not much, because really if you don't know
03:44
the drift.
03:44
Vector Field on the transport equation.
03:48
Well, you really cannot say much more than M is
03:51
a measure.
03:52
Maybe a probability measure if you don't know much about
03:57
M and think of this guy has come to some
04:00
power,
04:01
for example. Well, if you don't know anything about them
04:05
then there is relatively little that you can say about
04:10
the Hamilton Jacobi equation plus this equations of this structure.
04:15
Where you as terminal condition.
04:18
So at time T&M as initial distribution.
04:21
The reason is that as you'll see you is a
04:24
solution of a control problem.
04:26
So control problems take usually terminal values 'cause you want
04:32
to optimize something towards the future.
04:35
But you know the distribution of agents at the beginning
04:39
of time.
04:40
So part of the variables go forward,
04:43
part of the variable scope.
04:45
Backward in time. This creates one of the difficulties so
04:50
so things like Cauchy,
04:52
Kowalewski, local time, existence, kind of results are not available.
04:58
At least 4 W is very small,
05:00
so let me start by reviewing a little bit where
05:04
the Hamilton Jacobi equations come from.
05:08
So let's fix terminal time and fix an agent whose
05:13
state is X of T.
05:15
And well, what agents can do.
05:18
As I said, agency mutual games are active agents,
05:22
so they can choose a bounded measurable control and they
05:27
well in this simple model they just act on their
05:31
velocities.
05:32
Anne.
05:34
Now we have a cost.
05:36
When they choose velocities, they pay a price and this
05:40
price is aggregated in time by running cost,
05:44
which depends on their position depends on their velocity and
05:50
they face a terminal cost capital.
05:53
No, and. And the agents want to minimize and what's
05:58
well known in the theory of optimal control is the
06:03
phone you build.
06:04
Those gender transform of the Hamiltonian.
06:08
So for example, if my Lagrangian worthy square over 2,
06:12
then the Hamiltonian would be B ^2 /
06:15
2 as it is well known.
06:17
So you built the Legendra transform and.
06:21
In this Legendra transform you the the Maxim or the
06:27
Supreme is achieved as as a unique point an and.
06:32
We have something called verification theorem and the verification theorem
06:38
tells you the following if you solve the corresponding Hamilton
06:42
Jacobi equation with the appropriate terminal conditions.
06:47
Then you can build optimal trajectory in feedback for meaning.
06:51
You give me the X.
06:53
I'll tell you where you should go.
06:57
Out of the solution of the Hamilton Jacobi equation,
07:01
so two things you solve the Hamilton Jacobi question.
07:05
Then what you gain is.
07:08
In optimal feedback and you~ or your solution is the
07:12
value function in your in your control problem.
07:17
No.
07:20
But this is what one agent does,
07:22
what. What happens with when you have a crowd of
07:25
agents?
07:26
Well, when you have a crowd of agents and they
07:29
are following,
07:30
let's say a vector field that smile.
07:33
The simplest example. Well, a vector field induces a flow,
07:37
which is simply the map that takes initial conditions into
07:41
conditions at time T.
07:43
So you give me an ODI.
07:45
I can build a flow.
07:46
This flow Maps the initial conditions to the solution at
07:50
later times.
07:54
And once you have a flow,
07:55
then you have what's called the push forward,
07:59
which is a map that takes probability measures and builds
08:03
other probability measures in.
08:05
Let's say, the only possible way you take an initial
08:09
density for agents,
08:10
right? You have M 0 effects,
08:12
and then you see where the agents are at time
08:16
T,
08:16
and you integrate and this defines the probability measure probability
08:21
density depending on smoothness.
08:24
Um at later times and then you ask,
08:26
well, but can we give a carcass?
08:29
In principle, if you have a smooth enough flow,
08:32
you can use change of variables form and get the
08:36
form of four MXT.
08:37
But in fact what we are interested in characterizing what
08:41
happens in terms of differential equations and what happens is
08:46
that if you have a Lipschitz vector field.
08:49
Then you consider to push the push forward by the
08:53
flow and then the corresponding density.
08:56
In fact, is a weak solution of the transport equation.
09:00
And of course if you have more regularity than you
09:03
can push further.
09:07
No. So, So what I told you so far is
09:12
what a single agent does.
09:15
Solve the control problem and what happens when a crowd
09:19
follows a vector field,
09:21
but then you can ask well what happens if.
