 Итак, мы начнем вторую неделю на новую неделю и новые курсы мы начнем. Первый спитер сегодня профессор Антон Тальмайер из Люксимбург-Университет. Он говорит о ровном движении, о геометрии и антропеформе. Спасибо большое за интервью. И, в частности, для организаторов, все организаторы, для того, чтобы дать серьезные участия в этом месте. Если вы посмотрите на мой титул, вы увидите ровном движении, который definitivamente имеет связь с probability theory. Они являются геометрией, это дифферентальная геометрия. Есть также антропия, которая имеет математическое физическое, и так далее. Так что много вещей будет приходить вместе в моих сериалах. Я должен ополучиться, во-первых, для дли и, во-вторых, для того, чтобы дать presentation with a beam. Обычно я не люблю такие ровные движения, и, во-вторых, когда я пытаюсь организовать вещи, я понял, что есть так много notions, которые я должен ополучиться и, во-вторых, вы уже знаете, и это может быть лучше, чтобы перейти между Блэкбортом и слайдом. Так что у меня есть возможность увидеть мою афилецию, это университет Лаксенбург, и в случае, если вы не осознательны, есть новая университета в Лаксенбург, потому что, во-вторых, почти 12 лет, идея была построить исследовательную университету в Лаксенбург, но это еще не закончится, и позже, в этом году, мы перейдем к новой университете, в которой была инвестация более чем 1 миллион евро. Чтобы видеть такие сомнения в Европе, идти на исследовательство, это довольно невероятно. Если у вас есть возможность влезть нас в Лаксенбург, вы очень добры. В моменте, мы все еще выживали, но вместе с постоками мы уже 50 людей, поэтому это уже не больше, так маленькое. Я немного изменил программу, от того, что я узнал в прошлой неделе. Многие или многие у вас есть очень хорошие сомнения, и у вас есть очень много специальных систем, и, возможно, ТСРИ и так далее. Но, наверное, не так много в стокастике анализа, в особенности стокастике анализа, в многих фодах. И сегодня я дам вам, как раз в стокастике, в многих фодах, выживали, просто, чтобы разобраться немного о том, что нужно позже. И потом, я буду продолжать завтра, с вооружением многих фодов, и я буду обсуждать речи-флоры, речи-флоры, функциональные эквалиды, и, наконец, энтропия формула. Я начну, чтобы показать вам мою notion. М, это дифферентрирование многих фодов, и для меня все это смешно. Если я говорю, что дифферентрирование всегда имеет значение, я не интересен в технических моментах. Все это смешно, кроме того, конечно, бронезы. Они как бы необычные, как бы красивые, как обычно. Итак, здесь моя notion. Я возьму панель с тенцией М, и тенцией вектовых философии будут указаны тенцией ГАМАТИ. Эти панели с тенцией М. Так что для каждого панеля X, A of X лежит в тенцией панели X. Это usual setting. И я всегда буду идентифицировать вектовые философии с тенцией М. Для меня вектовая философия имеет значение от функции до функции, которая линяется, и имеет значение, что это тенцией М. Так что идентификация не usual. A of a function F. So A of F should be a function at the point X, while you take this tangent vector and transport it with the differential of the function F. Well, of course, there is a dynamical point of view to vector fields. You can look at the corresponding flow. So you start at the point X, for instance, and look at this equation. At each point you have a direction, and you go along this direction, and in this way you get the usual equation for a flow to a vector field. And what does it mean? This means if you compose things with smooth functions, particularly with compactly supported smooth functions, you end up with this equation. So here on the right-hand side, you have A applied to F, the vector field applied to this function. And the left-hand side, F puts everything to R, so it's just the derivative with respect to T. I take here compactly supported things, because, as you know, flows may not live for all time. They can explode in finite times, but when they explode, they are outside of the support of my function. So this always makes sense to write down. Well, if you integrate this equation, you get the following here. So you have F at phi T of X. This is the point where you are at time T when starting in X minus F the initial point minus this integral gives you zero. Just one point. This is usually, if you fix an X and look at the behavior or the dependence in T, this is called flow curve. And by defining PTF, F composed with phi T, you can see that you can recover the vector field from the flow. So if you know the flow, you can recover the corresponding vector field. Well, why am I telling you this? What I would like to do, or the first question I would like to discuss is this works for vector fields, but can we do the same or something similar also for second-order differential operators? So typical second-order differential operators on the manifold will be of this form. Of course, we can do it more generally, but I will stick to this Herm under form vector field plus a sum of squares of vector fields. So just to remind you how does this