 OK, zelo smo počučili na najbolj semnji semnjev. V tomom smo počučili na 9.30. Zato je to nekaj skazil. Zato je Anton Arnold z Vien. Zelo smo počučili o entropijne metode na ipahursivno in non-symmetrično-fokereplanku in vzelo vzelo. Čakaj. Čakaj. Čakaj. mnogo, da je organizacija za pravno invitacije, zelo v Frančesku, ki je injenačilo s njemem, in je vse početila. Znači sem, da sem tukaj prezent, da sem tukaj početil s mojem pjedešnjem pjedešnjem Jan Erb, in s početim Fransach Leitner. Tako, da sem tukaj početil, je tukaj pravne parabolice, In, v kvalitku na primeru, ki je to vzena, da je začeljena, z kvalitkom, da je začeljena, začeljena, da je začeljena je zrpraviljna, kot vsakulibrijem, in do bojenojga je, da je vsakulibrije začeljena. Zatim, da sem se počeril, zato je to, da se se prišeljemo, vs call plan convergence equation that we have here for the function F grocery plan coordination equation that we have here for the function F of position X and time t. I'm only going to speak about hole space problems. In this equation up here, I have a diffusion matrix D, which is constant in X and the main feature is that I assume it's degenerate so it's only positive Databan Net. Zanimidovo postavljev, da je hier n꽃va proda, če se zamišla na n꽃kvim mjelama c, na n꽃kvim mjelama n꽃kvim, je počkala srpiščati v zemljenj všem vreččih. A to bo, kar ima dne, ostalj tudi, da je, da se očite 이제 zančnikov o stavnih, z sedim stati, a then in the main part of the talk, I want to discuss the large time convergence to this steady state, possibly with sharp grades. And I want to give a rather complete theory for these equations under the assumption that the matrices C and D are constant in x. So let me put up an example where such type of focal plan equations appear in kinetic theory z plazmafizika. Zato, če je variabla X na prejvistih pagej, če je tudi x in v, x je pozitivna, v je velocitiva, če je momentum. Zato, če je spas, probabilitivna densitiva f, če je negativ, če je na zelo, če je hamiltonija, Zato je tudi tudi transport, ki je tudi funkcija F, in tudi je tudi influencije, da je nekaj potenšel, vsega vsega vsega, nekaj potenšel, ki je vsega vsega infinitiva, tudi tudi vsega vsega, in tudi na vsega vsega vsega, tudi nekaj potenšel, tudi tudi nekaj potenšel, tudi tudi nekaj potenšel, in nekaj, da je nekaj torque, je tudi vsega, da je tudi na zvore, tudi vsega vsega, trajilo je za laj pri vašlega vsega. To je tudi za delatorje, da je vsega pomega. Na zovem vsega vsega vsega vsega vsega. Tudi na vsega vsega vsega zvore. s velociti kvrti in tukaj je potenšel, potenšel je tukaj površel in tukaj vršel, da je tukaj stavil. Zatim, da sem tukaj površel konnečnju tukaj konečnju fokoplanku in tukaj fokoplanku, da sem tukaj v prvej sljede, da sem tukaj površel tukaj površel v tukaj matrični form. The first term that appears here is this degenerate diffusion term. So you see here you have this degenerate diffusion matrix here and this parameter sigma corresponds to the diffusion in velocity direction there is no diffusion in position direction. Then all the other terms this one this one in the last term are then just brought into this vector here q.a. driftvekta na prejojstva. If you assume for a moment that this external potential v vjelji would be quadratic, then this second vector here would be a constant matrix multiplied by x v, so then it would be exactly in the structure from the previous page. For this situation, say, with a quadratic potential, kor�厚 spt closing the last iron behavior, that you can write down explicitly the green's function, which will tell you what the exact and sharp decay rate is. However, if this potential is not quadratic then you won't have the green's function and then you will have to resort to other methods. And that is somehow the motivation for what I am discussing here. So, here is the outline of my talk. The notion of hypocursivity that is related to the degenerative equation, or degenerative equation equations, will give you the example of what it means. Then I will give you a short review of the standard entropy method and the standard entropy method. Tko sem videl, da sem bilo zelo, da sem bilo zelo, da sem bilo zelo, in potem sem bilo zelo, da se našli modljavacje in vse našli modljave zelo. Nel jaz sem pa da postoje, da sem da bo izgledan vse prav, da je to počet, da je to modljava in Kruppi pristik. So let me start with the example that Laurence also showed you this morning, the standard for kaplanck equation given here in n-space dimensions. So this operator appearing here on the right-hand side is a symmetric operator on the weighted L2 space where the weight is f infinity to the minus 1, f infinity just being the standard Gaussian. So the standard Gaussian spans the kernel of this differential operator. So for this operator you verify in two lines that it is dissipative, so this inner product in this weighted space is non-positive for all functions in some domain of this operator. Minus this operator is also coercive or in other words has a spectral gap. That means that this inner product is bounded below by the norm of the function squared whenever you are in the orthogonal of the steady state. That's the Gaussian. So that's the easiest example in this period. So as I told you before, I want to discuss here for kaplanck equations of this form the main feature that the diffusion matrix is now degenerate, so only positive semi-definite. So in this case you verify very quickly that minus this operator is not coercive in the sense of the previous slide. So this gave the motivation to Cedric Villani to come up with the following definition of hyper-coercivity, which is a notion that describes the large time behavior, the exponential decay in those situations where you