On some relations between Optimal Transport and Stochastic Geometric Mechanics
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Cruzeiro, Ana Bela
We formulate the so-called Schrodinger problem in Optimal Transport on lie group and derive the corresponding Euler-Poincaré equations.
 We'll talk about some relations between optimal transport and stochastic management mechanics. I must say that this talk will not be very technical in the sense that, well, let's say, for example, each time I write a PDE, I won't tell you if you're in what sense the solution might be found or if it's decent or not. I just wanted to focus on these relations and open the way for new problems, which some of them are being studied, some not yet. But it's more to explain the idea of these relations rather than be very technical. OK, so it is based on various joint works, some with Marc-Arnaudon, the president and Chin-Chen, those with Jean-Claude Zambrini. And there are also, I just referred that I'm not going to talk about this, but there are some extensions of these results which have applications with fluid, notably the infinitely dimensional counterpart of what I'm going to say. Part of them in collaboration with the RATU, part of them with the other authors, plus Christiane Lyonard, also here present. But these extensions somehow go into, OK, so there are some extensions that go, so these part of the work are extensions of stochastic geometric mechanics, and these are extensions of the optimal transport part. But they don't connect as the basic results in finite dimensions, at least as I'm going to explain. So I just wanted to refer in there that the endanguier view is unresolved. OK, so let me start by recalling the, probably everybody knows this here, but OK, since it's after lunch and maybe won't rest a little bit, so let's start slowly and recall the most counter of each automotive point problem in the flat case. So very, very simple just to recall the problem. So it consists in minimizing this cost functional, so the distance between the points. Let's say I'm in a flat case here, just for it to be simple. And they minimize in the joint distributions. So I'm integrating here in the cross-pace, the cross-product space. And these are joint distributions, such that the marginals along x and y, the coordinates, are given by priority by some, let's say, probability measures. And since this cost, let's say, in control terms, and since this distance can be also written as the infimum of the autonormal velocity of path, which started x and ended y, then if you average this quantity, you get a higher value. Therefore, the most counter of each problem is equivalent to minimizing this quantity. Pxy here means a probability measure on this path where the marginals are starting. So you start from x and you end up y. Therefore, the marginals, there are measures at x and y. And then by these integrations, this means that you are minimizing over probability space, probability measures on the path space of this quantity. OK. This is, in very short terms, what the marginals are of each can look like. And then, of course, you have the sortable arrangement, version of the problem. The other version of the problem consists in which, if everything goes right, in the sense that, as I said, I'm not here saying when can solve the problem or in what kind of assumptions, regularity assumptions I have. But if everything goes right, it turns out to be equivalent to minimizing the velocity. Or, of course, the velocity is computed along the integral flow with respect to the velocities, such that the loss of the underlying flows are, again, at initial n times given by some probabilities measures. And if these measures are absolutely continuous with respect to the Lebesgue measure here, and I mean the flat space, then this density function at each time, if it exists again, will verify this equation, which is the continuity equation. OK. So this much for the standard Moroshka-Tarovich problem, which is, of course, well, except that you deal with probability measures, but it's started deterministic control problems. Now, oh, OK. Again, also, I add if I solve the Hamilton-Jackoby equation for psi, and I have this continuity equation, then this problem can be solved via velocity, which is gradient upside. So this is the equation for the velocity, and this will be the equation for the upside potential function of psi. OK. So if you want to turn this problem into a stochastic problem, OK, this is one way to view it. But if you want to think about now the underlying pass that I was talking about not being deterministic pass, but being stochastic pass, so you have here a sort of random fluctuation, like even by some wrong in motion. This is the standard perturbation we put. I have put here just in the two or three slides some epsilon just to stress that this can be a perturbation, and that normally epsilon goes to 0. Everything goes to the monstrosity problem that I was talking before. So you recall the notion of the entropy of a measure with respect to another measure, which is this quantity here. So let's choose B to be the law of the Wiener processor, this Brownian motion. OK. And so Q, the law of the diffusion, which is driven by the Brownian motion, plus some drift. Now, it turns out, and this is a central thing in your theory, that the entropy of the law of this function with respect to the law of the without drift, so of the Brownian motion, is the entropy of the initial conditions, the initial marginals, plus 1 half of the, so the L2 norm of the drift here. And this is, OK, I'll come back to this a little bit later, but this is a consequence of the well-known theorem in psychotic analysis, which is the synopsis. So what is the sort of generalization of the most counter-efficient problem for stochastic path? It's called, in most, in many frameworks, in which the Schrodinger problem comes from Schrodinger and in relation to quantum mechanics. But anyway, it's consisting, minimizing the entropy function, I was talking about, subject to given marginals q0 and q1. If y, so remember that y, that y was the drift, so if epsilon is not there, this would be the derivative of the path, simply. If y is a function of the process, the density, again, if everything is solved and exists and then we are in the right regularity assumptions, then the density will verify the continuity equation per term by this term, which is given in terms of Laplacian, as the Laplacian generates the random motion. So we have the generalization. And if you have v equals to gradient of 5 solving, now the Hamilton-Jacques-Bellman equation, which has this extra Laplacian term here, then you have the analog, so you have here again an extra Laplacian, which gives the per-derivals. So you see that if you put brutally epsilon equals to 0 here, everything goes back to the most kind of each problem. Now, if the question whether you have, and in what sense you have limits of these solutions of this problem when epsilon