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In this short paper, we give a brief presentation of some contribution to microlocal analysis on the partial differential equations by Yoshinori Morimoto. The first part concerns the study of the degenerate elliptic equations, by using the microlocal analysis based on the theory of pseudo-differential operators. The second part is about the analysis of non-cutoff Boltzmann equations where the microlocal analysis contribute lots of progress, in particular, using Fefferman-Phong's uncertainty principle to prove the smoothing effect of solutions and Littlewood-Paley theory to study the existence of classical solutions for non-cutoff Boltzmann equations. So that we focus only on the contribution to the study of kinetic equations by Yoshinori Morimoto., "Workshop on the Boltzmann Equation, Microlocal Analysis and Related Topics". May 27～29, 2016. edited by Hisashi Okamoto, Yoshio Tsutsumi, Naomasa Ueki, Tadayoshi Adachi and Senjo Shimizu. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.

We study accretive quadratic operators with zero singular spaces. These degenerate non-selfadjoint differential operators are known to be hypoelliptic and to generate contraction semigroups which are smoothing in the Schwartz space for any positive time. In this work, we study the short-time asymptotics of the regularizing effect induced by these semigroups. We show that these short-time asymptotics of the regularizing effect depend on the directions of the phase space, and that this dependence can be nicely understood through the structure of the singular space. As a byproduct of these results, we derive sharp subelliptic estimates for accretive quadratic operators with zero singular spaces pointing out that the loss of derivatives with respect to the elliptic case also depends on the phase space directions according to the structure of the singular space. Some applications of these results are then given to the study of degenerate hypoelliptic Ornstein-Uhlenbeck operators and degenerate hypoelliptic Fokker-Planck operators., Comment: 46 pages. arXiv admin note: text overlap with arXiv:1411.6223

On etudie des bornes inferieures et superieures pour la probabilite qu’un processus de diffusion dans $R^{n}$ reste dans un petit tube autour d’un squelette deterministe jusqu’a un temps fixe. Les coefficients de diffusion $\sigma_{1},\dots,\sigma_{d}$ peuvent degenerer, mais on suppose qu’ils satisfont a une condition d’Hormander forte sur les coefficients et leurs crochets de Lie de premier ordre autour du squelette d’interet. Le tube est ecrit en termes d’une norme qui prend en compte la structure non isotrope du probleme: en temps $\delta$ petit, le processus de diffusion se propage avec vitesse $\sqrt{\delta}$ dans la direction des vecteurs de diffusion $\sigma_{j}$ et avec vitesse $\delta$ dans la direction de $[\sigma_{i},\sigma_{j}]$. On prouve d’abord des bornes inferieures et superieures en temps court (non asymptotiques) pour la densite de la diffusion. Ensuite, on prouve l’estimee de tube en utilisant une concatenation de ces estimees de densite en temps court.

This thesis mainly study the hypocoercivity and long time behaviour of kinetic equations. We first consider the kinetic Fokker-Planck equation with weak confinement force and a class of general force. We prove the existence and uniqueness of a positive normalized equilibrium (in the case of a general force) and establish some exponential rate or sub-geometric rate of convergence to the equilibrium (and the rate can be explicitly computed). Then we study convergence to equilibriumof the linear relaxation Boltzmann (also known as linear BGK) and the linear Boltzmann equations either on the torus or on the whole space with a confining potential. We present explicit convergence results in total variation or weighted total variation norms. The convergence rates are exponential when the equations are posed on the torus, or with a confining potential growing at least quadratically at infinity. Moreover, we give algebraic convergence rates when subquadratic potentials considered. We use a method known as Harris’s Theorem.; Cette thèse porte principalement sur l’hypocoercivité et le comportement à long terme d’équations cinétiques. Nous considérons d’abord l’équation cinétique de Fokker-Planck avec la force de confinement faible et une classe de force générale. Nous prouvons l’existence et l’unicité d’un équilibre normalisé positif (dans le cas d’une force générale) et établissons un certain taux exponentiel ou sous-géométrique de convergence vers l’équilibre (et le taux peut être explicitement calculé). Ensuite, nous étudions la convergence vers l’équilibre de la relaxation Boltzmann linéaire (également appelé BGK linéaire) et le équations de Boltzmann linéaire soit sur le tore ou sur tout l’espace avec un confinement potentiel.Nous présentons des résultats de convergence explicites au normes de variation total ou de variation totale pondérée.Les taux de convergence sont exponentiels lorsque les équations sont posées sur le tore ou avec un potentiel de confinement grandir au moins quadratiquement à l’infini. De plus, nous donnons taux de convergence algébrique lorsque les potentiels sousquadratiqué pris en considération. Nous utilisons le théorème de Harris.