09:24
Every agent is solving a control problem that depends on
09:29
the probability distribution of the agents and all of them
09:33
do the same thing.
09:35
Well, what you get is you have a distribution of
09:40
agents.
09:40
Never distribution of agents. You have a cost that depends
09:45
on the distribution of agents,
09:47
and agents seek to minimize a control problem involving this
09:53
cost.
09:53
They get the Hamilton Jacobi equation and out of this
09:57
Hamilton Jacobi equation they know where to go.
10:01
Because you, you know their optimal strategy out of these
10:05
two things,
10:06
you get the system that couples Hamilton Jacobi equations with
10:10
transport equations.
10:12
In this case. And of course you can get second
10:15
order equation,
10:16
2nd order Hamilton Jacobi questions if you.
10:21
If you will get instead of ODS stochastic differential equations
10:25
and so on.
10:28
What I want you and because most of our results
10:31
are in fact for 1st order,
10:33
mean field games. This this is something that.
10:38
Uh, we don't need to go to go into four
10:40
for the purposes of this stuff.
10:42
This talk?
10:45
This problem you are given with initial conditions on M
10:49
and terminal conditions on you and you say well look,
10:53
this is a nightmare in terms of of regularity theory
10:58
because.
10:59
Standard theory of Hamilton Jacobi equation tells you that you
11:04
is going to be Lipschitz and well.
11:07
This guy is DPH evaluated at Du,
11:10
Right? So if you have you Lipschitz then this is
11:14
going to be bounded and bounded is not such a
11:17
nice regularity for the transport equation,
11:21
so it's even doubtful that you can get a decent.
11:26
Kind of existence result, but you'll see that in fact
11:30
there are cases where where again and in our case
11:35
we have Hamiltonians with a specific form which have.
11:39
Think of the square over two as as something that's
11:45
uniformly convex.
11:47
We have so so this guy corresponds to a cost
11:50
of moving,
11:51
which in this case is going to be the square.
11:56
So agents pay a price which in this particular example
12:00
is quadratic in terms of cost of moving,
12:03
but then they have preferences about where they are and
12:07
this is encoded in a potential.
12:10
And then if you have a function that's increasing.
12:15
Well, what what you have is that.
12:18
So this is the Hamiltonian at the level of the
12:21
Lagrangian you get plus GFM because you switch the sign
12:25
between the Lagrangian and Hamiltonian.
12:28
And if you have plus G.
12:30
FM.
12:31
What you have is a engies.
12:33
Increasing. This means that your agents are crowded first.
12:39
Anne.
12:42
No this problem. Actually arose quite.
12:48
Before mean field games and and and let me show
12:52
you something that I'm.
12:53
I'm sure many of you are aware which is the
12:57
benamou brenier formulation of optimal transport,
13:01
so it's a. Seemingly unrelated problem,
13:04
which is this. You are given two probability measures.
13:09
And you seek to transport one probability into the other,
13:14
meaning you want to find a map that pushes forward
13:19
one probability measuring to the other that minimizes.
13:24
Minimizes the transportation costs. What's a quadratic transportation costs?
13:30
And you know this problem is a is a quite
13:33
long story from Kantorovich McCann.
13:36
Plenty of Villani and I'm sure there is someone in
13:40
the audience that made important contributions.
13:43
I apologize, you're not is not here,
13:46
but this. This is actually a very long,
13:49
very long history and the benamou.
13:51
Plenty of formulation is a continuous formulation of this optimal
13:56
transport and the idea is this well,
13:58
instead of minimizing the one shot.
14:01
Transport costs what they realize that you can say well
14:06
take velocity vector fields and minimize this kinetic energy term.
14:12
Let's say subject to the flow corresponding to V pushing
14:18
forward M 0 into M1.
14:20
And and and. And at all intermediate at all intermediate
14:26
times.
14:27
No, if you look at what are the optimality conditions
14:31
for this variational problem,
14:34
well they are in fact a mean field game.
14:38
Except that there is no coupling between these two equations.
14:43
So, so there the Hamilton Jacobi equation and the Fokker
14:47
Planck equation or the transport equation RN coupled or or
14:52
it's not true,
14:53
they are the Hamilton Jacobi equation does not depend on
14:57
them.
14:57
Of course this equation depends on them,
15:01
but they have some triangular structure and there is something
15:05
interesting 'cause if you solve this optimality pair,
15:09
let's say, then you get the velocity field as the
15:13
gradient.