don't have coercivity. So here we will use two Hilbert spaces. So first our differential operator L is considered on some space h and curly k, let me note the kernel of this operator. And now we consider some smaller Hilbert space h tilde, which is continuously embedded in the orthogonal of this kernel. Just to give you a vague idea at this point, think of the larger Hilbert space of some weighted L2 space as we had it on the previous page and think of the smaller Hilbert space as some weighted h1 space. In many examples this will be close to the correct setup. So then this differential operator L or minus L is called hyper-coercive on the smaller space, whenever there exists such an exponential decay rate, so you need two constants, a decay constant lambda, which appears here, and a multiplicative constant C that appears here in front, which typically will be larger than 1. So let me just give you an illustration of examples that on the one hand may be coercive, and on the other hand are not coercive, but at least hyper-coercive. So let's again go back to the simplest example with the standard Fokker-Planck equation that we saw this morning. So here in the standard Fokker-Planck equation we know the unique steady state, and this steady state can somehow be seen as a balance of two mechanisms. So on one hand the first term is a diffusion. So here I show you a cartoon just in two space dimensions. Diffusion wants to spread mass away from the origin towards infinity. On the other hand we have drift and with this sign here in front of the drift term this drift will move mass to the origin. So then we have a balance between these two mechanisms and the steady state appears somehow as a balance of these two mechanisms. So now let's look at the degenerate example. Here in the degenerate example diffusion matrix is singular and assume for a moment that these red terms in the drift matrix are not there, assumed that omega is zero. So then on the right hand side in this equation I have again my standard Fokker-Planck equation but only in the x1 direction. Which means in the x1 direction I have the equilibration just as on the previous page. But in the x2 direction I have no term there so there is nothing happening in the x2 direction so of course there won't be equilibration in the x2 direction. But now we switch on these two red terms and these two red terms just bring about a rotation in R2 so this rotation then mixes the good directions with the bad directions. The good directions are those where you do have this diffusion and the equilibration in the bad direction where you don't have it. And on the previous page I forgot to say so we know and Laurent showed that in his estimates that the sharp decay rate towards the steady state is one. So now in this example here in the x1 direction we have this equilibration with rate one no equilibration in the x2 direction. Now when you start to mix it's plausible that your resulting decay rate will be the average one half. This is of course no proof. The real story will be that you look at the drift matrix C that appears here on the right hand side which has the coefficients one minus omega omega in zero and you look at the eigenvalues of that and when omega is large enough this means when you rotate your system quickly enough then the real part of the eigenvalues will be equal to one half and this then will give you this sharp decay rate of one half in this two dimensional example. In the case that your rotation parameter is just a limiting value of one half then this matrix C here is similar to a Jordan block and then you lose an epsilon in this decay rate. So let me illustrate what is happening here from one more side to give you an idea of what is happening here. Here is again the Fokker-Planck equation from the previous page and I just put omega equal to one as an example. And for a moment I just want let's cancel this diffusion term and let's just look at this first order transport equation which has this divergence here of this drift term. Then you can solve that of course with characteristics and here I show you one characteristic curve in R2 this is this blue spiral that's then a solution to this simple linear ODE here and we will see towards the end of the talk that looking at this characteristic term really gives us information also about the Fokker-Planck equation. So what I am interested in here is the convergence towards the steady state on the level of the characteristic equation I am interested in the convergence of these characteristic curves towards the origin. So let's have a look what this blue spiral does and here on this graph I also show you some circles around the origin these circles are just level curves of the Euclidean norm. So if x should go to the origin I want to see how along this blue spiral how the norm, the Euclidean norm decays. What we observe is whenever this blue spiral intersect or cuts through the x1 axis then we have a strict decay of the Euclidean norm because it intersects this level curve with a nontrivial angle but whenever the blue spiral intersects the x2 axis then it's tangent to this level curve so there we don't have a strict decay of the Euclidean norm there we lose the strict monotonicity of the decay. So then on this ODE level one question to better understand the decay would be well let's introduce a different norm instead of the Euclidean norm rather this one where we put instead of the identity matrix another matrix P with level curves that are these ellipses and if you introduce this norm then this blue spiral will always intersect these curves with a nontrivial angle so then you will always have decay or a strict decay of your norm and one of the questions of course will be well how do we find