We consider an operator being a sum of squares of vector fields. It has the form, p , r ∈ N , P ( x , D x , D y , D t ) = D x 2 + x 2 ( p − 1 ) ( D y − x r D t ) 2 . This type of operator is C ∞ hypoelliptic by Hormander's theorem, [18] . Its analytic or Gevrey hypoellipticity has then been studied by a number of authors and is relevant in relation to the Treves conjecture. The Poisson–Treves stratification of P includes both symplectic and non-symplectic strata. In this paper we show that P is Gevrey ( p + r ) / p hypoelliptic, by constructing a parametrix whose symbol belongs to some exotic classes. One can also show that this number is optimal.

In this paper we will first present some results about the local solvability property of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration (which in turn is a generalization of the Kannai operator) exhibits a degeneracy due to the interplay between the singularity associated with the characteristic set of a system of vector fields and the vanishing of a function. Afterward we will also discuss some local solvability results for two classes of degenerate second order linear partial differential operators with non-smooth coefficients which are a variation of the main class presented above., Bruno Pini Mathematical Analysis Seminar, Seminars 2017

We study the null-controllability of parabolic equations associated to a general class of hypoelliptic quadratic differential operators. Quadratic differential operators are operators defined in the Weyl quantization by complex-valued quadratic symbols. We consider in this work the class of accretive quadratic operators with zero singular spaces. These possibly degenerate non-selfadjoint differential operators are known to be hypoelliptic and to generate contraction semigroups which are smoothing in specific Gelfand-Shilov spaces for any positive time. Thanks to this regularizing effect, we prove by adapting the Lebeau-Robbiano method that parabolic equations associated to these operators are null-controllable in any positive time from control regions, for which null-controllability is classically known to hold in the case of the heat equation on the whole space. Some applications of this result are then given to the study of parabolic equations associated to hypoelliptic Ornstein-Uhlenbeck operators acting on weighted $L^2$ spaces with respect to invariant measures. By using the same strategy, we also establish the null-controllability in any positive time from the same control regions for parabolic equations associated to any hypoelliptic Ornstein-Uhlenbeck operator acting on the flat $L^2$ space extending in particular the known results for the heat equation or the Kolmogorov equation on the whole space., Comment: 41 pages

The present contribution investigates the dynamics generated by the two-dimensional Vlasov-Poisson-Fokker-Planck equation for charged particles in a steady inhomogeneous background of opposite charges. We provide global in time estimates that are uniform with respect to initial data taken in a bounded set of a weighted $L^2$ space, and where dependencies on the mean-free path $\tau$ and the Debye length $\delta$ are made explicit. In our analysis the mean free path covers the full range of possible values: from the regime of evanescent collisions $\tau\to\infty$ to the strongly collisional regime $\tau\to0$. As a counterpart, the largeness of the Debye length, that enforces a weakly nonlinear regime, is used to close our nonlinear estimates. Accordingly we pay a special attention to relax as much as possible the $\tau$-dependent constraint on $\delta$ ensuring exponential decay with explicit $\tau$-dependent rates towards the stationary solution. In the strongly collisional limit $\tau\to0$, we also examine all possible asymptotic regimes selected by a choice of observation time scale. Here also, our emphasis is on strong convergence, uniformity with respect to time and to initial data in bounded sets of a $L^2$ space. Our proofs rely on a detailed study of the nonlinear elliptic equation defining stationary solutions and a careful tracking and optimization of parameter dependencies of hypocoercive/hypoelliptic estimates., Comment: minor revisions added

We report on recent results and a new line of research at the crossroad of two major theories in the analysis of partial differential equations. The celebrated De Giorgi-Nash-Moser theory shows H{\"o}lder estimates and the Harnack inequality for uniformly elliptic or parabolic equations with rough coefficients in divergence form. The theory of hypoellipticity of H{\"o}rmander shows, under general "bracket" conditions, the regularity of solutions to partial differential equations combining first and second order derivative operators when ellipticity fails in some directions. We discuss recent extensions of the De Giorgi-Nash-Moser theory to hypoelliptic equations of Kolmogorov (kinetic) type with rough coefficients. These equations combine a first-order skew-symmetric operator with a second-order elliptic operator involving derivatives in only certain variables, and with rough coefficients. We then discuss applications to the Boltzmann and Landau equations in kinetic theory and present a program of research with some open questions.