15:14
Of the the Hamilton question,
15:16
as you would get in a standard verification verification theorem,
15:22
there is only one thing that if you have never
15:26
seen it,
15:27
you'll find it puzzling which is this.
15:30
You are given M. As the beginning of end.
15:36
And end right? But you are not given any condition
15:40
for you,
15:41
so this is a problem where you.
15:45
Your problem really is this.
15:46
You try to find what is the cost that you
15:49
can impose.
15:50
What is the terminal cost that you can impose so
15:54
that?
15:54
Agents distributed according to M Zero will reorganize themselves as
16:00
agents distributed according to M1.
16:05
And.
16:09
As I pointed out, this is this.
16:14
Problem has been worked in the mean field game theory
16:18
and it's called the planning problem and it's a very
16:23
similar problem where you also give only initial and terminal
16:28
conditions for M and you don't give initial North terminal
16:33
conditions for you and you ask.
16:36
Well, can we find a terminal cost such that this
16:40
rational agents self organized and starting from you.
16:44
End up as new and this is called the planning
16:48
planning problem.
16:49
Now this problem was introduced by we answering these vectors
16:54
and collect funds has been studied by a number of
16:58
authors,
16:59
Anne, from variational methods by Dash 2 Camilli Grabber Bizarros
17:04
Sylvan.
17:04
To know these are two separate papers for the parabolic
17:10
case was studied by LSU OSU Perretta.
17:13
11 I sent them brought you actually study these problems
17:17
and establish estimates that are related to the ones that
17:22
will discuss here.
17:23
And Monostor is an missing.
17:27
Establish the existence of solutions when there is no no
17:31
potential,
17:32
but that will address that.
17:33
the Twitter.
17:36
OK, so let me explain the key question that we
17:42
want to understand and and.
17:45
And if you want to get the most general general
17:48
results,
17:48
you'll need to go to the to the paper.
17:51
I'll try to give you some ideas in the examples
17:54
and explain why this these are important points,
17:58
but our our key question is this.
18:00
You consider a planning problem.
18:03
And you ask, well, can we?
18:06
Prove that if we start with probability densities on the
18:11
Taurus on let's say 1 dimensional Taurus fixed ideas,
18:16
can we get that M is strictly positive everywhere and
18:22
the reason why?
18:24
And certainly this is an interesting question,
18:28
and a more generally another question that maybe you want
18:32
to do answer and will show how to do this
18:36
is given bounds on M0 given bounds.
18:39
An empty can we prove bounds on M at intermediate
18:44
types?
18:44
And this is not obvious right?
18:47
Because you don't know what the terminal conditions for you
18:51
are,
18:52
so so really proving bounds that intermediate Times 4 M
18:56
When you know M 0 and Mt seems a bit
18:59
of magic because you don't know part of the boundary
19:03
conditions.
19:05
Let me explain why this bounds are the lower bounds
19:08
are relevant.
19:09
In particular I, I guess getting bonds at intermediate Times
19:13
is is clear why.
19:14
This would be interesting from the point of view of
19:19
Piddies,
19:19
but from the point of view of this war bonds,
19:23
it's an observation by Lyons that if you look at
19:28
this problem.
19:29
When V0 so no potential,
19:31
you can take the first equation and say,
19:34
well look I can take the first equation and this
19:38
tells me what M is,
19:40
right? I mean in general you have a GOM,
19:43
but in principle convert because Jesus monotone function.
19:49
But when am I? I know what app is so
19:52
I can use this in the second equation and I
19:55
get well some expression which are reorganized and you see
20:00
that M is positive then the previous equation is elliptic,
20:05
right is uniformly uniformly elliptic so so.
20:09
So, so definitely this. This is something that we would
20:13
expect that we to get some some regularity theory.
20:17
Now, what do we expect in terms of lower bounds
20:21
for him?
20:22
And the answer is, well,
20:24
we asked Week F lower bounds for him if the
20:28
zero.
20:29
Uh. And and this is what we want to address.
20:33
The maybe here, well, it's actually the answer is yes.
20:37
And these bonds are false if V as large oscillation
20:41
and the reason why.
20:43
The potential plays a role is the following.
20:47
If you have think of the potential as the V
20:51
has been your special preferences.
20:54
So if we as high oscillation,
20:56
it means that are, well,
20:58
good places in bad places,
21:00
and the more the oscillation of V,
21:03
the more different the good places are from from the
21:07
bed differences,
21:08
so it's not completely impossible that agents will just avoid
21:13
some regions.