this matrix P how do we find the best matrix P and this is closely related to this paper of Torbo Moore and Schmeisser so let me first make some assumptions on the coefficients that appear in my Fokker-Planck equation so first and these coefficients are the diffusion matrix and my drift matrix so if you think back of this hyper-coercive example that I've shown you there we had a diffusive direction and a non-diffusive direction and the drift matrix C had to mix these two directions therefore it's plausible to make the following first assumption that no subspace of the kernel of the diffusion matrix should be invariant under C transpose because then you would lose this mixing and exactly this condition will also guarantee that this operator L is hyper elliptic in the sense of Hermandan so since this is the case we have the following convenient properties for the existence and regularity theory if you assume that you have an L1 initial condition your solution will be instantaneously C infinity or if you have an L1 initial condition that is non-negative instantaneously your solution will be strictly positive I will need a second condition the second condition requires that my drift matrix C here is positively stable which means that all eigenvalues of this matrix must be strictly positive so this relates somehow to the fact that here in this drift term you have somehow a confinement potential confinement potential such that in the drift term mass will be moved towards the origin and not away from the origin in all directions so somehow the punchline or the combination of the two assumptions if you have for this Fokker Planck operator hyper ellipticity plus confinement you have hyper coercivity again hyper coercivity means you have the chance to show exponential convergence towards the steady state so now some remarks concerning the existence and uniqueness of the steady state here is again my equation here we have the following simple theorem there exists a unique steady state and the steady state is of this Gaussian form it's a non isotropic Gaussian where the covariance matrix K that appears here is the unique solution of this matrix equation so C and D are input matrices in this equation and you have to solve it for the matrix K such an equation is called continuously up and off equation simple algorithms even in Matlab where you can compute K so you have the K, the unique K at hand now that we have our unique steady state and this matrix K I can simplify and normalize my problem so I make the following coordinate transformation from the X variables to Y variables if I make this transformation my new steady state and my diffusion matrix turns out then to be just the symmetric part of the drift matrix I can make one more simplification and now rotate my Y coordinate system such that the diffusion matrix is diagonal and singular so from now on I'll assume that these normalizations are done then computations will be slightly simpler so let me now give you a short review of the entropy method and with entropy method I just mean in the language of this morning's talk deriving entropy deriving the entropy entropy dissipation inequality here I am back to a simple Fokker-Planck equation simple in the sense that the drift term that appears here is the gradient of some scalar function A and here still I keep a constant matrix D which at the moment should be regular so we have already seen this morning if you start with an initial condition that is L1 and non-negative and say normalize to 1 then you will keep all these properties during the evolution for this equation here you can write down explicitly the steady state it's just the exponential of minus this potential appearing here you just plug it here into these two terms and to verify that moreover you can rewrite this Fokker-Planck operator in this slightly simpler form using the steady state and you verify again that this is a symmetric operator in this weighted L2 space so the weight is again the inverse of the steady state and for concerning this potential A of x think again it's a confining potential growing for example quadratically what I want to use for understanding the large time behavior will be relative entropies which should act or should be used as the opponent functionals just as we saw this morning so here I will use a slightly larger family of relative entropies so these relative entropies are somehow a non-symmetric distance between two probability densities F1 and F2 and here it's defined with this integral where I use scalar functions entropy generators so these are functions from r plus to r plus with the following properties here is a sketch of such functions psi they should be non-negative at the point 1 it should be 0 because when F1 equals F2 you want the distance to be 0 and it should be convex plus there is some technical assumption between the second third and fourth derivative of these entropies and I will show you later on how this technical assumption appears so concerning examples of such relative entropies first of all the logarithmic entropy that we have seen this morning or power law entropies where the powers can be between 1 and 2 so the admissible relative entropies due to this technical assumption somehow bounded below by the logarithmic entropy in bounded above by the quadratic one so we can use entropies within this range we've also seen already in the morning what the time derivative of this relative entropy is so here I consider the relative entropy of my solution the solution to this Fokker-Planck equation compared to the steady state so here on the right hand side we have minus the relative fissure information so what I denote here by capital I was denoted in the morning by capital D that's the entropy