Multi-dimensional Stochastic Differential Equations (SDEs) are a powerful tool to describe dynamics of several fields (pharmacokinetic, neurosciences, ecology, etc). The estimation of the parameters of these systems has been widely studied. We focus in this paper in the case of partial observations, only a one-dimensional observation is available. We consider two families of SDE, the elliptic family with a full-rank diffusion coefficient and the hypoelliptic family with a degenerate diffusion coefficient. The estimation for the second class is much more difficult and only few references have proposed estimation strategies in that case. Here, we adopt the framework of the optimal control theory to derive an estimation contrast (or cost function) based on the best control sequence mimicking the (unobserved) Brownian motion. We propose a full data-driven approach to estimate the parameters of the drift and of the diffusion coefficient. The estimation reveals to be very stable in a simulation study conducted on different examples (Harmonic Oscillator, FitzHugh-Nagumo, Lotka-Volterra).

International audience; To visualize how the randomness of a Markov process X is spreading, one can consider subset-valued dual processes I constructed by intertwining. In the framework of one-dimensional diffu-sions, we investigate the behavior of such dual processes I in the presence of hypoellipticity for X. The Pitman type property asserting that the measure of I is a time-changed Bessel 3 process is preserved, the effect of hypoellipticity is only found at the level of the time change. It enables to recover the density theorem of Hörmander in this simple degenerate setting, as well as to construct strong stationary times by introducing different dual processes.

We study spectral properties of a class of global infinite order pseudo-differential operators and obtain the asymptotic behaviour of the spectral counting functions of such operators. Unlike their finite order counterparts, their spectral asymptotics are not of power-log-type but of log-type. The ultradistributional setting of such operators of infinite order makes the theory more complex so that the standard finite order global Weyl calculus cannot be used in this context., Comment: 32 pages

In this paper we consider sums of squares of vector fields in R 2 \mathbb {R}^2 satisfying Hörmander’s condition and with polynomial, but non-(quasi-)homoge- neous, coefficients. We obtain a Gevrey hypoellipticity index which we believe to be sharp. The general operator we consider is \[ P = X 2 + Y 2 + ∑ j = 1 L Z j 2 , P=X^2+Y^2+\sum _{j=1}^{L}Z_j^2, \] with \[ X = D x , Y = a 0 ( x , y ) x q − 1 D y , Z j = a j ( x , y ) x p j − 1 y k j D y , X=D_x, \quad Y= a_{0}(x, y) x^{q-1}{D_y}, \quad Z_j= a_{j}(x, y) x^{p_j-1}y^{k_j}\,D_y, \] with a j ( 0 , 0 ) ≠ 0 a_{j}(0, 0) \neq 0 , j = 0 , 1 , … , L j = 0, 1, \ldots , L and q > p j , { k j } q>p_j, \{k_j\} arbitrary. The theorem we prove is that P P is Gevrey-s hypoelliptic for s ≥ 1 1 − T , T = max j q − p j q k j . s\geq \frac {1}{1-T}, T = \max _j \frac {q-p_j}{q k_j}.

We prove local hypoellipticity of the complex Laplacian $\square$ in a domain which has superlogarithmic estimates outside a curve transversal to the CR directions and for which the holomorphic tangential derivatives of a defining function are superlogarithmic multipliers in the sense of "A general method of weights in the $\overline{\partial}$-Neumann problem," [T. V. Khanh, Ph.D. Thesis, Padua (2009)].

In this paper we consider a model sum of squares of complex vector fields in the plane, close to Kohn’s operator but with a point singularity, ¶ P = B B ∗ + B ∗ ( t 2 ℓ + x 2 k ) B , B = D x + i x q − 1 D t . ¶ The characteristic variety of [math] is the symplectic real analytic manifold [math] . We show that this operator is [math] -hypoelliptic and Gevrey hypoelliptic in [math] , the Gevrey space of index [math] , provided [math] , for every [math] . We show that in the Gevrey spaces below this index, the operator is not hypoelliptic. Moreover, if [math] , the operator is not even hypoelliptic in [math] . This fact leads to a general negative statement on the hypoellipticity properties of sums of squares of complex vector fields, even when the complex Hörmander condition is satisfied.