goes to 0 is a very delicate problem that I'm not going to talk about, but some remarks. OK, so first that you can, by making a change, a logarithm change of variable, you can transform the Hamilton-Jacques-Bellman into a heat equation. And the other is that, again, when epsilon goes to 0, formally solving the problem converges to the optimal transporter problem. But this is a constant formula level if you want to make it rigorous. You have some works of Christian Lerner, for example, that work on this subject. OK, so you have these two problems, Mochan Korovic and the generalization, which is called Schrodinger problem. Now, what does it have to do with geometric mechanics or with stochastic geometric mechanics? OK, so now I will place myself into the context of league groups. And somehow I'm going to make the Schrodinger problem, but in the context of league groups. OK, so let's do framework. So you take g to be a league group. I'm taking here a right invariant metric, but you can take a left invariant and then, well, some changes signs and some changes of some operators. But anyway, take a right invariant metric. Now, we'll be able to have a g-b to connection, e to identity, and the algebra is called Karol G. Now you take normal basis of these three algebra, special one. It's not that you cannot take a more general one, but I want it to be a clean result as easy as possible. So I'm going to make these assumptions on this vector field of these three algebra. So right invariant and such that covariant derivative of h i with respect to h i is equal to 0 for n of this, and an ordinal basis. I'm supposing here that everything is fine in dimensional analysis because everything will work fine. For my point of view, the most interesting results is R when g is infinite dimensional, notably when g is the defiomophase group of some n group. I haven't studied anything, but I guess so. So one of the examples is the defiomophase group where morally any group will do. But if you want to do the analysis, initially to mention, you have to go case by case because some things are not well defined, some things are different, so it has to be case by case. I guess loop with an interesting case that you can deal with. I can tell you. Yeah, right. But in some sense, we were also motivated by two weeks and defiomophase groups who came out of this equation. We think like this at the beginning, so this was another group. But a priori, every group, the machinery in finite dimension essentially works, and then I'm going to state it here. And then if you want to adapt it to each instant marginal case, you have to work each case. But there is no fundamental reason for it not to work. There can be technical details. Kind of different. More of the, of course, things will be measured very much alike. OK, so what does the Brownian motion on some G-league group looks like? So the Brownian motion, which generated a given by the plus delta B. Well, it can be written like this. So G with a little 0 upstairs will be the Brownian motion. And the stochastic differential equation that it obeys, it can be given. So in terms of these bases, because I show here an orthonormal basis. So it's the tangent map with the right translation here. Again, it could be left of the sum in this direction of this orthonormal basis. And because of this condition here, it's the same to use here this little circle means Trotanovich's integral. OK, there must be some definition of some integral. There is Trotanovich and there is it. And as I'm making this assumption here, it's the same. So the contraction will be same. Again, it doesn't have to be so, but I'm choosing the left most simple model to explain the relations that one can see between these problems and not going into what difficult frameworks. But it's not an essential thing here. OK, so this G with a little 0 G-league not will be the standard Brownian motion on G with this in the right invariant case. And then a general diffusion process that we call G is the Brownian motion plus some drift. Again, as we saw in the flat case. This drift is an extension is indeed, OK, I can just call it like this. But I'm going to denote it by the nabla G D T just because it's more suggestive in the sense that, well, if again, if there was not this term, this would be just the derivative in time. And here it's a sort of regularized derivative in time. So this object normally you define it like this. So you have G, you transport it back with the parallel transport on the group. Transport it to the origin, you call this xi. And so this is the usual parallel transport over the gas expanse. So it was defined by Eto, I mean, to a shown to exist by Eto one time ago. And you call and at the origin you make some derivative in time, which is more or less the usual derivative, but with a conditional expectation. So it kills the marginal part. And then you transport it back to where you were. And so this justifies in some sense why one looks at this is sort of derivative, the drift. But OK, this will coincide for this case with the TLE RG of U. OK. Now, G's Sano's theorem says the law Q of this process with a drift U. And again, I will need some assumption on U, of course. And I will with G only assumptions. But if I have some essentially out to assumption on U, the G's Sano's theorem says that the law of this process is absolutely continuous. The law on the pass space, right? On the continuous assumptions, all of these values on the legal. The law is absolutely continuous to the law of the Brownian motion without the drift. So G not if they have the same initial margin distribution. And the density dQ with respect to P is given by the exponential of essentially this drift integrated against the Brownian motion minus 1 half of the square norm of the drift. So this is the basic thing that makes the connection, that allows to make the connection between entropy and the pentadentry and its expression. Because the entropy is therefore essentially if you have the same initial distribution. It is therefore even because you remember the entropy has a logarithm, so it drops down the exponential. And then it has an integral, I mean the expectation. And the expectation kills this martingale part, this stochastic integral part. Therefore, you are left just with this. And the entropy coincides exactly. So the entropy of the measure Q with respect to P on the pass space coincides exactly with the kinetic energy after all. I mean, somehow the general line in the sense that the derivative in time is not the derivative in time, is regularized, is the drift. But coincides is a sort of kinetic energy. Therefore, you see that the Schrodinger problem, so minimizing the entropy, coincists in minimizing what is essentially the generalization of the Elton-Rohmer-Legloss. With this regularization, because you are working with non-differential group but everything looks sort of natural. Now I will need some. How much time do I have? Three. Two three? Oh, OK. Oh, sorry. Two