 difference because I received an email saying that the first half of the talk should be really accessible to graduate students. So I decided to go from the quaternionic setting to the complex setting. So in this complex setting of course several things are already known. So part of the talk will be to remind some of the results that already exist and I will show you still some new results in some geometric context. So the topic of the talk is not exactly either in mixing or in control but it touches a little bit of both topics. So I thought I found that it was an interesting thing to speak about. So we are going to be interested in simple functionals which are related to the Brownian motion. So consider a Brownian motion is a complex plane which is started from zero. So you start from zero and you run a Brownian motion. So this is something that you can see as a limit of two-dimensional random walks. So this is a continuous analog of the two-dimensional random walks in the plane. We are interested in a very basic functional of the path which is an area which is swept out by z t. That is a moving point z t up to time t. So I'm looking at algebraic area meaning that it can take possibly negative values. That is loops are going to be counted differently the area within loops according to the orientation. So the formula for such functional is given simply by the integral of the area form in the complex plane which is x dy minus y dx. Sorry there is a typo here. Along the path of the two-dimensional Brownian motion path. So if the Brownian motion had smooth path then you understand that from Stokes theorem given the fact that the exterior derivative of x dy minus y dx is two times dx dy you compute exactly the area which is swept out by the path up to time t. So it turns out that Brownian motion path can be approximated for instance in the rough path topology by smooth path. So it makes perfect sense and you get a well-defined object. If you approximate the Brownian motion path by smooth path you compute the corresponding area and you take the limit of the corresponding areas. You are going to get something which is well-defined, probabilistically, which is the area which is swept out up to time t by the Brownian motion. And this gives for those of you who know about stochastic calculus, this gives the ito stochastic integral. That is this gives an integral in the ito sense. So this is a main functional. Yes? So that's a good question. So for instance you can integrate you know harmonic forms against any continuous path. So in that case continuity of the path is enough, no smoothness is required. However in this setting I'm not integrating an harmonic form so I need some older regularity of the path to make sense of this integral of a smooth form against one half minus epsilon older path. What is that? This is subtle because we have older for a parameter gamma, older which is more than one half. It will be the young integral. However Brownian motion path is one half minus epsilon older and this is why we need actually rough path. That is we need to go one level higher in the approximation. But anyhow we get by approximation or by directly by ito calculus an object which is a stochastic area. So this object actually was studied for a very long time by Paul Levy even before the notion of stochastic integral was made precise. That is he had an intuitive notion of stochastic areas and he was able to compute in particular the distribution. So the famous Paul Levy area formula is a formula that allows you to compute exactly what is the distribution at time t of the area which is swept out up to time t by the two-dimensional Brownian motion. It's even a little bit more precise by this because you see that the formula gives you a conditional Fourier transform that is not only do you know the distribution of st but you also know the joint distribution of st together with the position of the Brownian particle at time t. That is you know the distribution of the couple not only the distribution of st itself. So you have a nice formula given by a very nice function which is here and it was proved by Paul Levy even I told you before the notion of stochastic integral was made precise. So what he did he made a Fourier expansion of z t that is of the Brownian motion pass and he computed the Levy area of the Fourier approximations and he was able to prove you know after some harmonic analysis that there is an equality with the function which is given here. So the formula which is here can look to you a little bit complicated but it's actually a very nice formula which has the connections with a lot of area of mathematics. So there is a connection to rough pass theory that I just mentioned before because in rough pass theory the stochastic area of the Brownian motion plays a fundamental role. Essentially it says in rough pass that if you know how to construct the stochastic area of a pass then actually you know how to construct any integral with respect to this pass. So this is a continuity result which is fundamental in rough pass theory and if you look through the argument you know in rough pass the proof and the formula for the Levy area formula appears in some of the arguments. So there is also a connection which is more unexpected with analytic number theory and in particular with the Riemann zeta function. So