21:14
Right? There are regions that are so bad that people
21:18
will just avoid it.
21:19
Then you will have, and in fact this is something
21:23
that you can see.
21:24
In examples and let me give you just two examples.
21:29
One is a stationary case and you see that really
21:34
if you look at this second equation.
21:38
Either by just staring at it so everything is periodic.
21:41
So my solutions in this talk are periodic solutions.
21:46
As I put the Taurus before,
21:48
but just to be clear,
21:49
if you look at this equation is so well look.
21:52
If we're talking about classical solutions,
21:55
then U = 0 is going to be a solution
21:58
of this.
21:58
This equation for free for inspection and in fact is
22:02
the only possible solution,
22:04
because if you if you take this equation,
22:07
you realize that the gradient of you has to be
22:12
0.
22:12
Must be 0.
22:15
And unless MM is there.
22:19
No, if you is 0.
22:23
Sorry, then you go through this equation so well,
22:25
but this is zero then.
22:27
Well then I know what M is.
22:29
Right? But if I know what M is,
22:33
then this is what I want.
22:37
And look if, let's say for normalization.
22:40
I asked him to be a probability measure and I
22:43
can always add a constant to V so that its
22:46
integral is zero.
22:47
It doesn't change the oscillation,
22:49
you get that M = 1 plus fee.
22:52
And now you see that you have a contradiction in
22:55
V as a high oscillation,
22:57
because if we as I oscillation,
23:00
then M can have negative can take negative values and
23:04
this would be this would be a serious serious problem.
23:08
So so this model shows that.
23:11
I mean, it's it's really not a counterexample because this
23:15
is a stationary case and what I want is I
23:17
start with some M0I end up with F0 and one
23:20
to ask is in between M0,
23:22
right and and this is not the answer that I'm
23:25
giving to you,
23:26
but but this is somehow some plausible possible.
23:30
Possibly check right?
23:34
So then we started well.
23:36
Can we build time dependent examples and this is a
23:40
nice way to build examples because you build the answer
23:44
and then you figure out the question.
23:47
So what we did was the following thing.
23:49
We started with M that finishes.
23:53
And of course you bye bye inspection.
23:56
You can see it's zeros and then you say well
23:59
look.
24:00
But now we have empty,
24:01
sorry minus.
24:05
M T M UX sub X = 0 so
24:07
if you put them into this equation you may be
24:11
able to solve for you and you need to believe
24:15
me that this is what you get.
24:20
You get you and you so well,
24:21
but what about the Hamilton Jacobi equation?
24:23
Although I'm not going to copy question,
24:25
is this so? This gives you the V.
24:27
Right and what you realize is that this fee is
24:30
going to be unbounded,
24:32
right? So so but but but you can have at
24:35
least with V and bounded solutions for which M vanish
24:38
is right,
24:39
and it's not so far fetched to think that that
24:43
these things can happen.
24:45
Can happen in in some other way,
24:48
so.
24:51
The key observation and is is that.
24:57
The way to prove lower bounds on them is to
25:01
try to understand what happens to integral functions of EM
25:06
or or some density.
25:08
So so this should be up,
25:10
by the way.
25:15
So so displacement. So you take 2 probability measures and
25:20
you ask well what happens to functionals of M at
25:24
intermediate tax.
25:25
And this is something that was studied by by McCann.
25:31
To study gas models in the context of optimal transport,
25:35
and they actually showed that along solutions.
25:39
And it. I, I guess he didn't show along solutions
25:43
or benamou Premier problem because when I'm Ukrainian problem is
25:48
is what was afterwards.
25:50
But but this this is really what what happens along
25:54
solutions of Benamou Premier problem.
25:57
You have a number of potentials that are convex.
26:01
And and this is called displacement displacement convex.
26:05
For example, interaction potentials or internal energies with appropriate conditions
26:12
on on you.
26:12
So our goal. Is to identify functions.
26:18
Such that.
26:20
This function is convex. Where am solves 1st order mean
26:24
field games?
26:25
See if you have a convex function.
26:29
That is bounded by below.
26:31
So let's say I have zero.
26:33
I have capital TI have my integral of you.
26:37
Event.
26:39
And I know I'm 0 so I know this guy
26:41
here.
26:42
I know this guy at the terminal time.