dissipation exactly the same term now let me give you a brief review of the entropy method so again this is a procedure to show the entropy entropy dissipation inequality that we want to have in order to prove exponential convergence of the solution towards the steady state or another way how you could see that in the morning we saw that this inequality for the Fokker-Planck equation the key inequality was logarithmic sobolev inequality for the standard Fokker-Planck equation so another way how you could see what I am going to review now is how can you prove more general logarithmic sobolev inequalities for more general equations for more general entropies so such an entropy method always proceeds in two steps the first step is that you want to show the exponential decay not of the entropy but rather of the entropy dissipation so assume that initially the entropy dissipation functionally is finite and assume for the equation that I had on the previous slide so here the equation with this diffusion matrix and this drift potential A that these two input variables satisfy this so-called Bacrier-Marie condition we have the Hessian of our potential in the equation that should be bounded below by the inverse of the diffusion matrix at the moment the diffusion matrix is regular so this makes sense so under this convexity assumption one can prove that the entropy dissipation denoted here by capital I decays exponentially where the rate lambda 1 is related to this constant that appears there that I should mention that this approach goes back to Bacrier-Marie coming from probability theory and in this work with Markovic Toscani-Untarait was reformulated or represented in a more PD language and such approaches have become very popular in the last 15-20 years because they are very robust towards nonlinear perturbations or nonlinear modifications so now the second step of the entropy method what you really want to prove is the exponential decay of the relative entropy itself so under the same convexity assumption of your potential in the equation with respect to the drift matrix you have exponential decay of the relative entropy and let me just give you a hint of the proof in the proof you first think back of how you proved the analogous inequality for the entropy dissipation capital T capital I of T so there you derive in fact such a differential inequality for the time derivative of the entropy dissipation in terms of the entropy dissipation and once you have that you integrate this inequality in time starting at T going to infinity and then you recall that capital I the entropy dissipation is nothing but the time derivative of your relative entropy then you get the corresponding differential inequality for the entropy and then Gronwald's lemmer just gives you this exponential decay that you want to have plus some technicalities with density for example so this was a review of the standard method and now I want to show you how life changes when you switch from a symmetric Fokker-Planck equation to a non-symmetric or even degenerate Fokker-Planck equation so this slide is probably the key message that I want you to keep from today even if you don't follow all the technicalities what I show you here in this plot is the relative entropy on this red curve as a function of time for the standard Fokker-Planck equation for example with the quadratic entropy which is nothing but the L2 difference of the solution to the steady state in this weighted norm but the same thing would be true for the logarithmic entropy the entropy decays as a convex nice convex function with this exponential decay what else you would see here this dotted curve is the time derivative of the relative entropy and of course since the entropy decays this entropy dissipation is always strictly negative unless you have reached the steady state so this is also a notion that was introduced in the morning with this strict entropy framework and this fact that the entropy dissipation is always strictly negative this makes it possible to have such differential inequality between the time derivative of the entropy and the entropy now is here again a plot time dependence of the relative entropy for this degenerate prototype this two dimensional example that I showed you in the beginning so it does not decay in a convex way it rather decays in this wavy way and now let's look at this dotted curve here down here that's the time derivative of the relative entropy and what is really bad is that the time derivative can be zero here although we have not reached the steady state yet so on the level of the relative entropy it means you can have horizontal tangents to your curve so we are outside of this strict entropic framework from this morning so this means also that it's a priori impossible to have such a differential inequality between the time derivative of the entropy and the entropy but this was how we proved the exponential decay so we have to modify our approach let me mention that this wiggly decay of the relative entropy has also been known or is well known for more complicated examples here is a plot from a paper by Phil B. Moore and Pareski in 2006 for the inhomogeneous Boltzmann equation so let's just look at this dashed curve where they plot the relative entropy with respect to the global Maxwellian so again you see this wavy decay with horizontal tangents which is exactly the same phenomenon so let's go back to the analysis and what I want to discuss now is Fokker-Planck equations with a singular matrix D here so as I said the time derivative of the entropy can be zero for some states other than the steady state and this implies somehow that the Fisher information functional that we have here on the right hand side is not really useful