this connection was uncovered by Marcure and Jim Pitman and Philippe Bianne. This is an exact formula for the melin transform of the stochastic area that can be expressed in terms of the zeta function. That is there is a probabilistic representation of the Riemann zeta function in terms of the melin transform of the stochastic area. It's a very simple formula. That is essentially the zeta function is the melin transform of the stochastic area. So if you want to have some insight and some application to number theory for instance you can go to the paper by Pitman, Jor and Bianne. For instance you'll find an argument, a probabilistic argument that exists that there exist infinitely many primes which are equal to one modulo 4. It has also connections and I wanted to speak actually a little bit about that with sub-Riemannian geometry and the heat kernel on the Heisenberg group because it turns out that you have a very nice diffusion which is associated with the stochastic area which is defined as a three-dimensional diffusion where you put together the Brownian motion path zeta with the stochastic area. So a key insight about this is that the stochastic area itself does not satisfy the Markov property. It's not a Markov process. However if you had the path of the Brownian motion then you get a Markov process. This Markov process turns out to be the horizontal Brownian motion on the Heisenberg group. So this is a connection that was uncovered by Gavo in a famous paper in Acta Mathematica where he explained that the heat kernel on Heisenberg group actually is the probability density of Seismarkov process XT. So in other words Seismarkov process XT is the horizontal Brownian motion on the Heisenberg group. In terms of control it says that the horizontal Brownian motion on the Heisenberg group is controlled you know by its two-dimensional projection on the plane which is simply a two-dimensional Brownian motion. So the formula of Paul Levy admits many different proofs. Okay so I already told you about the original proof by Paul Levy which was very ingenious but which is not the shortest one. There exist many proofs depending on what aspect of the stochastic area you are interested in. So if you are more interested in sub-Riemannian geometry for instance you prefer maybe a PDE proof which tells you that the stochastic area, the distribution at 90, the fundamental solution of the heat equation associated to the sub-Laplacian on the Heisenberg group. So there is a purely PDE proof of this CRM which is due essentially to the observation that the sub-Laplacian on the Heisenberg group is unilaterally equivalent to the harmonic oscillator in R2. So using this unilaterally equivalence you'll find the formula for the heat kernel on the Heisenberg group. So this is the first proof. There is a proof by Paul Levy and there is a proof that I like a lot which is due to Marcure which is a purely probabilistic argument. It only uses probability, it does not use geometry, it does not use PDE and this is an argument which is very robust because we see that we'll be able to generalize it to a much larger framework than stochastic area. So the first observation is that Brownian motion, its distribution is invariant by the orthogonal group of the plane. That is you can do a rotation of the Brownian motion, you still get another Brownian motion. So if you are interested in computing such type of conditional expectation, it does not really matter exactly where the point Z is, it just matters on which circle it is due to rotational invariance. So what it means is that the conditional expectation given the position at time t is the same as the conditional expectation given the modulus at time z. So this is the first observation. I do a reduction of the dimension of the problem. That is I start with a three-dimensional problem. If I apply orthogonal groups, that is if I apply symmetries to this problem, I reduce the dimension of the problem and I have now actually if you think about it a two-dimensional problem. So using this invariance and then using stochastic calculus, especially a theorem that says that stochastic integrals are time-changed Brownian motion. So this is a classical theorem in itocalculus. So this is related to C's Dombies-Dubin-Schwarz-Martingale representation. You deduce that this conditional expectation after Martingale representation can be computed in this way. And then the next observation is that there is a very nice change of the probability on the pass-space of the two-dimensional Brownian motion that is going to transform the process z t into what is called an Einstein-Ulenbeck process and C's Gersonov transforms on the pass-space of the two-dimensional Brownian motion will reduce the computation of these additive functional of the square of the Brownian motion which is what we call the two-dimensional Bessel process that is it's a diffusion which generator is a two-dimensional Bessel operator. So it transforms its additive functional of the two-dimensional Bessel process into simply a square of z t that is into the square of a Gaussian process. So if you compare the proof that is the proof of the Laveria formula by Yor and the PDE proof they parallel each other that is the fact that the sublaplation on the Heisenberg group is unitary equivalent to the harmonic oscillator is exactly at the