for inviting Tobias for organizing the list for advertising. So let me directly jump in.

So today I try to select the topic, which I would say the most mathematical that I would have

in my group. And it's about the live walk bridges. I will explain in a second what it is.

And this work is basically is a two-man business. So with my long-time collaborator,

Sergey Denisov, who is now a professor in Oslo MET. So please, again, so we briefly discussed with

Tobias, please feel free to put the questions in chat and then he would read it for me. I am not

able to really look at the chat at the same time while I'm speaking, but feel free to interrupt.

Okay. Let me then just directly dive in into the topic of

live bridges. But with that, I have to start with live walks. So for those of you who were already

here during my entrance for lasing, they know a little bit about live walks, but I will repeat it

anyhow. So the model of live walk is a special kind of a random walk model, which was introduced

by these two very nice gentlemen. One is Yossi Klafter, who until recently was a president of

Tel Aviv University and Mike Schlesinger, who is a really high-rank officer in the office of novel

research in the US. I think in the role of, yeah. So very nice guys. And then in 1982, they published

the paper on live walks, which historically was motivated by them sitting together at the lunch

break and looking at how one's particular little fly is moving around in the air. And then they

said, look, this thing moves always with constant velocity instead of, it only changes the directions

instead of making jumps and waiting times, as was the dominating picture of continuous time random

walks at that time. So they said, let us try to build a model where a particle always moves with

a constant velocity. And this is the model of live walks where a walker chooses a random time

and the random direction. Then it moves along with this random direction for a fixed random time. In

this way, it creates its trajectory. So it's always in the move and this moving happens with

the fixed constant velocity, which is denoted by the V naught in here. And then the flight times

are randomly distributed. And typically we choose to describe them with the power law, whereby the

varying, the exponent tail of the power law over here, this gamma, then you can access different

regimes of the diffusion. And if you now imagine that the displacement during a single flight is

just proportional to the time of the flight via the constant velocity, then you can argue in terms

of these distances distributions. And then in case of gamma being large enough, when gamma is larger

than two, then you can calculate the mean squared travel distance of every individual displacement.

And then when it is finite, then you know that by summing up these intervals of a certain length,

where the variance of this length is finite, you can use the Gaussian. So the central limit

theory, which will tell you that after many such steps, the position of the particle is going to

be described by the Gaussian distribution. So that's the essence of the CLT. However, then the

question arises, what happens if you take gamma, which is smaller than two? And in that case,

if you go ahead and calculate the mean squared travel distance, then you would see that it's

going to be infinite. So which if that's possible, not we'll discuss later, but formally it is going

to be infinite. And in this case, the central limit theorem breaks down. And then, however,

in 1937, and then the two Russian guys, Gnidenko and Kolmogorov, some years later,

they really looked at these type of the distributions. And then they say, okay,

that there is actually not to worry, because there's a generalized central limit theorem,

which says that if you add up random quantities, which are power law distributed, they're also

converging to a single universal distribution, which is known as a Levy distribution, which

plays the same role as the Gaussian for the case of finite variances of individual random variables.

For the Levy distribution, there is no explicit expression in like that we can write down as a

Gaussian function only for some particular cases of gamma, you can do that. And in general,

the Levy distribution is represented with the help of its characteristic function, which is a Fourier

transform of its PDF. And then it has this typical stretched exponential form minus k to the power

gamma. And you see actually when gamma is equal to two, that's the diffusive regime, you would

recover the characteristic function of a Gaussian process. Then for what is important for this PDF

of the Levy distribution is the individual steps, which actually sum up together to end up