three. Oh, OK. Because otherwise, I will get the three myself. No, no, no. I think it's two three. OK. Then it's OK. So some notations. You denote by alpha the invariant measure on G. P of x, the law of the Brownian motion, G, 0, starting on x. And the transition semi-group corresponding to the Brownian motion by P and P2 by starting with probability measuring. We assume that U and sigma are absolutely continuously respect to alpha. And we denote by m mu sigma, all the probability margins on G times G with probability measures or probability distribution such that the marginals at initial time and final time are the given mu and sigma probability measures. And also some technical thing that pi is absolutely continuous with respect to mu plus P1, which is this transition. Again, for notations of this slide, sorry, it's just sort of notations. If you have some probability distribution pi in this space, you write P pi dw, the integral of the transition kernel of the Brownian motion with respect to this marginal P pi, which is pi, which is the probability measure on the past page. Now, there is a very general theorem in entropy theory that says that if you have probability measure in this space, such that the entropy with respect to mu times P1, so P1 is the transition of the semi-group, is finite, then there is one that attains the influence, so it minimizes the entropy with respect to q with respect to P2. So this one solves exactly the problem that I was stating, the Schrodinger problem. Or if you want, it minimizes the section function, which is essentially given by the energy. Moreover, so q is P, remember, P pi is finite. This, moreover, q is P of pi, and pi attains the influence of the entropy of P with respect to this measure with respect to P1. And also, it is a law of a marginal process. So this is a very general theorem. But if you, the answer to the existence of a minimum. Again, in this assumption, it's not everything is finite. OK, so we have the problem, Schrodinger problem, defined in a general e-group. We have stated it in terms of entropy. We know how to minimize. We know there exists a minimum of this entropy. And now we want to know what kind of equations think so. And then, OK, so again, I have already had said this, but if you recall, if you recall this entropy is given by this, I call it the action functional energy. OK, so we have this stochastic Euler-Panquerier reduction theorem. G is a process of this time, a diffusion process of this kind, like before. And a process like this is a critical point of value, a, so in particular, a minimum of this point, if and only if the drift as a vector, so the drift is defined by the direction of the vector field, you satisfy these equations. So now, if you recall, now I mean this geometric mechanics part. If you recall geometric mechanics in the deterministic case, you have exactly this, this equation here, and you don't have the k. Now, what is the k? It's something else that we have here. k is, well, the expression is here, but it's essentially the R-Moch Laplace operator. So that's the reach. That's the reach, yeah. So the thing that appears here in the equation of motion for the velocity is the operator produced with the R-Moch. So just give you a hint of the proof, but doesn't make any sense. The point is that, at least if you, when you are in finite dimensions, everything goes smoothly. You have this existence result, and at the same time, you have that it solves this equation. Then you have to work in order to obtain what kind of regularity you have for you and so on in what kind of sense. For example, in infinite dimensions, things get complicated, but you have to go step by step. Do the same thing. So essentially, the proof goes like this. You have a critical point, so you have to define the variation. So you vary it. We are in the right invariant framework. So you vary it on the left with respect to exponential functions. So essentially, we are variating in the direction of v's. v's, any vector field, c1, such that in order to keep the final and initial end end points fixed, you consider v of 0 equals to v of t equals to 0. So you differentiate. And then you apply this differentiator. So you have these variations of the initial pass g. And by it or for where you compute the differential of these variations. So this is essentially eto-formal. So there is this stochastic part. This one is already, these two are already in the third to be this one. So you only have this. Yeah, I mentioned here that there's no part coming from contention terms because I'm using the assumption that allows to have eto integral equals to the tannobit integral. Then, OK, so then you derivate in epsilon all these objects. And you plug this differentiator into the action function. And you get this part as it is already in the case of finite dimensions. Nothing more. Sorry, I did a deterministic case. Nothing more. Plus a term which is, well, not very elegant term. But if you work out the details in case of righty variant, it turns out that this thing is, I mean, it's Riemann and Calcoules. It turns out this object is indeed the drama watch of greater depth than I was in action. And if you put, of course, I already said that. If you put all the HG theory we recovered, you have a Poincare reduction theorem. So this must be seen as a stochastic geometric counterpoint. OK, you also have a possibility of making of changing variables. Again, you have to define exactly what this means and it goes through the Hamilton-Jakob development equation here with Laplace-Batrony operator. And some extra remark, and then I'll go in here. If in the assumptions of Cesar's theorem, there is an extra assumption, which is that pi, this pi not that exists, and this makes the entropy finite. If it's absolute, then you have to respect for new Cesar of P1, P1 being the transition semi-group of the Brownian motion. Then actually, the pi that solves the minimizer is of this form. And this psi, so again, it has a problem structure. And you can go through a pair of equations from this mu and sigma, which are keeping at the end, to this psi and phi through these equations. And this is called, OK, I don't know what this is called, but this is a problem that was solved in finite dimension, at least by Brownian also a long time ago. And it's kind of comfortable because instead of giving probability measures, you give initial and final functions. So sometimes it's easier to deal with. So just to end up with some reference, OK, so the geometric result, which I mentioned here, extensions to include effective quantities and going to infinite dimensions and so on is here. It's in archive. But this is extensions again in the stochastic geometric mechanics part. Concerning the optimal transport problems, Schrodinger problem, there are general things here. I mean, the general statements and properties of the Schrodinger problem, you can see this here. And here, this is the extension, which is also quite recent. But in the direction of the