work. If you have A i squared applied to F, well, this is you take A i applied this to F, this is a function, and you apply one more time the vector field. So this is the way how this works. And the question is, is there a notion of a flow to L? You may say first, well, that's a stupid question. What do I mean for a vector field? I have at each point a direction. I can go along this direction. And what should I do here with these sum of squares? And it turns out, indeed, there is a flow, which has very nice properties. But what one has to do is, the flow lines are no longer deterministic. They depend now on an additional parameter, which is a random parameter. So there are stochastic processes. So here is now, I start at the point x, and this is the position, where I am at time t. This will be a random variable. So I fix some probability space with a filtration. This is an increasing sequence of sigma algebras, such that xt is measurable with respect to ft. So ft will be the events, which are available at time t. And I call this flow process, just in generalization, to flow curve I had before. And what is useful, or more common is to call it L-diffusions, if, well, what should I write? If you remember the integrated formula I had at the beginning, I took a test function. I looked at f at the position at time t when starting at x. Minus, and here I had a vector field. For the vector field I write now L. Because if L, if my second order partial differential operator is a first order, it's just a vector field, then it will be the same thing. So if I have here a vector field, I should say this here is zero. But it turns out that would be too much to ask that this here is zero. The right condition is that this is a martingale. And I should remind you what is a martingale. A martingale is a process which is purely fluctuating, which does not have any drift. So in explicit terms, that means if you take this process and you look at the increments from s to t, then you get exactly what I wrote here. And it would be too much to say this has to be zero. But if you take a conditional expectation with respect to the sigma algebra f of s, I told you these are the events which are available at time s. So if you look at the behavior of this thing, only taking the information which is available at the beginning of the interval, then it should be zero. So maybe just a small reminder about conditional expectations. What does this mean? If you have a probability space, and let's say we take a sub-sigma algebra in f, then, of course, you may look at the L2 functions of omega fp. And you may also look at the L2 functions omega, but now you take the smaller sigma algebra and you restrict your measure to c. Then you can immediately check this is a closed subspace, closed linear subspace in a Hilbert space. So if you like, you have here your L2p. And inside sitting here is the L2p restricted to c. These are the L2 functions which are c-measurable. Measureable with respect to the smaller sigma algebra. So if I give you now some element in L2p, what you can do is you can look at the orthogonal projection onto this space. You get a unique element, which is called the conditional expectation with respect to c of this element x. So what you do is you project it down to a subspace, making a random variable x measurable with respect to a smaller sigma algebra. You take a Corsair look at what you had before. And this is exactly what we are doing here. So if you project it to what you know at the beginning of the interval, then this thing here will look exactly as you know from deterministic flow lines to a vector field. Just some simple consequences. We have that this NTF of x is a martingale. So if you take expectations, and of course the usual expectation is nothing, but the conditional expectation with respect to the trivial sigma algebra. Then you see that if you have a martingale, the expectations do not depend on t. So remember here this at time zero, this is zero. So expectation of this thing here at time t is the same as expectation at time zero, which gives you zero. So what you see is, if you take expectation of what you had before, you get this equation. Now the right-hand side obviously is differentiable in t. So you get this formula, take derivative of this, it will be turned out to be a semi-group, you get Pt of L of F. And so in particular if you take the derivative at time t equals zero, you can recover your operator from the flow. So if you know this stochastic flow, then you can use the formula as before. The difference is you have to average over all paths. So this means you take the point small x, you run your process up to time t, you evaluate the function f at this random point and then you average over all possible trajectories. And then you take derivative. Okay, well that's just what I said. And as before technical points, well we already know from deterministic flow, in general we cannot expect that they are defined for all times. So there may be explosion in finite time. So there may be some