heart of the Gersonov transform which is here. That is you'll find that a change of probability measures is constructed exactly on the unitary map between Heisenberg group and the space of the harmonic oscillator. So this is an observation which later is important in different frameworks. So my goal will be to explain some natural generalization we can think of of stochastic areas. So after the plane maybe one of the simplest space on which we can think about a notion of area which is swept out by a path is a Riemann sphere that is Cp1 the complex projective space of complex dimension one which is the same as S2 that is the two dimensional sphere. So in that framework or I will speak directly in the framework of Cpn which is a complex projective space of dimension n there is going to be a notion of area because to construct area of curves that is area of which is swept out by paths you know on a manifold what you need is an area form. So this area form x dy minus y dx where does it come from? That is y is it what we call an area form in the plane. A way to see that is that what I'm speaking about right now is how do we define an area which is swept out by a path on a n dimensional manifold. It's a geometric question. So this is my question that I want to answer now. What are the manifolds for which the notion of area swept by a path makes sense. That is what do we need you know to define an area form. Okay so we need one form with exterior derivative that is if you apply the Stokes theorem give you something that when you integrate on two dimensional spaces you get something that looks like an area. That is what is the structure we need on the manifold to have this notion. So at least a convenient framework is a framework of simplectic manifold for which you have a canonical two form. This canonical two form will define area processes for paths. So if I am on a manifold and I am able to find a one form with exterior derivative give you a simplectic form on the manifold then maybe I have a good notion of area which is swept out by a path. So a very good framework on which simplectic forms are canonical are Keller manifolds. So Keller manifolds are simplectic manifolds that is manifolds for which you have an area form and this area form is intrinsically associated with a distance on the manifold. So this is a perfect framework to have areas swept out by paths on Keller manifolds because we have a canonical area form and this area form is intrinsically associated with geodesics that is is intrinsically associated with the distance on the manifold. So I close it I close it this way by choosing a geodesics between zero one by geodesics. Yeah exactly. So what are the simplest examples of Keller manifolds? So if you are in dimension two that is in real dimension two well you have in that case you know the complex plane you have the two-dimensional sphere and you have hyperbolic space. In higher dimensions the story is a little bit more complicated because even dimensional spheres they do not carry Keller structures. So the model spaces in Keller geometry are not the spheres they are what we call the complex projective spaces. Okay that is the analog of spheres are complex projective spaces in this geometry and we see that the analog of hyperbolic spaces are complex hyperbolic spaces. So I'm going to be interested at least in the first part of this part of the talk in the case of I'm looking for a Brownian motion in the complex projective space I'm going to explain you how we construct such object so don't worry too much about it and then I'm going to look at the area which is swept out by this Brownian motion on the complex projective space. So I use a lot of words that maybe some of you are not familiar with so let's go back to the basic so I'm going to work in cpn what is cpn cpn you can see it as a manifold or as a set as a set of complex lines in cn plus one that is you look at all the lines going through zero in cn plus one you have an n-dimensional manifold which is called the complex projective space so cp1 for instance it's not difficult to see even geometrically that is going to be isometric to the two-dimensional real sphere however cp2 is not isometric to the four-dimensional sphere so the isometry between complex projective spaces and Euclidean spheres only hold in dimension one so there is a natural way to parameterize line going through zero in cn plus one this is to use Hino-Mogenius coordinate that is you consider the set of coordinates which is given in cn plus one by wj is zj over the n plus one in this way you are able to parameterize all the lines going through zero but one which correspond to the n plus one is equal to zero which correspond to the point at infinity in the complex projective space so this set of coordinates sees in a objective way all the point in cpn but for the point at infinity that correspond to the line the n plus one is equal to zero so for us you know if we consider probabilistic properties we need only set of coordinates which are defined almost everywhere because we will see that Brownian motion if you don't start it at the point at infinity it's never going to visit it so this is the property of the Brownian motion pass you know that in dimension higher than two it does not visit points it visit any open set but it does not visit points therefore you know