optimal transport problem, that in order to cover some equations of fluids, like the Navier-Solvstein problem. OK, so I think I finished here. Thank you for your attention. Thank you. Thank you. Thank you. Any questions? So do I get a thread that you get the best stock situation in this way? Yes. So could you explain? There was a paper by Hugo Gomez, I think at some point in the book. Sorry, may I precise in what sense I obtain Navier-Solvstein equation? OK, so we obtain Navier-Solvstein in the sense of the theorem of the stochastic color Poincare theorem. So if you have a critical point, then it solves Navier-Solvstein equation, a representation in this sense. Concerning existence, we don't obtain it because we are infinite. As I mentioned, there's a few problems. We don't obtain, exactly, we obtain some approximations by using the Schrodinger problem, but with imposing this at each time, this effect at the density with the conservativeness. And then we have some sort of, this is studied here. I mean, the real existence of the, this is studied here. And well, there are some restrictions. OK, now please go ahead. I was just wondering, there was a paper by Hugo Gomez at some point about the real process. I know, it has nothing to do with it, and it's not the same variational principle. It is a stochastic principle. It is a stochastic principle. It is also in the form of representation. Again, he doesn't show that this gives the solution of nothing though, as I don't hear in this stochastic method approach. But the action functions are completely different. I mean, variational principle is different. It's a follow up on the same question. There are many approaches, many different approaches to the model. No, it's in the model. For instance, there is this model by Italian form. Yeah, but then this is not for Navistok, say, or not for, this is not for PDE. This is for a stochastic PDE. That we can also do that here. But that's something else. Then I don't see the relation with transport problem, by the way. Yes, this is my question. No, because there the Lagrangian itself is randomized. So there are randomness at two levels in variational approach. And again, what you do here. There are the paths that are random, right, diffusion process. And the Lagrangian, because Lagrangian here, it is essentially the three, so the square. So the Lagrangian itself is computed on stochastic paths, but it's not stochastic form. Now, if you perturb stochastic Lagrangian to, then you have stochastic PDE. You don't have Navistok, for instance. You have stochastic Navistok. So that's different. And then I didn't speak about this in particular, because I don't see the relation with optimal transport. So I want it to be in a framework where there things could be viewed from one side or the other. So that was the most simple. And especially if you are in finite dimension, because then NaviCy, more or less, can be really constructed and result in problem. It's a simple problem. Did you apply this for a G taking SO2 or SO3, like having a [?]. We have examples. Yeah, we have examples. Those would be. But again, what those have been for the optimal transport, like the rigid body, optimal stochastic rigid body, or the rigid body in transport. We can do this. We can do this. I can give you formulas and so on. Now, the point is that I don't understand very well what does this mean in terms of rigid body. I wish Dario was here, because then he would make a speech about this. Physically, I don't understand very well what this means, but we can do that. That sounds perfectly OK. It's a finite dimensional example. No problem. If you have infinite dimensional G by finite dimensional noise, can you then say something about the analysis? Infinite dimensional G, but a finite dimensional noise. So finite, about the time. Yeah, yeah. So it's a cylindrical noise, right? So only a number of directions are perturbed, it's like me. Yeah. Yeah, it's the same problem. I mean, the same problems are already there. You can have only one direction perturbed. And how is it that the analysis is complicated? In order to prove the existence of the solution, yes. In order to prove the counterpart of the stochastic point-career result, it's not that complicated. You have to work out and see that things don't explode. OK, it's not that complicated. But the existence. Because the existence, either you go through the entropy approach, or you go through the PDE approach. I don't know. Maybe there is a third way. I don't see. The PDE approach, you finally end up with the same problems that PDE people have. The entropy approach, you end up with others. But that does not depend on the number of directions that you randomize. Perhaps. So I want to ask a different question. But maybe you comment on what you just said. Perhaps in some cases, the PDE approach becomes a linear approach. On the lines that you have, the sense that if you ask, don't work, don't do the reduction. Just work in the group. Is there a chance that some of these problems, in some cases, the interventions may become statistical means? And then, or something like that, something that would be. You mean at the Lagrangian level? Because at your level? Yes. Yes. Yes. The Lagrangian level. The group level. I mean, whatever the fact. Because the group level. Yeah. So now I'm coming back to you. What I said earlier, it seems to me that indeed, probably, trying out first, before even considering the few morphos groups, trying out some of the techniques here of what the analysis looks like in the case of the group group, some interesting problems. Yeah. Thank you. Because I didn't think about studying the group group, but you need to study it. Even then, if things can be worked out in the group in the Lagrangian cycle, the maybe even the path space is all right. Then I see some kind of connection to my account. Yeah. That's the obvious option. Yes. Yeah. And then the techniques will be new. The techniques new tools will be available. So I guess that I don't see existence in any of the terms there much easier to obtain. This is after all the Bama group. Yeah. Yeah. So that's very good. I would suggest that we shift further questions to the coffee break. We have 20 minutes time. And I guess there's something urgent that you would. Well, I just wanted to add to what there is a nice equation which is one of the other equations with the Laplacian that is exactly on loop groups. So we have one of the simplest examples on loop groups. The oil punker, he's talking about this. The Laplacian equation is a new thing. And probably you can recover it. This is a good question. OK. I'll talk to you. Let's give the big thanks. So we'll. And we'll start again in 20 minutes. So let me see if we can all talk to each other. Yeah.