random time, usually that's a stopping time, depending on x, which gives the maximum lifetime of my flow. Well, I give you a first example to remind you of Ito's formula. So if you take now as our manifold just Rn, and for the operator, the standard Laplace operator, and well usually I speed up my process by the factor 2, so Brownian motion for me is always related to Laplacian. Then the Ito formula tells me what I get if I compose xt with a function, you see I have two terms here, this is differential of, so stochastic differential, this is with respect to dt. But this is not important, the important point is this here, if you take f of xt minus f of x0, this here I said if you integrate it, your martingale just goes away, so there is only the Laplacian state. So it means if I subtract the integral over the Laplacian f of f at xsds, integrated from 0 to t, I will get the martingale. And this is precisely my notion of stochastic flow to the operator given here as the Laplacian. So you quite often hear people saying Brownian motion is the stochastic flow to the Laplacian. This is exactly what this slide shows you. Well, you may say, what are these things good for? And to give you a bit an idea, I will first just assume that we have such an object, which I called stochastic flow, and show you some consequences. For instance, look at the following. Suppose you want to solve the directly problem. That means you take some domain d and on the boundary you give you a function, you want to find a continuation of this function you have on the boundary. While I take here continuous, but I can do it much more general, to the interior and there it should be harmonic, or L harmonic. So L of u in the interior should be 0 and it should have the right boundary condition. Ну, suppose we have such stochastic flow for some technical reasons, I will take smaller subdomains dn, which are exhausting d from inside, and I take my process, maybe starting at x, and I run it up to the first time, I hit the boundary of dn. So that will happen at the time x, x tau n of x, x. So I start at x, and I run the process up to the first time, I hit the boundary of dn. And well, if n goes to infinity, this will go up to the first exit time of the process from the boundary, the first hitting time of the boundary when starting at x. Well, now coming back to the definition, and now you see why I took these dn's. I can, for each n, I can choose un, such that un on this domain dn agrees with the original u. Let's suppose we have a solution to the Dirichlet problem, and that un is still compactly supported in dn. Then my functions un are test functions, and so I can go back to the defining property of my flow. I know this here is a martingale, right? So if this is a martingale, I stop it at the time when it's exiting dn, it will still be a martingale. So from what I said, I can take expectations of this here, will give me zero, but be aware l of un Well, up to at xr, and r is less than the first exit time of dn, but l of un is on dn the same as l of u, and l of u is zero. So this thing here goes away, and I get this formula. Here, u of x, so if I have a solution to the Dirichlet problem, it must have this property. Now the right-hand side depends on n. I ship n to infinity, so while u is certainly bounded, because it's continuous on dbar, so I can use dominated convergence, and I get the formula here on the right-hand side. This formula still depends on t. So I can also ship t to infinity, and here I make a hypothesis. Let's suppose that the first exit time, x is finite. So the process will exiting when starting at x in finite times my domain. It will not stay forever in d. So let's make this hypothesis. Then if we take the limit as t goes to infinity, while this here continuous process just goes to x at the time of exiting, but there I'm sitting, the x tau x, there I'm sitting on the boundary. And on the boundary my u equals phi. So I can replace here the u by phi. And so what we found is the following, that my solution u of x, if there is such a solution, has this representation. So you take the boundary function, and let your L diffusion starting at x running up until it hits the boundary, and then take the expectation over all phi's. I come to that here in just one moment. So what it means is I recover that my u of x equals this expression, just an integral over the boundary function with respect to this exit measure. So that means you take, if you have d here, and you take some part on the boundary, then starting at x, you take u of x is the probability that your diffusion will exit d via a. And with respect to this measure I have to integrate. Well, this is interesting because what we just showed with more or less no work is no matter what the manifold is, what your space is, no matter what your operator is, no matter what the boundary is. If there is a solution and the corresponding L diffusion will exit the domain in finite times, then the solution will be unique. And it will come by this formula. Here u of x equals that. And