the probability that Brownian motion sees this point at infinity is zero so if you have a set of coordinates which is defined almost everywhere this is perfectly fine for the Brownian motion so then how do we construct Brownian motion so for those of you which are a little bit more geometric oriented that is you have in mind for instance the case of the Heisenberg group there is a nice submersion which correspond to the projection I spoke about the projection of this three dimensional path into the plane so this projection is very famous in physics this is what is called the hopf vibration so there is Riemannian submersion from the two n plus one dimensional sphere into cpn which is simply given by this inhomogeneous coordinates that is if you take a point on the two n plus one dimensional sphere which you can see embedded inside of cn then if you consider this map what you have is a Riemannian submersion that is it's going to preserve the metric the Riemannian metric of s to n plus one is going to induce in a very nice way a Riemannian metric on cpn that is you are going to project geodesics for instance the same will be true for brown and motion sees submersion since projection map is going to transform a brown and motion on the two n plus one dimensional sphere into what we call the brown and motion on cpn okay so this is a perfectly nice way you know to construct brown and motion on cpn is that you take a brown and motion on the two n plus one dimensional sphere and you project it through this hopf vibration onto a brown and motion on the complex projective space cpn it will not work there will be a time change so that's a very good question actually if you start from a brown and motion in cs plus one first you can project it on the sphere if you divide you know by the modulus but if you project it on the sphere you don't get a brown and motion on the sphere it's going to be a time changed of a brown and motion on the sphere so if you don't want a time change that is if you don't want this conformal factor you need to start from the sphere itself yeah so this is a standard construction of the brown and motion on cpn to start from the brown and motion on the two n plus one dimensional sphere and you project so you get a mark of process which is associated with the Laplacian on cpn okay that is from the general picture of diffusion theory you know if you have a mark of process you have an associated generator and broadly speaking brown and motion on a riemannian manifold is the mark of process which is associated with the Laplace-Beltrami operator of the manifold so this construction yields the mark of process which is associated with the Laplace-Beltrami operator on cpn so as I told you from the very beginning cpn is nice so this is the space I want to work in because I can make a geometric sense of what is the area which is swept out up to time t in that framework so the formula which is given here I give it to you as a definition you integrate a given one form alpha along the path of this brown and motion on cpn so what is this one form alpha well you can read the formula for the one form alpha in C's integrand that is this is a formula which is given in inhomogeneous coordinates this is a one form such that the alpha that is the exterior derivative of alpha is almost everywhere equal to the symplectic form of cpn so there are some computations which are involved here to see that when you integrate this one form and you use stokes identity you get exactly the area form on cpn which is the symplectic form so there is a little point that I wish to mention here is that you know in compact Keller manifold it's a general theorem that the Keller form is never exact it's always closed but it's never exact however we don't really need that the one form alpha satisfies d alpha is equal to the Keller form almost everywhere but we just need that d alpha is equal to the Keller form almost everywhere we don't need that this is true everywhere so if I remove one point which is a point at infinity which is never visited by the brown and motion my formula is perfectly valid I also put a little bit under the rug that is one form alpha is not unique okay so I choose one which is canonical in inhomogeneous coordinates but you know I can reproduce my computations for any form alpha that satisfies d alpha is equal to the Keller form almost everywhere so it's just going to induce some different in constant you know in my computations so I start from this one that you can take as a definition so this one is the analog of the levy area formula in cpn in my exotic space cpn so there was a nice theorem that relates you know this stochastic area to the interpretation of gavel that I spoke about so I started you know with the brown and motion in the plane if you add the stochastic area you have a horizontal brown and motion in hazenberg group so there is a similar fact that is if you have a brown and motion on cpn you add the stochastic area what you get actually is the horizontal brown and motion on s2n plus one so if you take a process on cpn you add the area you get a process you know on the 2n plus one dimensional sphere which is going to be associated with the horizontal laplacian of the hopf vibration so in other words the situation is exactly identical to the situation of the hazenberg group that is the