26:45
I know presumably that this function is convex in time,
26:49
so I can bound it by a line if I
26:51
know that this functional is bounded by below.
26:55
Then that's it. I mean,
26:56
this dysfunctional here is going to be bonded for all
27:00
times in between zero and capital T by only the
27:04
initial data.
27:05
And it turns out, as you will see,
27:07
that in fact it you have things like the LP
27:10
norms or 1 /
27:11
M and so on are displacement,
27:13
displacement, convex and this gives you a number of bonds
27:17
for these problems.
27:22
And for optimal transport, the condition is this.
27:26
Essentially you take the Legendra transform.
27:30
Of this potential, and you ask when is this?
27:34
I guess pressure nondecreasing, and if you are in dimension
27:41
1.
27:41
For example, powers. Like Z to the power cube will
27:45
work,
27:46
provided that Q is less than zero or Q is
27:49
greater than 10 one.
27:54
So as I as I said,
27:56
convexity of the internal energy gives that the internal energy
28:03
is bounded at intermediate times.
28:06
For example, an example if is the L2 norm at
28:11
XT or square of the L2 norm is bounded by
28:15
the L2 norm at T.
28:19
Times the L2 norm at 0.
28:25
So so you gain bounds almost out of well.
28:28
I guess it's not out of nothing because there is
28:32
a fair amount of work to prove displacement,
28:35
convexity. So for any and it turns out this is
28:39
something that I did with Tommaso Tama.
28:42
Na Chu was my my student.
28:46
Is that when in fact,
28:47
if you have potentials that satisfy McCann's condition then you
28:52
have displacement convexity for the for the mean field games.
28:57
And and really, the proof is you differentiate and you
29:02
integrate by parts and I'll get back to the end
29:06
if we have we have time.
29:12
So so so you integrate by parts and you re
29:16
organize reorganize your your identity is in in the clever
29:20
way.
29:21
One thing that we obtained also and this is interesting
29:25
is the the bounds also form in field games with
29:29
congestion.
29:30
So midfielders with congestion corresponds to problems where they'll grangent
29:36
is El~ is M to the Alpha L of let's
29:39
say so.
29:39
So you have a situation where.
29:42
Moving in high density areas is expensive.
29:45
It's more expensive than than what you would expect,
29:48
and this is something where you can also get this
29:53
this same box.
29:54
No um.
29:58
What's?
30:00
Now these bonds require actually a fair amount of integration
30:05
by parts.
30:06
And certainly in the case of this mean field games,
30:11
because actually apples three Ari Sebastian Munoz proved that actually
30:18
the story is leonce.
30:21
Sketched in his lectures, inchoate fans that this system was
30:25
well posed in the solutions are smooth,
30:27
smooth, mean seat to something.
30:31
Anne and then Sebastian Munoz actually proved wrote a nice
30:35
nice proof of of distinct and so this I'm not
30:38
sure if the paper has been published,
30:41
but it's it's available on online,
30:43
so all of these computations can be actually properly properly
30:49
justified.
30:50
What he didn't prove and will address at the end
30:53
is the case with the potential.
30:55
'cause it indicates the potential you actually need.
30:59
Displacement, convexity to prove the existence of solutions.
31:05
So so, So what we did in the in the.
31:12
In this thing with Tommaso is we realized that we
31:17
could extend displacement,
31:20
convexity, and in fact the logarithm of some functions is
31:26
convex also.
31:28
So, so we we could prove that the logarithm of
31:32
some of these this LP norms,
31:34
let's say, is in fact,
31:36
is in fact convex because we have.
31:41
Identity is that that give you that a logarithm of
31:45
integral of M2 Q is convex,
31:48
and if you have this.
31:51
Then you can take. The curious.
31:58
Of the OR it could take the exponential and then
32:02
you can take the cube root of what convexity of
32:06
the logarithm gives you and obtain that.
32:09
In fact, the LQ norms as claimed in the in
32:14
the theorem.
32:15
Satisfy.
32:17
This inequality here an in this inequality you can pass
32:22
queue to Infinity.
32:24
So in fact you obtain your tender bound that that
32:29
is valid up to our Infinity.
32:32
So you can get uniform bounds on M and this
32:35
means that in this is kind of a natural result.
32:38
If you think about this,
32:40
'cause these agents are crowd averse and this means that
32:44
well,
32:44
if you start with the agents reasonably well distributed,
32:48
they will not in the in the middle pile together
32:52
at the end and create unbounded and bounded densities.