for our large time analysis and the the reason why we have this problem that this relative Fisher information that's this integral here disappears for states other than the steady state is the fact that the diffusion matrix is only positive semi-definite because there are states F other than the steady state where the gradient of this quotient solution at some time over the steady state lie in the kernel of the diffusion matrix and then it's already zero although you're not at the steady state yet so one remedy to get out of this trouble is to say well then let's just modify this functional a little bit was the problem here the problem was the diffusion matrix so let's just replace the diffusion matrix by a positive definite matrix P then let's see if this gives us a better picture or a better tool to study the large time behavior and the goal would be as in the standard entropy method that we want to find a differential inequality between the time derivative of this modified functional this modified entropy dissipation functional and the functional itself and if we are able to find such a differential inequality for the good choice of a matrix P then we have exponential decay of this modified functional which is not intrinsically linked to our equation however this matrix P is a regular matrix so it bounds from above our singular diffusion matrix d with some constant so if this functional decay is exponentially to zero then also the true functional will decay to zero exponentially and this is what we want to have so the main question is how do we find this matrix P and the answer will be given by this small algebraic lemma which appears at first glance a bit disconnected from the discussion so far so think of a matrix Q that is positively stable so this means all eigenvalues have a positive real part and assume that all eigenvalues that give rise to this minimum are non-defective non-defective means the geometric multiplicity is equal to the algebraic multiplicity so let me give you here an idea of the spectrum of such a matrix so here I have the real and the imaginary axis here I have my critical vertical line with this parameter mu and assume for this matrix Q so I'm plotting here the spectrum of Q there are some eigenvalues assume the matrix is real so eigenvalues will come in complex conjugate pairs so assume the spectrum looks like this so the statement or the assumption up there is all eigenvalues on this critical line here should be non-defective then the statement is there exists a positive definite matrix P not unique such that this matrix inequality holds let me just anticipate a little bit and put it into perspective what this matrix inequality will mean to us this parameter mu that appears there which is defined by the matrix Q there and in the end we will have in our application Q will be the drift matrix C this parameter mu will be our exponential decay rate and this matrix inequality will be for our Fokker-Planck equation the replacement of the Bakriemerik condition of this convexity condition so what happens if there are some defective eigenvalues on this vertical line then in fact you lose an epsilon on your decay rate you can still find matrices P to have this inequality but you lose a little bit just a glimpse of the proof how to construct this matrix P in the non-defective case just take all eigenvalues eigenvectors of the transpose of this matrix Q sum up these tensors of these eigenvectors and you can compute and verify in two lines that this gives a visible matrix P I said before this matrix P is not unique but independently of P you will always get the same decay rate for your equation in the end so now let's come to the entropy method for these degenerate equations and as I said before the entropy method comes in two steps first exponential decay for the entropy dissipation functional and then the second step for the entropy or the relative entropy itself here is our modified entropy dissipation functional which is an auxiliary functional with a matrix P that is that is obtained from the previous lemma so here we have the following result the parameter mu is obtained from the drift matrix C under some nice assumptions on the initial condition we have exponential decay of our modified entropy dissipation functional if all eigenvalues on the critical line there are non-defective if there is a defective eigenvalue you lose an epsilon in the decay rate there is one point that we should keep in mind already at this point the decay rate mu that appears here in the exponential is determined by the spectral gap of the drift matrix C which is in our equation so let me give you a hint of how to prove that so we have to compute the time derivative of the modified entropy dissipation function S and this time derivative can be written as a sum of two terms so in this sign here there are two to three pages of computations which I skip so first we have a term here that looks very similar to the functional that we started from so we have a second derivative of the entropy generator psi and here we have this quadratic form of the gradients of solution over steady state with the matrix in between so here we have almost the same the U is just a shorthand notation for this gradient solution over steady state and we just have this combination of the matrix P which was put in our functional and the drift matrix C but for exactly this combination of matrices we have this algebraic lemma from before so it is this point where I use now this modified Bakrie-Marie condition and if I use this inequality and back to this functional S so I have my differential inequality because this term here this remainder term has a sign therefore I can throw it away so let me add a little comment about this