Thank you very much Enrique.

I'm just trying to get the spotlight here.

Now you can see a red dot, right?

That should work.

Okay, so I will give an overview over the work that I did in the last years, mostly

in Vienna and also in Shenzhen and in Shanghai.

And I will talk about two linear kinetic models.

First, the Fokker-Blank equation with a linear drift, sometimes also called the Kolmogorov

equation, has some other names.

And second, the Goldstein-Taylor model.

They both share some similarities that I'm interested in, which are defects that are

appearing and we will talk about this in both cases.

And secondly, the main challenge is the hypercursive dynamics that are present in both of these

models.

First, I'm going to use our first from spectral theory, mostly for the Fokker-Blank equation

and entropy methods for the Goldstein-Taylor equation.

Our main question of interest will always be how fast do solutions converge to the equilibrium?

So let's start with the Fokker-Blank equation.

So to make the entrance easy, if we first neglect the second term and we just say this

matrix D here is the identity, then we have nothing else but the heat equation here.

For where F is now, in our case, the solution of the probability density of an average particle

of a large particle system that is under the influence of two forces.

So first, it's the collision force.

This is modeled by, as I said, this diffusion term, the heat equation term.

But here we have the specialty or the challenge that we only have a partial heat equation.

So the matrix D here is only semi-definite and we can have a non-trivial kernel of this

matrix.

And then the second part of our Fokker-Blank equation is this drift part.

This is C multiplied with the position vector X multiplied with the function F. And we are

here generally in Rd.

This is the transport term.

And in the symmetric case, this corresponds to a potential.

So we have once the partial diffusion of our average particle and secondly, a force field

or potential.

And to get the convergence to equilibrium for this case with only positive semi-definite

diffusion.

Hello?

Yes.

There is some sort of overlay on the right side of the screen.

I think it is from the zoom window, actually.

So it has the form of the pictures of the speakers.

So yeah, now you're moving around one thing.

This is a black box, but there's also one on the right, which is probably the wind.

Yeah, this is the one you are moving right now.

Yes.

Okay.

So you can see this also interesting.

We can just see a black box.

Okay.

Okay.