32:56
So.
32:58
And then you can estimate.
33:02
You can get the number a number of other of
33:05
other estimates out of out of this thing.
33:08
Let me just.
33:11
Address the the one the 1 dimensional case if you
33:15
are in one dimension.
33:17
Then you can get negative bonds.
33:20
The negative powers of of Q.
33:24
And and in fact, what you can do is to
33:28
show that 1 /
33:30
M. Is bonded.
33:33
And and and this requires a little bit of of
33:37
the limiting limiting argument,
33:40
because apriori you don't know whether 1 /
33:44
M is is bounded or not.
33:46
But you can you can,
33:48
you can get it. OK so.
33:52
And we really explored this this thing to a fair
33:56
amount of detail,
33:57
and then I'm not.
34:00
I'm not expecting everyone to.
34:03
Appreciate that the numerology in this in this side but
34:08
we we kind of understand the reasonably well.
34:11
What are the various combinations of congestion,
34:15
polynomial growth on G and polynomial growth on the non
34:19
linearity?
34:20
This guide that that will give you displacement,
34:23
displacement convex is so all of this of this quote
34:27
could be analyzed in a fair amount of detail and
34:31
and in particular byte by the previous results.
34:34
We can also get the.
34:36
Feed Ella Nfinity bonds out of this thing,
34:41
no? The in in this second part of the talk
34:44
what what,
34:45
what I want to address is the case of mean
34:48
field games with the potential now with the potential you
34:53
don't get convexity,
34:54
displacement, convexity, director. So so if you try to do
34:59
this computations,
35:01
you don't get something positive because the potential still remains.
35:06
And as far as I can tell.
35:10
I mean the proof by Sebastian Munoz or or the
35:13
ideas of Lyons don't automatically give you M greater than
35:19
0.
35:19
Even if these guys are bounded bounded away from from
35:23
zero at the initial and terminal and terminal time Ann,
35:28
and somehow they cannot give it to you.
35:31
I mean, these proofs would need to be changed in
35:36
some substantial way because it's most likely.
35:40
Course without any conditions on on V that that you
35:45
get this so.
35:47
Actually, what we found is that there are some smallness
35:50
conditions on.
35:51
In this case, is the Laplacian of Phi.
35:55
And these are two assumptions that somehow combine two things.
35:59
Combine how are proportion of the is and how large
36:03
the time is and the way to explain this is
36:06
look if you if you want to avoid the region,
36:10
you need to move.
36:12
But moving is expensive. Right,
36:15
if you have a short time,
36:16
you need to move fast.
36:18
So now it's it's a game between the size of
36:21
V and how much time do you have to move
36:25
to avoid the region.
36:26
But then what? Our estimates somehow explain or?
36:30
Or this is the explanation?
36:32
I mean what the estimates show is that if you
36:36
have the right balance,
36:38
then either the time is small enough for what portion
36:42
of these is small enough then then you are in
36:46
in in business and and you cannot.
36:49
Health this this area is void of pigeons.
36:54
So so if you have this thing.
36:57
Then of course this is a theorem under the previous
37:00
assumption,
37:01
so we have a potential and so under.
37:04
Previous
37:08
assumptions.
37:15
So, So what we can do is to prove that
37:18
in fact the LP bounds.
37:20
Are they happy? Norms are bounded,
37:23
and in fact that they if the dimension is one,
37:27
we have bonds in one over him,
37:29
so we have we don't have vacuum vac information.
37:33
Let me see. So what is the key inequality well?
37:39
Bikini quality is this is you keep track of what
37:42
you have and and you try to control what whatever
37:46
remains.
37:46
That depends on fee and you get the inequality and
37:50
you say,
37:50
well, this really looks like.
37:53
An elliptic equation, and in fact this is really what
37:58
it is,
37:58
it's it's. It's an elliptic equation in one dimension with
38:03
boundary boundary conditions,
38:06
and you ask when can we.
38:09
We get.
38:11
When can we get fonts on this solution?
38:15
And it turns out that if Epson.
38:19
You see small compared to two epsilon T.
38:22
Then you have a uniform bound on on the solutions
38:26
to this to this disease question.
38:28
So this is more or less what's what's going on
38:33
here.
38:34
And so out of. This differential inequality what what you
38:40
gain is exactly abound on the other solution,
38:44
so just this gives you the.
38:48
The LP bonds. Anne. And now let me consider a
38:53
special special example with the polynomial power here.
