 Thank you for, thank you for all the organizers, so Bastien, Pascal, Céline, Eileen, Xiao. And thank you for being here to listen to me. I'm going to talk about joint work with my supervisor, Stephen Sere, at Quittaye in France. The topic is about the regularity properties of some random measures. So I already talked about this at another conference for young people, so I'm glad to re-talk it to some other young people that I didn't know yet, so it's a good occasion to me. Okay, so same point of a class of macro processes of these drums. Basic problem is the following. So let us consider a continuous time run, stochastic processing, RD. So X in 01. So let us define the occupation measure of this process, which is simply the time spent by this process at any fixed marable setting, RD. So the question is, the regularity about this measure is random measure. First, the direct, the most basic question is the absolute continuity of this measure. So in some other terminology, it is also called occupation densities. In the Markovian context, it is often called local times. So when the local time does not exist, we can consider another kind of regularity properties for these measures. It is called local dimensions. So for this measure, we consider a point in the support of this measure. We search for, we want to observe some kind of power law for the occupation measure of both centered at this point, X. It is simply the positive real number H such that we have this relation, okay? It is, first of all, it's not always well defined because this limit may not exist. And another observation is that for some, for many measures, it might happen that this exponent H may depend on the position of this X. So I will give some examples. The first example, the most interesting example is the burning motion. It's studied, it's been well, it's been studied. So they reconsider the local time, the existence of local times. And in dimension one, it does exist. And in higher dimensions, a brand motion does not have low times. And we consider local dimensions as was defined in the last slide. So Perkins and Tyler have, can say have, has proved that for all the points in the support of this measure, browning motion has local dimension too. It says two things. First, the limit, the limit, log, log limit does exist. The second thing is that for all the points, the regularity exponent H is the same for all the X. So for another kind of Levy process, this is a special case of Levy process. For another class of Levy process, for example, the increasing stable Levy process also called spondenators. Alpha stable spondenators, we can also consider the question. So its occupation measure has local dimension alpha. In this case, not for all the X in the support, but for me almost every point in the support. Now this is a result by who and Tyler in the late 90s. So does there exist, expect exceptional sets, exceptional points? The quick answer is yes. I'll talk about this later. Okay, let us just mention another related question. For browning motion in higher dimension, we don't have local times, and the local dimension is a constant for all the point, but we can consider some fluctuation for the regularity of this measure with logarithmic order. So this is a work in the early of the 2000. We can see a check of paper by Dan Bove Paris. Roseanne is a 2D. Okay. So the framework that I adopt to consider this problem is a multi-fractal analysis. The basic, the goal of this framework is to distinguish different local behavior of the measure, considered measure, by a description of the set of points with a given regularity exponent. Means that with a given power law of H, this kind. So the definition that makes sense, always makes sense, is the upper local dimension. We all, instead of taking limit, we take this limb sweep for this log log quality. We can always, we can also define this lower local dimension, which is only the limb inf here. On the limb exist, we call it a local dimension. Okay. So the definition of this, of the, what we call upper multi-fractal spectrum is the mapping to associate each value of H, possible H, the power law, to the host of dimension of this level set for this regularity exponent. Okay. So the host of dimension is just, okay. It's by word, it describes the whole thing a setting in a metric space is. Okay. This is the right notion in this context and in many others. But we can also consider, for example, packing dimensions of this kind of side. Okay. So finally, what is called a same point? The same point recorded for alpha stable boundary nature for mu almost every point, we have this power law with alpha. In this case, the power law exists, the limit exists for almost every point. And, but there are many other points. We have another power law with bigger exponent H. In this case, our power to H is much smaller to that one. So we call it, in this case, we call this the point X, a same point. Who entirely in 97 proved that the host of dimension of set of points with given regularity H is this function. So if I draw a picture of this spectrum, it'll be like this. This is the host of dimension of the set. This is all the possible value of H. For the other values, larger than two alphas, smaller than alpha, it has host of the, this is only a simple empty set. And this alpha corresponds to the host of dimension of the range of the alpha stable spawn nature. Okay. So with all this in mind, let us just give a few words on the process I studied. So the goal is to describe same points of the jump diffusion, jumping SDEs, by using the notions of multi-fractal analysis. The kind of difficulty or differences that for this kind of more general model process, we don't have stationary increment. Okay. This is the difference between process I started and that of who and Tyler. So the definition of this kind of process given by Bayes Bayes introduced this in late 80s. This is a Markov process with generator of this kind. We remarked that if beta is a constant function, we recover what we call alpha stable spawn nature. Of course, here we only keep the large jumps. The large jumps does not influence the path, the sample path properties of a process. We remove it. Okay. So the SDE satisfying by this process is this kind. It's written as a stochastic integral with respect to some Poisson-Pont process, some Poisson matter generated by a Poisson-Pont process with this intensity. Okay. So let us remember that we want to study the host of dimension of this set of singularities. So we need a translation from this formulation and the way to do the computations. Okay. So the first thing that we can say is that we consider a point X in the support of this mirror. The spot of this mirror is just the range of this process with this closure. We consider a point in the support. We consider this quantity. We consider a point in the set with this property. So this limit sweep is bigger than h minus epsilon. So for if in many scales tending to zero, we have this inequality. And this describes the time spent by this process m inside the ball with radius rn. It is upper bounded by something like this. It means that the process cannot be too slow. So if we translate it to the increment of the process, it means that the two side increment of the process is larger than the increment power two, some one over h, essentially, multiplied by beta. Okay. So we need some increments, we need some increments estimates to do some further computations. So the second key estimate is the following. So if we consider the increment, this is not exactly the increment of our process m, but the increment truncated the large jumps bigger, bigger than the time increment power two, one over delta. If we truncate all the large jumps of a certain scale, then we have this uniform control for the increment of the process. And the larger jumps are much smaller to control. Once we have this, we can give good estimate for this, we can good estimate for the increment. So the observation is the following. For smaller increment, we have for smaller jumps, accumulation of smaller jumps, we have this upper bound. And for the total increment, we have this lower bound. It means that there's has some large jumps. It's a smaller jump because the smaller jump accumulation only gives this order. And I'm sorry, this is the big order. So definitely there are some large jumps. So our conclusion is that there are two large jumps beside the time t such that mt, so every mt in this set, the t should satisfy this two jumps configuration beside it. Okay. So if we highlight all this kind of double jump configuration, like what we did for the percolation, fractal percolation, we consider at each scale, the time, the increment, the intervals, with this kind of configuration. And by considering it's descendant, also consider, we also choose to keep all this double jump configuration, we get a set, a fractal set. And this will give an upper bound for this point. Remind that we do not want to compute this host of dimension. In fact, we want to consider the set of x such that we have this regularity h. So there's still some a step from time to space. In this case, we'll need a nanolog of the dimension doubling theorem for Brownian motion. Okay. The lower bound is much more involved. I'm not going to talk about it. It concerns the construction of condosets inside this all this level set of singularities. So result is the following. The upper multifractal spectrum of this matter is this, in fact, this is a random mapping, which is the superposition of all this kind of curves. It gives also a random fractal effect. So thank you for your attention. Thank you. Is there any questions? If there are no questions, thank you again. Thank you.