this is quite a remarkable statement already. So if you have any chance to solve the problem, there is only one way to do it. Right? And well, we have this hypothesis and as you asked, what does it mean? Well, it means or it's basically a question that the operator should not be too degenerate. So if you start at x you should not stay in d for all time. Of course, if the operator is zero you will not move at all. But it might also be possible that if you have a very degenerate operator that you do not go out. And for the exercise session I prepared some cases where you can look at the behavior. And of course, if this is the only way how to solve the problem you may say, well, let's define or look at this function then we have to see whether it satisfies the right boundary condition and for that you will need at the exit time when starting at x goes to zero in a weak sense weekly if you approach a boundary point. Then you this function will then have the right boundary conditions and you have to check this is differentiable that you use techniques available in stochastic analysis and you get the solution to the directly problem. Of course, this formula you can easily implement on a computer. So what you do, you start in x and you simulate the trajectory of your diffusion and for instance take small ends you let them run off at the point x and you kill them as soon as they reach the boundary and you do this for some time then you look at the distribution of the dead ends on the boundary this gives you your measure and this you use to approximate this integral okay second very brief application if we have such an object which I called L diffusion we can immediately solve the heat equation so this is the parabolic problem time derivative of function u equals the operator applied to you specifying some initial condition in this situation fix capital T and let time run backwards then this is a space time process so x t is on M here is time running backward and it's easy to check this will be diffusion to this parabolic operator and then just do the same thing as before if you suppose that there is no explosion that when starting at x your process lives forever you go back to the definition of an L process substitute this here now on the looking at this integral you have L minus time derivative but applied to u so if u is a solution to the heat equation this will be zero so you find the formula here that u x at capital T is this expectation then you shift T to capital T which you can do because u was supposed to be a bounded solution so this converges then if small t goes to capital T to x capital T and t minus small t goes to zero but u at time zero is just my f so I just showed under the simple hypothesis that there is no explosion for my diffusion I have uniqueness of bounded solutions to the heat equation again no assumptions on L nothing just a simple observation that or it all comes from the fact that or the defining property of an L diffusion that I get a martingale well now the question how to construct stochastic flows up to now I did not say anything about that and of course you know how to construct deterministic flows you have to solve differential equations ordinary differential equations and here you have to solve stochastic differential equations so very briefly what is a stochastic differential equation well this is an abstract definition so an SDE stochastic differential equation on M this will be a pair of mapping M times E to the tangent bundle which should be a homomorphism of vector bundles over M that just means if you project it down to M you get the identity so this is this is the trivial bundle E is take Rn and this is the tangent bundle right and I write it like this XE goes to A of XE and that means if I fix X and look at this as depending on E I get a linear map from E to the tangent space TXM if I fix E and look at the dependence on X I get a vector field if I substitute here X I get a tangent vector in TXM well more coming closer to SDE we write such a system also as TX equals A of S should be capital X here and this is what people call Stratonovich differential for the moment that's just completely formal I have not yet explained what this means this is just a symbolic writing of what I said before well if if E has the dimension R it may take a basis then from what I said if A I is now this vector field I get from the standard coordinate vector EI here I can write it in symbolic way like that well but what does it mean I have to tell you what is a solution to such an SDE I will not explain what DX is X will be a process on the manifold so I will write a composition of F with some function or with a test function and the meaning is this here take F of XT that should be F of the initial point plus and you apply the differential of F to this A so look I wrote again you apply the differential you come back to R well very quickly the most important example is the following take for E the R plus one dimensional EI space for Z take the first component T and the other