39:01
And then what we can do in this case is
39:06
in fact show that you have both L Infinity bounds
39:11
and well,
39:12
Infinity bounds in in 1 /
39:15
M.
39:17
And.
39:20
The way you do it is is is really this.
39:25
So.
39:28
The idea is that you first.
39:31
Try to find.
39:34
Abound for something.
39:38
So you try to find the R for which the
39:41
integral 1 /
39:42
M to some power R is bounded,
39:45
and then the key argument is in the iterative argument.
39:50
OK, so let's let's go over this.
39:55
So you see, The thing is this.
39:57
The thing is that you get the differential inequality on
40:02
2nd derivatives.
40:03
This guy is positive, so it's good.
40:06
And this guy here? Well,
40:08
you need to cook it up in in some way,
40:11
right? So if you choose Q small enough,
40:14
depending on C&T and so on,
40:16
what you get is that for one R this guy
40:21
is going to be pumped.
40:23
And and then what you have is that this term
40:27
here.
40:28
You can use to produce some iterative inequalities.
40:35
And out of these iterative inequalities,
40:38
you get some iterations that allow you.
40:41
2 built.
40:44
A sequence QN.
40:47
Where you you bound each guy by by the previous
40:52
ones iteratively?
40:53
And this is really where from where this this result
41:00
is not false now.
41:01
So so. The proof is similar for for the L
41:06
Infinity TL Infinity bound.
41:08
So so in this case that.
41:12
The LCD bound you actually get it for for all
41:15
D greater than one,
41:16
and in fact this is a result that's extremely related.
41:20
Work to what fury percent and project proved so 50%
41:24
from Baraja proved the similar a similar result using completely
41:29
different methods,
41:30
but I I think in the end is the same.
41:34
Same spirit, let's say that there is some hidden convexity
41:38
in these models.
41:42
And no, so the last.
41:45
The last topic that I want to address is the
41:50
existence of solution.
41:52
So a lot of what we did here was really
41:56
the the getting that the solutions are bounded.
42:02
Provided they exist right? So in fact what we realize
42:07
is that what we have is this lower bounds are
42:11
exactly what's needed to complement the work by Sebastian Munoz,
42:17
and in this case we can get that for.
42:23
Fully normal nonlinearities you you get.
42:29
You get the classical classical solution to this to this
42:34
planning to this planning problem.
42:37
OK, so so. Let me just tell you one more
42:41
thing so so I I put here some further credits
42:44
and the reason is that actually I benefited a lot
42:49
from working with these two people.
42:51
So one was left on their back and and and
42:55
and let me tell you how do we do.
42:57
We came up with with with this displacement,
43:01
convexity and at some point I was I was involved
43:05
in this project of looking at symbolic computations for PDS.
43:09
And what we what we were interested is in finding
43:13
more or less automatic ways to decide when expressions can
43:18
be integrated by parts to be positive.
43:21
For example, you have an expression like this and you
43:25
want to decide can I integrate this by parts and
43:29
get in this case in negative and negative quantity and
43:33
it turns out.
43:36
That this is a purely algorithmic project and I was
43:40
working with her safari and and in the implementation of
43:44
this of this code and we were looking for an
43:47
example that that would be interesting and nontrivial.
43:51
An in fact was was level that was my postdoc
43:54
at time and now is stuck in UCLA.
43:56
They said, oh, you know you guys should should experiment
44:01
with the with the displacement,
44:03
convexity and see if you can get.
44:06
Can get something and this is in fact how we
44:09
are.
44:09
We get this thing so I think I think certainly
44:12
they deserve a little bit of credit.
44:14
Well, fair amount of credit for for for suggesting suggesting
44:18
these ideas.
44:19
So I think this is more or less what what
44:23
I wanted to tell you guys.
44:25
So I'm sure questions and comments are be.
44:29
Since I see that there is a question in the.
44:34
Kadiogo, thank you so much.
44:36
Beautiful talk. So let's get people have questions.
44:42
Oh, I see that there is one question in the.
44:47
Can the crowd aversion function G in the Hamiltonian be
44:51
considered as an elastic potential within the Hamiltonian?
44:54
Yeah, this this is really what it is right?
44:57
Because if you.
45:00
I mean, it's some kind of internal potential from from
45:04
the system that that you have some external potential that's
45:08
prescribed to you.