 We'll talk about some relations between optimal transport  and stochastic management mechanics.  I must say that this talk will not  be very technical in the sense that, well, let's say,  for example, each time I write a PDE,  I won't tell you if you're in what sense the solution might  be found or if it's decent or not.  I just wanted to focus on these relations  and open the way for new problems,  which some of them are being studied, some not yet.  But it's more to explain the idea of these relations  rather than be very technical.  OK, so it is based on various joint works, some  with Marc-Arnaudon, the president and Chin-Chen,  those with Jean-Claude Zambrini.  And there are also, I just referred  that I'm not going to talk about this,  but there are some extensions of these results which  have applications with fluid, notably  the infinitely dimensional counterpart  of what I'm going to say.  Part of them in collaboration with the RATU,  part of them with the other authors,  plus Christiane Lyonard, also here present.  But these extensions somehow go into, OK,  so there are some extensions that go,  so these part of the work are extensions  of stochastic geometric mechanics,  and these are extensions of the optimal transport part.  But they don't connect as the basic results  in finite dimensions, at least as I'm going to explain.  So I just wanted to refer in there  that the endanguier view is unresolved.  OK, so let me start by recalling the, probably  everybody knows this here, but OK,  since it's after lunch and maybe won't rest a little bit,  so let's start slowly and recall the most  counter of each automotive point problem in the flat case.  So very, very simple just to recall the problem.  So it consists in minimizing this cost functional,  so the distance between the points.  Let's say I'm in a flat case here, just for it to be simple.  And they minimize in the joint distributions.  So I'm integrating here in the cross-pace, the cross-product  space.  And these are joint distributions,  such that the marginals along x and y, the coordinates,  are given by priority by some, let's say, probability measures.  And since this cost, let's say, in control terms,  and since this distance can be also  written as the infimum of the autonormal velocity of path,  which started x and ended y, then if you average this quantity,  you get a higher value.  Therefore, the most counter of each problem  is equivalent to minimizing this quantity.  Pxy here means a probability measure  on this path where the marginals are starting.  So you start from x and you end up y.  Therefore, the marginals, there are measures at x and y.  And then by these integrations, this  means that you are minimizing over probability  space, probability measures on the path space of this quantity.  OK.  This is, in very short terms, what the marginals  are of each can look like.  And then, of course, you have the sortable arrangement,  version of the problem.  The other version of the problem  consists in which, if everything goes right, in the sense  that, as I said, I'm not here saying when can solve the problem  or in what kind of assumptions, regularity assumptions I have.  But if everything goes right, it turns out  to be equivalent to minimizing the velocity.  Or, of course, the velocity is computed along the integral  flow with respect to the velocities,  such that the loss of the underlying flows  are, again, at initial n times given  by some probabilities measures.  And if these measures are absolutely continuous  with respect to the Lebesgue measure here,  and I mean the flat space, then this density  function at each time, if it exists again,  will verify this equation, which is the continuity equation.  OK.  So this much for the standard Moroshka-Tarovich problem,  which is, of course, well, except that you  deal with probability measures, but it's  started deterministic control problems.  Now, oh, OK.  Again, also, I add if I solve the Hamilton-Jackoby equation  for psi, and I have this continuity equation,  then this problem can be solved via velocity, which  is gradient upside.  So this is the equation for the velocity,  and this will be the equation for the upside potential function  of psi.  OK.  So if you want to turn this problem into a stochastic problem,  OK, this is one way to view it.  But if you want to think about now the underlying pass  that I was talking about not being deterministic pass,  but being stochastic pass, so you  have here a sort of random fluctuation,  like even by some wrong in motion.  This is the standard perturbation we put.  I have put here just in the two or three slides  some epsilon just to stress that this can be a perturbation,  and that normally epsilon goes to 0.  Everything goes to the monstrosity problem  that I was talking before.  So you recall the notion of the entropy of a measure  with respect to another measure, which is this quantity here.  So let's choose B to be the law of the Wiener processor,  this Brownian motion.  OK.  And so Q, the law of the diffusion,  which is driven by the Brownian motion, plus some drift.  Now, it turns out, and this is a central thing  in your theory, that the entropy of the law of this function  with respect to the law of the without drift,  so of the Brownian motion, is the entropy of the initial  conditions, the initial marginals, plus 1 half of the,  so the L2 norm of the drift here.  And this is, OK, I'll come back to this a little bit later,  but this is a consequence of the well-known theorem  in psychotic analysis, which is the synopsis.  So what is the sort of generalization of the most  counter-efficient problem for stochastic path?  It's called, in most, in many frameworks,  in which the Schrodinger problem comes from Schrodinger  and in relation to quantum mechanics.  But anyway, it's consisting, minimizing  the entropy function, I was talking about,  subject to given marginals q0 and q1.  If y, so remember that y, that y was the drift,  so if epsilon is not there, this would be the derivative  of the path, simply.  If y is a function of the process, the density, again,  if everything is solved and exists  and then we are in the right regularity assumptions,  then the density will verify the continuity equation per term  by this term, which is given in terms of Laplacian,  as the Laplacian generates the random motion.  So we have the generalization.  And if you have v equals to gradient of 5 solving,  now the Hamilton-Jacques-Bellman equation,  which has this extra Laplacian term here,  then you have the analog, so you have here again  an extra Laplacian, which gives the per-derivals.  So you see that if you put brutally epsilon equals to 0  here, everything goes back to the most  kind of each problem.  Now, if the question whether you have,  and in what sense you have limits of these solutions  of this problem when epsilon goes to 0  is a very delicate problem that I'm not  going to talk about, but some remarks.  OK, so first that you can, by making a change,  a logarithm change of variable, you  can transform the Hamilton-Jacques-Bellman  into a heat equation.  And the other is that, again, when epsilon goes to 0,  formally solving the problem converges  to the optimal transporter problem.  