remainder term here we have a trace of two matrices X and Y both of them are 2 by 2 matrices the matrix X has this form it involves the second, third and fourth derivative of solution over the steady state so now we want this to have a sign therefore we had this technical assumption at the very beginning on a special combination of these derivatives and here is this technical assumption and for this matrix Y which is some junk here with Košić-Schwarz you can also show that this has a sign so this term is a remainder with a sign so as I said the entropy method always comes in two steps first exponential decay of the entropy dissipation secondly exponential decay of the relative entropy let me briefly recall how we did that step in this say standard entropy method we had a differential inequality for the dissipation rate and then we just integrated it in time and we were done we cannot do that here because this functional S is nothing intrinsic for our equation that is the time derivative of the relative entropy so if we integrated it in time we don't get the relative entropy still we have the following theorem so under some nice assumption on the initial condition we have here the relative entropy of your solution at time t with respect to the steady state is bounded by a term that decays exponentially with the same exponential at time t which decays exponentially this is something we already know so the question is only how to prove this inequality here so here this is what we wanted exponential decay of the solution so here the whole line of inequalities here we have the this inequality here so let me anticipate for the experts here this inequality here is nothing but a logarithmic sobolev inequality or a generalization of a logarithmic sobolev inequality so how do I get this generalized logarithmic sobolev inequality in order to show that I use an auxiliary Fokker-Planck equation it has a priori nothing to do with our equation that we want to study here I cook up this equation the following way I want to have it the same steady state as my true equation so the G infinity should be the same as F infinity therefore I put here the steady state into the equation and I want to have it the diffusion as a diffusion matrix my positive definite matrix P from the functional then in this equation my steady state is just the standard Gaussian by construction this is a symmetric Fokker-Planck equation that I've shown you in my review at the beginning so for this symmetric Fokker-Planck equation the scalar confinement potential A of X is just this quadratic term so the Hessian of this confinement potential is just the identity matrix which can be bounded below by the inverse of the matrix P because it's positive definite so therefore this auxiliary equation has an exponential decay towards zero and hand in hand with this exponential decay goes a generalized logarithmic sobolev inequality here it is logarithmic or convex sobolev inequality which is an inequality between the relative entropy and this functional s, this modified Fisher information functional and this is exactly the term that we want to plug in here so let me repeat the equation g actually has nothing to do with my original Fokker-Planck equation it's just auxiliary and I use this convex sobolev inequality for each fixed point in time and then just plug it in here to the original equation and then I close this inequality let me make one more remark why is this auxiliary function g the right one this auxiliary function g and this evolution equation the right one well the modified Fisher information s is just the true entropy dissipation for this auxiliary equation and therefore this is the reason how the story matches so where do we stand we have here exponential decay of the relative entropy this was my goal so I could stop here but I still have some more minutes so this inequality is okay but not nice in the sense because I have here exponential decay of the relative entropy but I have to invest a higher order functional of the initial condition here the s involves the gradients and in the original entropy method you just invest as in information on the initial condition the relative entropy initially so let's improve on that and the way I want to improve here is I want to use parabolic regularization it's a degenerate parabolic equation but nevertheless it regularizes so in order to show you a regularization let me briefly discuss the structural property of the equation the Hermann order of my Fokker-Planck equation is an index M in natural number such that I have this matrix inequality so let's pretend for a moment that the diffusion matrix D is positive definite then I don't need this pre and post multiplication with C and the matrix D is bounded below by multiple of the identity of course but if the D is singular equal 0 will not work and you have to go higher up and add more of these terms until you get a bound below by this identity matrix and the smallest index M such that you get this inequality gives you this index and somehow quantifies how complicated the structure how degenerated it is and how many turns times you have to mix good and bad diffusion directions in the Hermann notation how many iterated gradients do you need to span your whole space once you have fixed this index then you have this regularization result that tells you start initially just with a finite relative entropy and then instantaneously with this inverse power law you even get finite modified Fisher information think of this regularization inequality as a regularization from some weighted L2 space into some weighted H1 space I should say this regularization goes back to work of herro in the book of Cedric Villani it was worked out for the quadratic and