area process it's not a mark of process by itself but if you add the pass that is a position of the pass at time t then you get a mark of process with a very nice geometric interpretation because this mark of process which is given by this formula is exactly the process which is associated with a sub laplacian on the 2n plus one dimensional sphere okay so this is exactly the parallel with the hazenberg group situations so in particular if I am able to compute the probability density of this pass of this random pass I am able to compute the heat kernel of the sub laplacian on the 2n plus one dimensional sphere okay so this is the heat kernel interpretation of the computation I'm going to show you so this is a very nice theorem and now I want to apply the proof by your that I gave you at the beginning for the hazenberg group that is how do I compute you know the probability density of this pass that looks very complicated so the key point will be to perform Gersonov transformation on the pass space paralleling the fact that the sub laplacian on s2n plus one is going to be unitary equivalent to a harmonic oscillator on the 2n plus one dimensional sphere so I told you in hazenberg group if you start from the sub laplacian you get an operator you know self a joint operator which is a sub laplacian which is unitary equivalent to the harmonic oscillator so here this is the same if you start from the sub laplacian on s2n plus one you have a self a joint operator which is unitary equivalent to a harmonic oscillator on s2n plus one so the potential of the harmonic oscillator on 2s plus one is the sine square of the distance okay the potential of the harmonic oscillator euclidean space is the square of the distance in the sphere the potential is the sine square so if you look at this unitary equivalence you can guess a Gersonov transform on the pass space by using the yaw method so in particular there will be two steps in the proof a first step is a dimension reduction of the problem that is I'm going to let an isometric group act on my problem to reduce the dimension so good for us cpn is what we call a hank one symmetric space meaning that the isometry group of the manifold act transitively on the space so I have a lot of symmetries on cpn okay there was a transitive action of the isometry group so the law of the Brownian motion is always invariant by isometry that is if you apply an isometry to Brownian motion you get another Brownian motion so if you apply the action of this group on the law of the Brownian motion you can reduce the problem which was 2n plus one dimensional into a two-dimensional problem about a two-dimensional generator it means that I am looking at the sub laplacian on the 2n plus one dimensional sphere in radial coordinates in cylindric coordinates so there is a formula which is directly given here that follows from some geometric analysis of the sub laplacian and the Brownian motion on the sphere and in particular the important point that I'm going to start with is the fact that his stochastic area process as in the Hasenberg group it turns out that this is a time-changed Brownian motion so in the Hasenberg group you know my stochastic area ST was a time-changed Brownian motion this is the same situation in the complex project space the stochastic area is a time-changed Brownian motion so just by looking at the proof by your and adapting it to this uh framework using the ingredients that I told you which is this unitary equivalence between two operators we can make the relation between the stochastic area in cpn with the Jacobi heat cannon that is there will be a canonical family of operators that are going to integrate this problem which is associated with the family of Jacobi operators which if you are familiar with the special functions they are related to the radial parts you know the laplacian on a symmetric spaces so these are diffusion operators which are integrable that we call integrable in the sense that we are able to compute with them we know what are the heat kernels at least there is an expression of the heat kernels in terms of special functions and there is a formula for the green functions and so on so these are integrable systems which are always associated with these families of Jacobi operators so in particular you know the formula that is the analog of the levy area formula can be read in terms of these uh Jacobi functions so this formula for instance implies a formula for the heat kernel associated with the sub laplacian so if you want a pd interpretation of this probabilistic representation this is what it is it's a little bit more precise than the heat kernel formula because it gives a joint conditional distribution of the third component given uh the first ones so in particular there is a nice mixing okay to use this word because of the workshop which is associated with the process and which is a nice uh convergence okay that is if you look at the area which is swept out by time t by the Brownian motion on the complex project space if you divide by t that is you take an average in time you have a convergence when t goes to infinity exactly to the Cauchy distribution of parameter n so this is a multi-dimensional Cauchy distribution with parameter n where n is exactly the dimension of the space okay so this is a very nice result you know about the area