But this is a constant formula level  if you want to make it rigorous.  You have some works of Christian Lerner, for example,  that work on this subject.  OK, so you have these two problems,  Mochan Korovic and the generalization,  which is called Schrodinger problem.  Now, what does it have to do with geometric mechanics  or with stochastic geometric mechanics?  OK, so now I will place myself into the context of league  groups.  And somehow I'm going to make the Schrodinger problem,  but in the context of league groups.  OK, so let's do framework.  So you take g to be a league group.  I'm taking here a right invariant metric,  but you can take a left invariant and then, well,  some changes signs and some changes of some operators.  But anyway, take a right invariant metric.  Now, we'll be able to have a g-b to connection,  e to identity, and the algebra is called Karol G.  Now you take normal basis of these three algebra,  special one.  It's not that you cannot take a more general one,  but I want it to be a clean result as easy as possible.  So I'm going to make these assumptions on this vector  field of these three algebra.  So right invariant and such that covariant derivative of h i  with respect to h i is equal to 0 for n of this,  and an ordinal basis.  I'm supposing here that everything  is fine in dimensional analysis because everything  will work fine.  For my point of view, the most interesting results  is R when g is infinite dimensional,  notably when g is the defiomophase group of some n group.  I haven't studied anything, but I guess so.  So one of the examples is the defiomophase group  where morally any group will do.  But if you want to do the analysis,  initially to mention, you have to go case by case  because some things are not well defined,  some things are different, so it has to be case by case.  I guess loop with an interesting case that you can deal with.  I can tell you.  Yeah, right.  But in some sense, we were also motivated by two weeks  and defiomophase groups who came out  of this equation.  We think like this at the beginning,  so this was another group.  But a priori, every group, the machinery in finite dimension  essentially works, and then I'm going to state it here.  And then if you want to adapt it to each instant  marginal case, you have to work each case.  But there is no fundamental reason for it not to work.  There can be technical details.  Kind of different.  More of the, of course, things will  be measured very much alike.  OK, so what does the Brownian motion on some G-league  group looks like?  So the Brownian motion, which generated a given  by the plus delta B. Well, it can be written like this.  So G with a little 0 upstairs will be the Brownian motion.  And the stochastic differential equation that it obeys,  it can be given.  So in terms of these bases, because I  show here an orthonormal basis.  So it's the tangent map with the right translation here.  Again, it could be left of the sum in this direction  of this orthonormal basis.  And because of this condition here,  it's the same to use here this little circle means  Trotanovich's integral.  OK, there must be some definition of some integral.  There is Trotanovich and there is it.  And as I'm making this assumption here, it's the same.  So the contraction will be same.  Again, it doesn't have to be so, but I'm  choosing the left most simple model  to explain the relations that one can see  between these problems and not going  into what difficult frameworks.  But it's not an essential thing here.  OK, so this G with a little 0 G-league  not will be the standard Brownian motion on G  with this in the right invariant case.  And then a general diffusion process that we call G  is the Brownian motion plus some drift.  Again, as we saw in the flat case.  This drift is an extension is indeed, OK,  I can just call it like this.  But I'm going to denote it by the nabla G D T  just because it's more suggestive in the sense that,  well, if again, if there was not this term,  this would be just the derivative in time.  And here it's a sort of regularized derivative in time.  So this object normally you define it like this.  So you have G, you transport it back  with the parallel transport on the group.  Transport it to the origin, you call this xi.  And so this is the usual parallel transport  over the gas expanse.  So it was defined by Eto, I mean,  to a shown to exist by Eto one time ago.  And you call and at the origin you  make some derivative in time, which is more or less  the usual derivative, but with a conditional expectation.  So it kills the marginal part.  And then you transport it back to where you were.  And so this justifies in some sense  why one looks at this is sort of derivative, the drift.  But OK, this will coincide for this case with the TLE RG of U.  OK.  Now, G's Sano's theorem says the law Q of this process  with a drift U. And again, I will  need some assumption on U, of course.  And I will with G only assumptions.  But if I have some essentially out to assumption on U,  the G's Sano's theorem says that the law of this process  is absolutely continuous.  The law on the pass space, right?  On the continuous assumptions, all of these values  on the legal.  The law is absolutely continuous to the law  of the Brownian motion without the drift.  So G not if they have the same initial margin distribution.  And the density dQ with respect to P  is given by the exponential of essentially this drift integrated  against the Brownian motion minus 1 half of the square norm  of the drift.  So this is the basic thing that makes the connection,  that allows to make the connection between entropy  and the pentadentry and its expression.  Because the entropy is therefore essentially  if you have the same initial distribution.  It is therefore even because you remember  the entropy has a logarithm, so it drops down the exponential.  And then it has an integral, I mean the expectation.  And the expectation kills this martingale part,  this stochastic integral part.  Therefore, you are left just with this.  And the entropy coincides exactly.  So the entropy of the measure Q with respect to P  on the pass space coincides exactly  with the kinetic energy after all.  I mean, somehow the general line in the sense  that the derivative in time is not the derivative in time,  is regularized, is the drift.  But coincides is a sort of kinetic energy.  Therefore, you see that the Schrodinger problem,  so minimizing the entropy, coincists  in minimizing what is essentially  the generalization of the Elton-Rohmer-Legloss.  With this regularization, because you  are working with non-differential group  but everything looks sort of natural.  Now I will need some.  How much time do I have?  Three.  Two three?  Oh, OK.  Oh, sorry.  Two three.  Oh, OK.  Because otherwise, I will get the three myself.  No, no, no.  I think it's two three.  OK.  Then it's OK.  So some notations.  You denote by alpha the invariant measure on G. P of x,  the law of the Brownian motion, G, 0, starting on x.  