logarithmic entropies and here we have it for all entropies in between using the similar spirit so with this regularization we can now show you the somewhat smoother result so assume that this index mu which is the smallest real part of the eigenvalues of the drift matrix so that's the spectral gap here is positive then we have exponential decay of the relative entropy you only need on the right-hand side the initial relative entropy again if you have defective eigenvalues you lose an epsilon in the decay rate so with all what we have already seen up to now the proof is really simple here is the relative entropy at time t this is bounded due to a convex oblev inequality by this modified Fisher information at time t for the modified Fisher information we've already seen that it decays exponentially but I don't go back now until t equals zero I leave a short initial time layer of length delta and there I use this regularization result to get a bound on the right-hand side with the initial relative entropy let me say that the rate that we obtain in this way is sharp this constant, this multiplicative constant see not necessarily so I motivated this whole analysis by the kinetic Fokker-Planck equation so let me get back to this kinetic Fokker-Planck equation here it is let me recall that the steady state is this exponential term here and in case that this given external potential capital V is quadratic the exponential decay of the relative entropy is completely discussed by what we have seen so far let me skip that formula but if the potential is not quadratic we need some more analysis and some more tools and in fact what I have shown you so far can be extended in some cases to more general potentials so here I just show you the result in one dimension assume that your potential up there the external potential it's a quadratic term plus a perturbation a perturbation that is such that the variation between the second derivatives so the maximum here the maximum of the second derivative and the minimum of the second derivative is bounded in some sense by the friction constant in the kinetic Fokker-Planck equation then we still have exponential decay of the relative entropy the story is somehow like this without this perturbation term you have a certain decay rate and then you can give up part of this decay rate and allow for a perturbation the potential so this is somehow the compensation that appears in there and I should say it's not a say an epsilon perturbation but you can quantify in relation to a parameter in your equation and let me also say again in this perturb setting you don't have a grains function anymore so there you really need to use tools so far I've shown you a modified entropy method that works for degenerate Fokker-Planck equations with a diffusion matrix that is singular which brought about a variant of the entropy method so one question that one may ask is well let's try to apply that to Fokker-Planck equations that are not degenerate and let's see if we learn something more so here we have a Fokker-Planck equation in the same structure as before but the diffusion is now not degenerate but the equation is not symmetric not symmetric means the drift term is not just the gradient of a confinement potential so there is this rotation contribution also included in there you can verify that the steady state here is the standard Gaussian and what we want to find is such a decay inequality and exponential decay inequality for the relative entropy so you want to find a decay rate lambda and the multiplicative constant c so at this point we have two methods we have the standard entropy method and we have these modified entropy methods and we want to compare the results that we obtained so the red curve is again the evolution the time evolution of the relative entropy for this example so you see as before there is this wavy decay the equation here is not degenerate therefore we don't have horizontal tangents but we have this wavy decay so let's first look at the standard entropy method here the the steady state is the standard Gaussian so we have a quadratic confinement potential its hashian is just the identity matrix which is bounded below by the inverse of this regular diffusion matrix and then you get here a decay rate which is known to be sharp and let us call us now it's the sharp local decay rate with a lambda that is equal to one quarter what means sharp local decay rate well if you start here t equals zero you have an exponential function here you have a constant that is one and your decay estimate this function here is the blue dotted curve and they are tangent this is tangent to the true entropy function this red curve here in this sense it's sharp at every point in time but clearly for large time this is a very bad estimate then on the other hand we have this variant of the entropy method which gives us a sharp global decay rate in the sense of capturing the sharp envelope of this wavy decay with a decay rate which is 5 eighths which is clearly better in this case we can also show that at least in two dimensions we always catch here this sharp envelope so I have a few more minutes left so far this picture and the comparison of these two methods look quite nice there is just one question left why does this work so let me again give you an algebraic answer based this algebraic answer on the following simple question let's compare the spectral gap of a matrix Q let's think of a matrix that is again positively stable and let's compare the spectral gap of such a matrix non symmetric matrix with the spectral gap of its symmetric part so that's one half the matrix plus its transpose why do I raise this question well in fact the entropy method works mostly with quadratic functionals