which is swept out by Brownian motion up to time t if you take its average in distribution it converges to a Cauchy distribution with parameter n where n is the dimension of the space so it's a very clean result at the end about the long time behavior of the area which is swept out by the Brownian motion on cpn okay so this is a story for cpn as you can imagine i don't want to stop here and i want to understand if there is a general picture you know to the crm i just presented in this specific case of cpn and before i speak a little bit about that i want to look at the hyperbolic analog of cpn okay so you know that in euclidean geometries that is in rimanian geometries are model spaces which are euclidean space spheres and hyperbolic space in keller geometry you have complex spaces complex flat space complex projective spaces and complex hyperbolic spaces okay that is a complex hyperbolic space is not exactly the same as the real hyperbolic space so i'm going to look at this area construction in the framework of the complex hyperbolic space to see what goes through you know and in particular if there is a limit crm for the stochastic area which is swept out by the broian motion on chn so as a set chn is very easy to describe this is just a unit bowl in cn of course you have a metric which is different the angle you know in chn geometry are going to be the same as euclidean angle but the distance is computed differently that is you have a distortion of the distance and in particular it's a non-compact space even though you imagine as a unit bowl you should imagine that you have points with distance from the origin goes to infinity okay because if you go closer and closer to the boundary actually you slow down the motion and you can never reach the boundary in this geometry so i want to show you a construction of the complex hyperbolic space uh which is the the exact analog of heisenberg group and hop fibrations so in that case the fibrations is also very well known from physicist because this is the anti-desicciter fibrations so it's a little subtle because the projection that we are going to consider that is rimanian submersion actually it's not rimanian anymore it's going to be a semi-rimanian submersion that is you have to start from a Lorentzian metric on a hyperboloid to get the geometry of chn so let me just describe uh broadly you know how this construction goes on so you start from this complex hyperbolic uh anti-desicciter space which is a complex hyperboloid which is embedded in chn plus one so this is an hyperboloid that has signature n minus one minus one that is you have two directions in the anti-desicciter space that have a sign minus for the metric okay so this is called the anti-desicciter space and this is famous in physics okay because these are solutions of Einstein relativity and there is an associated submersion from this complex hyperboloid to the complex hyperbolic space chn which for me in these talks is a open unit ball so this submersion is given exactly as it was for the hopf fibrations you consider inhomogeneous coordinates zi divided by zn plus one so this gives a well defined way to construct bronyl motion on the complex hyperbolic space you are going to start from a process on the complex hyperboloid and you project down to a process on the complex hyperbolic space so if you want to use a language of semi-riemannian submersion this is a riemannian submersions the fibbers actually are totally geodesic some manifolds of chn plus one the fibbers are isometric to the circle s one and you have a very nice projection so you should think of it as being a projection map i project down a process so in this geometry i also know what is an area because the complex hyperbolic space as the complex projected space is a killer manifold so i have a symplectic form that allows to compute area form i know what is area which is swept out by a path so i give you directly sorry the formula for the one form in homogeneous coordinates you should see that it looks really like the one on cpl the only difference is that i have instead of a sin plus i have a sin minus that essentially the formula is the same so for this form alpha this one is a little bit nicer than for the form on cpn because this form alpha is the exterior derivative is everywhere the color form on chn so i don't need this is almost everywhere defined so this one is everywhere defined so the picture is the following you have brown and motion and she's complex hyperbolic space okay i told you how to construct it and you have an area process so you should believe me that it plays a role of c's quantity that we looked at in the plane so i would like to tell the same story for this area process as i did in heisenberg and in cpn so in particular i would like to see if this is related to some sub-rimanian geometry on the top space which is an anti-besitter space so this is indeed the case you can construct you know a mark of process out of the stochastic area on chn which is going to be the horizontal brown and motion on the anti-besitter space so you have on this complex hyperboloid a canonical sub-rimanian structure which is the one coming from the structure the ambience structure in cn plus one so for this sub-rimanian structure on the anti-besitter space you have a sub-laplation and what i'm telling you is that this process