And the transition semi-group corresponding to the Brownian  motion by P and P2 by starting with probability measuring.  We assume that U and sigma are absolutely  continuously respect to alpha.  And we denote by m mu sigma, all the probability  margins on G times G with probability measures  or probability distribution such that the marginals  at initial time and final time are the given mu and sigma  probability measures.  And also some technical thing that pi  is absolutely continuous with respect to mu plus P1, which  is this transition.  Again, for notations of this slide, sorry,  it's just sort of notations.  If you have some probability distribution pi in this space,  you write P pi dw, the integral of the transition kernel  of the Brownian motion with respect  to this marginal P pi, which is pi,  which is the probability measure on the past page.  Now, there is a very general theorem in entropy theory that  says that if you have probability measure in this space,  such that the entropy with respect to mu times P1,  so P1 is the transition of the semi-group, is finite,  then there is one that attains the influence,  so it minimizes the entropy with respect to q  with respect to P2.  So this one solves exactly the problem  that I was stating, the Schrodinger problem.  Or if you want, it minimizes the section function, which  is essentially given by the energy.  Moreover, so q is P, remember, P pi is finite.  This, moreover, q is P of pi, and pi attains  the influence of the entropy of P with respect  to this measure with respect to P1.  And also, it is a law of a marginal process.  So this is a very general theorem.  But if you, the answer to the existence of a minimum.  Again, in this assumption, it's not everything is finite.  OK, so we have the problem, Schrodinger problem,  defined in a general e-group.  We have stated it in terms of entropy.  We know how to minimize.  We know there exists a minimum of this entropy.  And now we want to know what kind of equations think so.  And then, OK, so again, I have already  had said this, but if you recall,  if you recall this entropy is given by this,  I call it the action functional energy.  OK, so we have this stochastic Euler-Panquerier reduction  theorem.  G is a process of this time, a diffusion process  of this kind, like before.  And a process like this is a critical point of value,  a, so in particular, a minimum of this point,  if and only if the drift as a vector,  so the drift is defined by the direction of the vector field,  you satisfy these equations.  So now, if you recall, now I mean this geometric mechanics  part.  If you recall geometric mechanics in the deterministic case,  you have exactly this, this equation here,  and you don't have the k.  Now, what is the k?  It's something else that we have here.  k is, well, the expression is here,  but it's essentially the R-Moch Laplace operator.  So that's the reach.  That's the reach, yeah.  So the thing that appears here in the equation of motion  for the velocity is the operator  produced with the R-Moch.  So just give you a hint of the proof,  but doesn't make any sense.  The point is that, at least if you,  when you are in finite dimensions,  everything goes smoothly.  You have this existence result, and at the same time,  you have that it solves this equation.  Then you have to work in order to obtain  what kind of regularity you have for you and so on  in what kind of sense.  For example, in infinite dimensions, things get complicated,  but you have to go step by step.  Do the same thing.  So essentially, the proof goes like this.  You have a critical point, so you  have to define the variation.  So you vary it.  We are in the right invariant framework.  So you vary it on the left with respect  to exponential functions.  So essentially, we are variating in the direction of v's.  v's, any vector field, c1, such that in order  to keep the final and initial end end points fixed,  you consider v of 0 equals to v of t equals to 0.  So you differentiate.  And then you apply this differentiator.  So you have these variations of the initial pass g.  And by it or for where you compute  the differential of these variations.  So this is essentially eto-formal.  So there is this stochastic part.  This one is already, these two are already in the third  to be this one.  So you only have this.  Yeah, I mentioned here that there's  no part coming from contention terms  because I'm using the assumption that  allows to have eto integral equals to the tannobit integral.  Then, OK, so then you derivate in epsilon all these objects.  And you plug this differentiator into the action  function.  And you get this part as it is already  in the case of finite dimensions.  Nothing more.  Sorry, I did a deterministic case.  Nothing more.  Plus a term which is, well, not very elegant term.  But if you work out the details in case of righty variant,  it turns out that this thing is, I mean,  it's Riemann and Calcoules.  It turns out this object is indeed  the drama watch of greater depth than I was in action.  And if you put, of course, I already said that.  If you put all the HG theory we recovered,  you have a Poincare reduction theorem.  So this must be seen as a stochastic geometric counterpoint.  OK, you also have a possibility of making  of changing variables.  Again, you have to define exactly what this means  and it goes through the Hamilton-Jakob development  equation here with Laplace-Batrony operator.  And some extra remark, and then I'll go in here.  If in the assumptions of Cesar's theorem,  there is an extra assumption, which  is that pi, this pi not that exists,  and this makes the entropy finite.  If it's absolute, then you have to respect  for new Cesar of P1, P1 being the transition  semi-group of the Brownian motion.  Then actually, the pi that solves the minimizer  is of this form.  And this psi, so again, it has a problem structure.  And you can go through a pair of equations  from this mu and sigma, which are keeping at the end,  to this psi and phi through these equations.  And this is called, OK, I don't know what this is called,  but this is a problem that was solved in finite dimension,  at least by Brownian also a long time ago.  And it's kind of comfortable because instead of giving  probability measures, you give initial and final functions.  So sometimes it's easier to deal with.  So just to end up with some reference, OK,  so the geometric result, which I mentioned here,  extensions to include effective quantities  and going to infinite dimensions and so on is here.  It's in archive.  But this is extensions again in the stochastic geometric  mechanics part.  Concerning the optimal transport problems,  Schrodinger problem, there are general things here.  I mean, the general statements and properties  of the Schrodinger problem, you can see this here.  And here, this is the extension, which is also quite recent.  