Thin points of certain Markov processes with jumps
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Transcript: English(auto-generated)
00:16
Thank you for all the organizers.
00:21
So Bastien, Pascal, Celine, and Lingxiao. And thank you for being here to listen to me. I'm going to talk about a joint work with my supervisor, Stephen Saway at KTEI in France. The topic is about the regularity properties
00:41
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 theme points of a class
01:01
of Markov processes of this joint. Basic problem is the following. So let us consider a continuous time run, a stochastic process in R-D. So X in zero one. So let us define the occupation measure of this process,
01:22
which is simply the time spent by this process at any fixed measurable set in R-D. So the question is the regularity about this measure, this random measure. So first, the most basic question
01:42
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,
02:03
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
02:23
for the occupation matter of ball center at this point X. It is simply the positive real number H such that we have this relation. First of all, it is not always well defined because this limit may not exist.
02:46
Another observation is that for many measures, it might happen that this exponent H may depend on the position of this X.
03:02
So I'll give some examples. The first example, most interesting example, the most interesting example is the Brownian motion. It has been a while, it has been studied. So we consider the local time,
03:21
the existence of local times, and in dimension one, it does exist. And in higher dimensions, Brownian motion does not have low times, and we consider the local dimensions as was defined in the last slide. So Perkins and Tyler have proved that
03:41
for all the points in the support of this measure, Brownian motion has local dimension two. It says two things. First, the limit, the log-log limit does exist. The second thing is that for all the point,
04:00
the regularity exponent, this H, is the same for all X. So for another kind of Leibniz product, this is a special case of Leibniz process. Another class of Leibniz process, for example, increasing stable Leibniz process, also called Spartanators.
04:21
Alpha stable Spartanators, we can also consider as a question. Its occupation matter has local dimension alpha. In this case, not for all the X in the support, but for almost every point in its support. This is a result by Hu and Tyler in the late 90s.
04:43
So does there exist exceptional sets, exceptional points? The quick answer is yes. I'll talk about this later. Okay, let us just mention another related question. For Brownian motion in higher dimension, we don't have local times,
05:02
but 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.
05:20
So this is a work in the early of the 2000s. We can see a check of paper by Danbo Paris, Rosen and Zetui. Okay, so the framework that I adopt to consider this problem is a multi-fractal analysis.
05:41
The basic, the goal of this framework is to distinguish different local behavior of the measure, considered measure by this description of the set of points with a given regularity exponent, means that with a given power law of H, this kind.
06:03
So the definition would make sense, always make sense is the upper local dimension. We, instead of taking limit, we take this limit for this log-log quantity. We can always, we can also define
06:22
this lower local dimension, which is only the limit here. And the limit 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.
06:47
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 a word, it describes the whole thing
07:05
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, kind of site.
07:22
So finally, what is called a same point? The same point recorded for alpha-stable coordinator for almost every point, we have this power law with alpha. In this case, the power law exist, the limit exist for almost every point.
07:44
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.
08:01
So we call it, in this case, we call this the point X, a same point. Hu and Tyler 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,
08:22
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,
08:41
more than alpha, it has the host of, this is only a simple empty set. And this alpha corresponds to the host of dimension of the range of the alpha stable spot nature. Okay, so with all this in mind, let us just give a few words on the process I studied.
09:03
So the goal is to describe same points of the jump diffusion, jumping SDEs by using the notions of multi-factor analysis. The kind of difficulty or differences is that for this kind of more general Markov process,
09:20
we don't have stationary increment. Okay, this is the difference between process I started and that of Hu 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.
09:43
We remarked that if beta is a constant function, we recover what we call alpha stable spot nature. Of course, here, we only keep the large jumps. The large jumps does not influence
10:01
the sample path properties of a process. We remove it. Okay, so the SDE satisfying by this process is this coin. It is written as a stochastic integral with respect to some Poisson point process,
10:20
some Poisson error generated by a Poisson point 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
10:43
and the way to compute, 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 matter. We spot this matter is just the range of this process
11:00
with some closure, with this closure. Okay, we consider a point in the support, we consider this quantity, we consider a point in the set with this property. So this limb sweep is bigger than H minus epsilon. So for even many scales tending to zero,
11:24
we have this inequality. And this describes the time spent by this process N inside the ball with radius R N. It is upper bounded by something like this. It means that the process cannot be too slow.
11:44
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 to one over H,
12:00
essentially multiplied by beta. Okay, so we need some increments, some increments estimates to do some further computations. So the second key estimate is the following.
12:21
So if we consider the increment, this is not exactly the increment of our process N, but the increment truncated the large jumps bigger, bigger than the time increment power to one over delta.
12:43
If we truncate all the large jumps of a certain scale, then we have this uniform control for the uniform control for the increment of the process. And the larger jumps are much smaller to control. Once we have this,
13:00
we can give a 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.
13:23
It means that this has some large jumps. Let's go smaller jump, because the smaller jump accumulation only gives this order. This is the big order. So definitely there are some large jumps.
13:41
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. So if we highlight all this kind of
14:03
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.
14:21
And by considering its 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.
14:43
In fact, we want to consider the set of X, such that we have this regularity H. So there's still a step from time to space. In this case, we need an analog
15:03
of the dimension doubling theorem for Brownian motion. The lower bound is much more involved, we're not going to talk about it. It concerns the construction for control sets inside this always a level set of singularities. So, result is the following,
15:24
upper multifractal spectrum of this matter is this, in fact, this is a random mapping, which is the superposition of all these kind of curves. It gives also a random fractal effect.
15:43
Thank you for your attention. Thank you.
16:05
Thank you. Thank you very much. Thank you.