components are a standard Brownian motion on R R then look at such homomorphism of vector bundle write it in this R plus in terms of this R plus one vector fields and then you end up with an equation like this here so if there are no AI this would go away then you get something you probably would have seen before so this is always capital X that's the process that would just be the flow to the vector field A0 and you may work it out what this tells you and this has been the defining property then Stratonovich differential we convert to E2 and then we get an AI squared so the upshot is the following we see if we have a solution then this here will be a martingale which tells you that solutions to my SDE will be stochastic flows to this Hermonda type sum of square operators so here it is again if you solve such a Stratonovich type equation you will get diffusion to this operator right? and there is a general theorem about existence of solution that tells you basically if you fix an initial condition then there is a unique solution living up to some maximal lifetime theta and uniqueness holds if you have another solution which lives up to a time psi then psi must be less or equal than the maximal lifetime and on the interval where both solutions are defined they agree so just take this as information that we can solve SD ease and construct such processes so of course if we say now Brownian motion on M are L diffusions to the Laplace-Beldrami operator we can always write the Laplace-Beldrami operator as a sum of squares so we can construct Brownian motions but the bad news is there is no canonical way of writing the Laplacian as a sum of squares and this requires to go to the frame bundle I think there are still 5 minutes OK so very very quickly look at the bundle of orthonormal frames over M so at each point X you associate the frames in the tension space to this point where a frame is for me a linear isometry from Rn to the tension space so I get the frame when applying it to a basis to the standard basis in Rn I get this frame and the connection as you know gives us a splitting of this vector bundle over P so we get a notion of horizontal lift and in terms of the horizontal lift we can decompose the tension bundle in a vertical sub bundle in a horizontal sub bundle so for each U in the tension space to the frame bundle I have a horizontal space at U which I just get by lifting things up here to T of P and the vertical space is canonically given is everything which goes by the differential of pi down to zero so what I have here is a way of horizontal lift I can lift tension vectors on M up to the frame bundle which starts will be lie in the horizontal sub bundle and I get an isomorphism like that the point is that the frame bundle is a trivial bundle the frame or the tension bundle to the frame bundle is trivial over the frame bundle which is of course not true itself for instance the horizontal sub bundle is trivialized by the standard horizontal vector fields which you get just by lifting up these basis vectors to P and in terms of these standard horizontal vector fields you may look at the operator sum of squares here h i sum of squares which is the horizontal Laplacian on the orthonormal frame bundle yes well this is again a manifold so I have an operator which is sum of squares so I know how to define a flow to such an operator solve this equation here no longer on M but on the frame bundle and then I project the solution down to M and I get a bro to I start with Z I get U and the projection down to the manifold will be X and actually I can recover the Z from the X which is not so important and well in this way I get out of X here which is my driving process of the SDE something on the manifold which I call stochastic development along with U which moves by parallel transport along the trajectories of X and here is the important thing Z is a Brownian motion if and only if this U on the frame bundle is an L diffusion to the horizontal to Pochner's horizontal Laplacian on the third equivalent condition then the manifold will get Brownian motion well here probably one picture tells you more than lot of formulas we started with Z living on Rn then taking a frame here we identify Rn with the tension space at some point X on the manifold then this in the tension space what we are doing is now we roll the manifold without slipping along this pass here and we get a trace on the manifold and in that way the original frame we use to identify Rn and tension space at this point moves along the curve by parallel transport if this here would be a differentiable pass this is a classical procedure you may say well our pass are nowhere differentiable if this is a Brownian motion but how we did it was to replace ordinary differential equation by stochastic differential equation and as I said this gives also parallel transport so the u was taking values in the frames so you take a tangent vector at Xs go back to Rn and apply u that gives you well I think time is over so thanks a lot sorry for rushing a bit