45:10
If you look at the Lagrangian,
45:12
let's say you have maybe some kinetic energy and you
45:16
have a G of M of X.
45:18
F of T and did an and this guy can,
45:22
um.
45:23
Comes from from from the internal internal system.
45:26
So so in some sense that's the true.
45:29
What it is, yeah.
45:34
OK, thank you. Can you put a question?
45:47
Diogo
45:49
just a curiosity.
45:51
So for the G, so G is always.
45:54
I mean it can be thought as a potential,
45:57
an exponent of M, right?
45:59
So me some power Alpha.
46:00
So what happens when Alpha increases or decreases?
46:04
Does it improve the regularity or or so the bigger
46:08
the Alpha,
46:09
the wars, the regularity or vice first there is any
46:13
interaction clear interaction between this?
46:17
So let's see.
46:20
It's easier to prove so so.
46:24
It's easier to prove or it's easier to prove more
46:29
regularity if.
46:30
The behavior at zero is more singular actually,
46:35
so so things improve. Surprisingly,
46:39
with things like the logarithm of them,
46:41
let me give you an example.
46:47
Actually, let me yeah, let me give you this example.
46:51
So if you look at this stationary right?
46:55
Versions of this thing called zero and this this is
46:58
the undergraduate question.
46:59
Of the following problem, we minimize over you the integral
47:04
of E to the DU Square over 2 plus VFX.
47:10
And in fact, if you take some G of the
47:14
you square.
47:16
Plus we affects and you consider an integral functional like
47:20
this and you write down what they are.
47:23
LaGrange equation for this thing is is well in weak
47:26
form is G prime.
47:29
Slept with.
47:31
It's G prime of distinct DUDV.
47:36
Equals zero, which then you can do the following game.
47:41
We define this thing to be M and then you
47:45
get diversions of MD U = 0.
47:48
Together with.
47:51
Do you square over 2 + 3 effects equals G,
47:56
prime inverse?
48:01
Because you you just, I mean there is you square
48:05
over 2 plus next year right?
48:07
So you just invert invert this relations so.
48:11
Single guys like this sometimes have a really.
48:19
It's a really nice regularising behavior,
48:23
and you would think that it would be much worse,
48:27
right? But the and the meaning is this is that
48:31
if you have place if you if the agents.
48:35
Love really love empty places then there can't be empty
48:40
places.
48:41
I see OK, so this is what the Walker it
48:44
means.
48:44
So if you take a logarithm of 0 is minus
48:47
Infinity.
48:48
So if there is an empty place,
48:50
it's infinitely good place. So people will go there in
48:54
this place won't be empty anymore,
48:56
so so that's the thing.
48:59
Interesting interpretation of the of the regularity theory beneath the
49:06
DPD.
49:06
So there is someone asking whether I have a sample
49:11
PD and part of the solution,
49:14
yes, but not at hand.
49:16
So you can. You can look at my papers with
49:20
11 and Mariana Presearch.
49:28
And there is actually a detailed analysis of 1 dimensional
49:32
models there are.
49:34
I mean there are a number of people that did
49:37
numerics on mutual games.
49:38
They probably even have better.
49:42
Better graphics and and and numerics than me.
49:45
But if you want let's say a simple model where
49:48
you can really understand what's what's going on in mean
49:52
field games,
49:53
then then maybe this is a good place to start.
49:56
OK, any further question. OK,
49:59
if we don't have any further question,
50:02
please join me to think yoga for this very insightful
50:08
talk once again.
50:10
And I will pass the floor too,
50:13
maybe. Young 'cause I think I still had to leave
50:17
exactly.
50:17
I think Standard first let me extend my thanks also
50:20
to do both of you to Diego and Eduardo.
50:22
Thank you for thank you.
50:25
Chairing and delivering the talk today.
50:29
So well as I just posted chest,
50:31
there will be a talk next week and just a
50:33
reminder will be on Thursday and not on Monday next
50:36
week.
50:36
So it will be on the 4th of March but
50:39
it is same time of the day.
50:42
And that will be on the.
50:44
High order numerical methods, affectionate effusions in polygons,
50:48
by Professor Marcus MyLink from Vienna.
50:51
As please feel free to join us again next week.
50:55
OK, thanks everybody for joining.
50:57
Thanks Diogo, thanks, Eduardo, thank you everybody.
51:00
Have a nice day or evening or whatever it is
51:03
in your time zone.
51:04
Thanks a lot you so much bye bye.