But in the direction of the optimal transport problem,  that in order to cover some equations of fluids,  like the Navier-Solvstein problem.  OK, so I think I finished here.  Thank you for your attention.  Thank you.  Thank you.  Thank you.  Any questions?  So do I get a thread that you get the best stock  situation in this way?  Yes.  So could you explain?  There was a paper by Hugo Gomez,  I think at some point in the book.  Sorry, may I precise in what sense  I obtain Navier-Solvstein equation?  OK, so we obtain Navier-Solvstein in the sense  of the theorem of the stochastic color Poincare theorem.  So if you have a critical point, then it  solves Navier-Solvstein equation, a representation  in this sense.  Concerning existence, we don't obtain it  because we are infinite.  As I mentioned, there's a few problems.  We don't obtain, exactly, we obtain some approximations  by using the Schrodinger problem,  but with imposing this at each time,  this effect at the density with the conservativeness.  And then we have some sort of, this is studied here.  I mean, the real existence of the, this is studied here.  And well, there are some restrictions.  OK, now please go ahead.  I was just wondering, there was a paper by Hugo Gomez  at some point about the real process.  I know, it has nothing to do with it,  and it's not the same variational principle.  It is a stochastic principle.  It is a stochastic principle.  It is also in the form of representation.  Again, he doesn't show that this gives the solution of nothing  though, as I don't hear in this stochastic method approach.  But the action functions are completely different.  I mean, variational principle is different.  It's a follow up on the same question.  There are many approaches, many different approaches  to the model.  No, it's in the model.  For instance, there is this model by Italian form.  Yeah, but then this is not for Navistok, say, or not for,  this is not for PDE.  This is for a stochastic PDE.  That we can also do that here.  But that's something else.  Then I don't see the relation with transport problem,  by the way.  Yes, this is my question.  No, because there the Lagrangian itself is randomized.  So there are randomness at two levels in variational approach.  And again, what you do here.  There are the paths that are random, right, diffusion  process.  And the Lagrangian, because Lagrangian here,  it is essentially the three, so the square.  So the Lagrangian itself is computed on stochastic paths,  but it's not stochastic form.  Now, if you perturb stochastic Lagrangian to,  then you have stochastic PDE.  You don't have Navistok, for instance.  You have stochastic Navistok.  So that's different.  And then I didn't speak about this in particular,  because I don't see the relation with optimal transport.  So I want it to be in a framework where there  things could be viewed from one side or the other.  So that was the most simple.  And especially if you are in finite dimension,  because then NaviCy, more or less,  can be really constructed and result in problem.  It's a simple problem.  Did you apply this for a G taking SO2 or SO3,  like having a [?].  We have examples.  Yeah, we have examples.  Those would be.  But again, what those have been for the optimal transport,  like the rigid body, optimal stochastic rigid body,  or the rigid body in transport.  We can do this.  We can do this.  I can give you formulas and so on.  Now, the point is that I don't understand very well  what does this mean in terms of rigid body.  I wish Dario was here, because then he would make a speech  about this.  Physically, I don't understand very well what this means,  but we can do that.  That sounds perfectly OK.  It's a finite dimensional example.  No problem.  If you have infinite dimensional G by finite dimensional  noise, can you then say something about the analysis?  Infinite dimensional G, but a finite dimensional noise.  So finite, about the time.  Yeah, yeah.  So it's a cylindrical noise, right?  So only a number of directions are perturbed, it's like me.  Yeah.  Yeah, it's the same problem.  I mean, the same problems are already there.  You can have only one direction perturbed.  And how is it that the analysis is complicated?  In order to prove the existence of the solution, yes.  In order to prove the counterpart of the stochastic  point-career result, it's not that complicated.  You have to work out and see that things don't explode.  OK, it's not that complicated.  But the existence.  Because the existence, either you go through the entropy  approach, or you go through the PDE approach.  I don't know.  Maybe there is a third way.  I don't see.  The PDE approach, you finally end up with the same problems  that PDE people have.  The entropy approach, you end up with others.  But that does not depend on the number of directions  that you randomize.  Perhaps.  So I want to ask a different question.  But maybe you comment on what you just said.  Perhaps in some cases, the PDE approach  becomes a linear approach.  On the lines that you have, the sense that if you ask,  don't work, don't do the reduction.  Just work in the group.  Is there a chance that some of these problems,  in some cases, the interventions  may become statistical means?  And then, or something like that, something that would be.  You mean at the Lagrangian level?  Because at your level?  Yes.  Yes.  Yes.  The Lagrangian level.  The group level.  I mean, whatever the fact.  Because the group level.  Yeah.  So now I'm coming back to you.  What I said earlier, it seems to me that indeed, probably,  trying out first, before even considering  the few morphos groups, trying out some of the techniques  here of what the analysis looks like in the case  of the group group, some interesting problems.  Yeah.  Thank you.  Because I didn't think about studying the group group,  but you need to study it.  Even then, if things can be worked out  in the group in the Lagrangian cycle,  the maybe even the path space is all right.  Then I see some kind of connection to my account.  Yeah.  That's the obvious option.  Yes.  Yeah.  And then the techniques will be new.  The techniques new tools will be available.  So I guess that I don't see existence  in any of the terms there much easier to obtain.  This is after all the Bama group.  Yeah.  Yeah.  So that's very good.  I would suggest that we shift further questions to the coffee  break.  We have 20 minutes time.  And I guess there's something urgent that you would.  Well, I just wanted to add to what  there is a nice equation which is one of the other  equations with the Laplacian that is exactly on loop groups.  So we have one of the simplest examples on loop groups.  The oil punker, he's talking about this.  The Laplacian equation is a new thing.  And probably you can recover it.  This is a good question.  OK.  I'll talk to you.  Let's give the big thanks.  So we'll.  And we'll start again in 20 minutes.  So let me see if we can all talk to each other.  Yeah.
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