is a mark of process associated with this sub-laplation what is nice is that there is a formula okay so i want to point out this fact again is that this type of formula they provide formulas for the brown and motion for the horizontal brown and motion on c's sub-rimanian spaces okay formula that involves c's area which is swept out by pass okay so this is how the situation is and based on this representation the story follows exactly what was happening in the context of the complex projective spaces that is if you apply a group of isometry because the good news is that chn as cpn is a rank one symmetric space and therefore the space of symmetries is extremely large on the manifold so you can reduce the dimension of your pd essentially to one dimensional or two dimensional pd's which are connected again to jacobi functions but in that case they are going to be instead of trigonometric jacobi they are going to be hyperbolic jacobi functions and in particular you know uh worthwhile nothing fact is that the area which is swept out up to time t is again a brown and motion which is time changed so there is an ergodic serum which is going to be associated with this area process and you observe two things about this ergodic serum is that the speed is different first so it's going to be uh an almost sure central limit serum okay so i pointed out in the slide the convergence in distribution but it's possible to prove that the convergence also holds almost surely so there is a factor one over square root of t which is different from the one over t that was in the sphere case and so this is the first thing the speed is different this can be understood geometrically by the fact that there is a difference between brown and motion and compact spaces and brown and motion on non-compact spaces brown and motion non-compact space is always going to converge to equilibrium that is at the end of time you have the uniform measure on the space okay if you think of it physically speaking if you propagate heat you know and you wait at infinity the distribution of heat is uniform on compact spaces this is not the same okay that is you have dissipation of heat and at the end you end up at zero okay probabilistically speaking the brown and motion on chn is transient with probability one it's actually going to go to infinity so this is going to affect of course the speed at which the area grows okay and if you take see the effect into account you can understand where the factor square root of t comes from so this is the first observation the area does not grow in the same right you know in imperbable spaces than it does in the spherical spaces the second observation which is an observation for which I do not have an extremely satisfying answer is that the limiting distribution does not depend on the dimension it's a very surprising fact so remember that in the case of cpn you add exactly a Cauchy distribution with parameter n so n of course is what the dimension of the space that is a limiting distribution sees how big dimensionality speaking the space is in the hyperbolic space you forget about the dimension and the limit theorem is always the same so this is a universal result whatever the dimension of the space is the limit is going to be the same it's a normal distribution within zero and variance exactly equal to one so these are the stories for stochastic areas on complex spaces so these are model spaces in complex geometry you can play more or less the same goal the same game in quaternionic geometry so in that case it's a little bit more even more involved geometrically speaking that's why I switched finally this talk you know from the quaternionic to the complex setting but we can't define quaternionic areas okay and we can construct as we did and find formulas for processes which are associated with sub-laplacian on sub-rimanian quaternionic space like the quaternionic hub fibrations or the quaternionic antideciter fibrations there is an exact analog of the formulas I told you in the complex case so just to finish in the last 10 minutes I would like to speak now you can reboot if you want because I'm this is a part two of the talk so I'm going to restart from the beginning you don't need to worry too much about all the geometry I was speaking about and this is winding functional okay I told you I'm interested in very simple functional of the path that allow me to understand you know how it behave one you can think of is area which is swept out the other one if you are in the complex plane is how it goes around zero okay that is you have a random path this time I'm not going to start it from zero you start from a point z not and you are interested how does it oscillate around zero how does it win around zero so there is a probabilistic random variable that is going to count how the process goes around zero this is a winding okay so instead of an area form now we'll get a winding form so this is winding form is given in the complex plane as follows so is this in that case you know coming back to your question this form this winding form is harmonic so actually I don't need stochastic calculus to define the integral of this form around the Brownian motion because I can integrate you know from general theory of forms harmonic forms against any continuous path so this form is what I'll call the winding form because