and here I don't mean so much the quadratic entropy this is a special case I rather mean the fissure information or the modified fissure information which is quadratic in terms of the gradient of solution over the steady state and in computing in terms of the derivative of this functional we get here matrices which are not symmetric and therefore when trying to estimate these terms we will only see in the estimates the spectral gap of the symmetric part of whatever matrix is in there so let me come back to the question let's compare the spectral gap of a matrix with its symmetric part so there's first the following simple lemma for every matrix q the smallest eigenvalue of the symmetric part is smaller equal to this value mu but typically it's strictly smaller so what does this mean in this plot here so here let's say the spectrum of the matrix q is contained in this vertical strip we have a non-symmetric matrix now if I want to compare to the spectrum of the corresponding symmetric part then the spectrum will be typically contained in an interval that is larger so that will be the spectrum of the symmetric part of the matrix q which is in the end then our drift matrix goes because that will mean in my estimates that I will lose this part in the estimate of the exponential decay and if I'm unlucky the situation is like this and I cannot prove exponential decay but there's a way out again a very simple algebraic lemma there exists a positive definite matrix p which will give me a similarity transformation a similarity transformation of my matrix q with actually the square root of p in such a way of course this transformed matrix q tilde has the same spectrum as q but in such a way that when I now look at the smallest eigenvalue of the symmetric part of this transformed matrix I don't lose on the spectral gap I get exactly the same spectral gap if there are defective eigenvalues on this line I will lose an epsilon so this means there's a similarity transformation that my original matrix q it's similar matrix q tilde and most of all the symmetric part of this q tilde have the same spectral gap and this is what I want to catch for my estimates how this very simple algebraic result works let me get back to the drift characteristics to the simple linear ODE that I've shown you at the very beginning there I told you this was the story with the blue spiral and changing the Euclidean norm to a modified norm that was indicated with this red ellipse and this modified norm was this quadratic form here with the matrix p so let's compute the time derivative if you compute the time derivative you get here matrix c and now I just re-write this term in a crazy way I re-write it in a crazy way because this term here is just the similarity transformation from my previous slide and here it's transpose so here I have the symmetric part or twice the symmetric part of the similarity transform part and the lemma from the previous side just told us that the smallest eigenvalue here is the smallest eigenvalue or the smallest real part from up there so this part is bounded below by 2 mu but if I use this then I have here this simple inequality for this p norm which gives me exactly this decay rate 2 mu which is the sharp rate but this is only an ODE story so the last question is how is this simple ODE question related to my Fokker-Planck equation let me recall I'm only interested here in Fokker-Planck equations that are non symmetric so you don't have total orthonormal eigenvalue basis but this Fokker-Planck equation has a surprising property it has an infinite sequence of flow invariant eigenspaces of growing dimension that are orthogonal so of course you have the kernel and then above that you have a first eigenspace if you are in R2 of dimension 2 which is orthogonal to the kernel and above that you have a 3 dimensional eigenspace which is again a flow invariant and orthogonal to all the previous ones and so forth so in the first eigenspace of dimension 2 say the evolution in that finite dimension space is exactly given by this characteristic equation therefore there this if you go to the next higher eigenspace there the evolution is determined by a tenserth version of this ODE and then the next one in a triple tenserth version of that and so forth therefore this simple ODE determines the whole evolution of this Fokker-Planck equation and this shows us this simple computation and this matrix P that appears here in the norm which is the correct behavior so I am already a bit over time let me just summarize I have shown you a modified entropy method for degenerate Fokker-Planck equations first with only linear drift terms the key tool was to modify the entropy dissipation functional then I have shown you how to extend that to a kinetic Fokker-Planck equation not necessarily quadratic and what we can learn from that story also for non-symmetric Fokker-Planck equations and then let me just point out for the case where eigenvalues here are defective I have given you decay estimates where you lose an epsilon in the decay rate and that is not the true story because if you think of ODEs if you have defective eigenvalues your true behavior will be polynomial in time times in exponential so here for the Fokker-Planck equation of course the story is the same so the truth is you don't lose an epsilon in the exponential term you rather get a polynomial in the simplest case quadratic or higher order and this is a preprint to be as well who is here and amit enough and here are some references where this method is written up applications to kinetic BGK models where you have a drift part and the reaction part is joined work with Fransach Leitner and Eric Kallen so sorry for taking longer than plant thank you for your attention