of this representation if you have a smooth path in the complex plane you can look at its radial part which is its modulus and you can look at the angular motion the angular motion is always given by the integral along the path gamma of this one form alpha okay so for every smooth path we have this representation so in particular the integral of alpha along gamma is exactly the angular motion of the path it explains how my particular z t you know wins around zero okay so it's natural because of this representation to call alpha the winding form and for a probabilist it's going to be interesting to study the winding process that is what happens you know when I start instead of gamma with a two-dimensional Brownian motion how does it go around zero so this is a very nice story which is associated to it I cannot tell you all the details but there have been many many contributions you know around Spitzer your local Bertrand Verner and so on so this is the starting point of a beautiful theory you know about the study of two-dimensional Brownian motion I'm just going to speak about the Spitzer theorem which is a limit distribution of the windings okay so with a very nice theorem so which is due to Spitzer in 1958 let's say that if you start from a two-dimensional Brownian motion you look at the angular motion okay you have a renormalization by a factor of two over lock t and it's going to converge to a Cauchy distribution of parameter one okay so again you see the Cauchy distribution appearing as a limit distribution of these angular motion of z t so what is funny of course is the terms the renormalization speed which has to be as locked also it's nice to have an exact formula for the limit distribution which is a Cauchy distribution so this winding formula was generalized you know to the case of cp1 by Henry McKean a little later so so in cp1 there is also a winding form okay that is the winding form on cp1 a good way to see it that is to construct it is that cp1 it's a fancy name for the Riemann sphere that is the sphere s2 s2 is telegraphically equivalent to c okay the c's projection preserves angle it's a conformal map that is windings are going to be preserves by c's conformal map by the stereographic projection so if you start from the winding form on the complex plane you pull it back on the Riemann sphere by using the stereographic projection you get a nice one form on the Riemann sphere which is my winding form okay that is you know how to compute angles essentially on two-dimensional spheres using stereographic projection okay so if you do this construction for the Brownian motion you have McKean theorem that goes for the windings of the Brownian motion in cp1 so the factor normalization is 1 over t and in that case we have a Cauchy distribution with parameter 2 so the 1 over t should not surprise you too much because the space is compact so you can imagine that Brownian motion is going to wind much more than in the complex plane okay because you go around a fixed point much more often and the last theorem is about the complex hyperbolic space the complex hyperbolic space enjoys the same feature it's conformally equivalent to c that is you have a stereographic projection that is you have a nice map that preserves angle you know between the complex plane and the hyperbolic space this allows to define what are the winding form that is you know how to compute angles you know in the hyperbolic space so you can look at the angle of the Brownian motion that is the winding process of the Brownian motion in the hyperbolic space I told you that Brownian motion in hyperbolic space is transient it's going to go to infinity okay so in particular the winding is going to be finite okay you're going to have a convergence of the radial motion when t goes to infinity to a Cauchy distribution whose parameter is given by the starting point so these planes very very well if you have a Brownian motion on a hyperbolic space how does this Brownian motion goes to infinity it's going to follow you know a direction which is given by this Cauchy distribution with a parameter depending on the initial point of the Brownian motion so there is an analog story for the Brownian motion on quaternionic spaces that is we can explain what is a quaternionic winding that is it's going to instead of counting you know an angular motion around zero you are going to to look at a winding form which is valued in the lig group su2 okay that is what plays the role of s1 in quaternionic geometry is the lig group su2 and you can define one forms which are valued in the Lie algebra of su2 that plays the exact analog of the winding form so you have nice limit theorems you know also in the quaternionic spaces which are the quaternionic projective space and the quaternionic anti-ducitor space and the quaternionic flat space all these geometric probabilistic results what what I like about them is that they are all related to sub-Riemannian geometry that is they tell you something about the underlying sub-Riemannian geometry of the space in particular you know they give formulas for the diffusions and they gave some very good insight into the heat kernel of the space so you can view at this formula from several several different points of view which are all interesting in themselves so thank you for your attention and I am done for today