We compare two approaches to the predictive modeling of dynamical systems from partial observations at discrete times. The first is continuous in time, where one uses data to infer a model in the form of stochastic differential equations, which are then discretized for numerical solution. The second is discrete in time, where one directly infers a discrete-time model in the form of a nonlinear autoregression moving average model. The comparison is performed in a special case where the observations are known to have been obtained from a hypoelliptic stochastic differential equation. We show that the discrete-time approach has better predictive skills, especially when the data are relatively sparse in time. We discuss open questions as well as the broader significance of the results., 25 pages

We investigate the classical solvability for some classes of linear, degenerate equations in divergence form with prescribed Dirichlet data. Since the boundary value problem is characteristic according to Fichera on a part of the boundary, some typical nonlinear phenomena at these points are observed as boundary gradient blowups of the classical solutions in space directions. The regularity results explain the lack of hypoellipticity for special right-hand sides or boundary data for linear degenerate parabolic equations.

This paper investigates the regularity of solutions of convolution equations in the frame of classes of ultradifferentiable functions and ultradistributions. We improve previous work by Bonet, Chou, Fernández, Galbis, Meise and others.

We study C ∞ and analytic hypoellipticity for an invariant class of operators with multiple characteristics, which generalize the Gilioli-Treves model.

We consider the linear Fokker-Planck equation in a confining potential in space dimension $d \geq 3$. Using spectral methods, we prove bounds on the derivatives of the solution for short and long time, and give some applications., corrected version of the outdated article "Uniform bounds and exponential time decay results for the Vlasov-Poisson-Fokker-Planck system"

With a view to numerical applications we address the following question: given an ergodic Brownian diffusion with a unique invariant distribution, what are the invariant distributions of the duplicated system consisting of two trajectories? We mainly focus on the interesting case where the two trajectories are driven by the same Brownian path. Under this assumption, we first show that uniqueness of the invariant distribution (weak confluence) of the duplicated system is essentially always true in the one-dimensional case. In the multidimensional case, we begin by exhibiting explicit counter-examples. Then, we provide a series of weak confluence criterions (of integral type) and also of a.s. pathwise confluence, depending on the drift and diffusion coefficients through a non-infinitesimal Lyapunov exponent. As examples, we apply our criterions to some non-trivially confluent settings such as classes of gradient systems with non-convex potentials or diffusions where the confluence is generated by the diffusive component. We finally establish that the weak confluence property is connected with an optimal transport problem. ¶ As a main application, we apply our results to the optimization of the Richardson–Romberg extrapolation for the numerical approximation of the invariant measure of the initial ergodic Brownian diffusion.

In this thesis we address two problems. In the first part we consider hypoelliptic diffusions, under both strong and weak Hormander condition. We find Gaussian estimates for the density of the law of the solution at a fixed, short time. A main tool to prove these estimates is Malliavin Calculus, in particular some techniques recently developed to deal with degenerate problems. We then use these short-time estimates to show exponential two-sided bounds for the probability that the diffusion remains in a small tube around a deterministic path up to a given time. In our hypoelliptic framework, the shape of the tube must reflect the fact the diffusion moves with a different speed in the direction of the diffusion coefficient and in the direction of the Lie brackets. For this reason we introduce a norm accounting of this anisotropic behavior, which can be adapted to both the strong and weak Hormander framework. We establish a connection between this norm and the standard control distance in the strong Hormander case. In the weak Hormander case, we introduce a suitable equivalent control distance. In the second part of the thesis we work with mean reverting stochastic volatility models, with a volatility driven by a jump process. We first suppose that the jumps follow a Poisson process, and consider the decay of cross asset correlations, both theoretically and empirically. This leads us to study an algorithm for the detection of jumps in the volatility profile. We then consider a more subtle phenomenon widely observed in financial indices: the multiscaling of moments, i.e. the fact that the q-moment of the log-increment of the price on a time lag of length h scales as h to a certain power of q, which is non-linear in q. We work with models where the volatility follows a mean reverting SDE driven by a Lévy subordinator. We show that multiscaling occurs if the characteristic measure of the Lévy has power law tails and the mean reversion is super-linear at infinity. In this case the scaling function is piecewise linear, Dans cette thèse on aborde deux problèmes. Dans la première partie on considère des diffusions hypoelliptiques, à la fois sur une condition d'Hormander forte et faible. On trouve des estimations gaussiennes pour la densité de la loi de la solution à un temps court fixé. Un outil fondamental pour prouver ces estimations est le calcul de Malliavin, et en particulier on utilise des techniques développées récemment pour faire face à des problèmes de dégénérescence. Ensuite, grâce à ces estimations en temps court, on trouve des bornes inférieures et supérieures exponentielles sur la probabilité que la diffusion reste dans un petit tube autour d'une trajectoire déterministe jusqu'à un moment fixé. Dans ce cadre hypoelliptique, la forme du tube doit tenir compte du fait que la diffusion se déplace avec une vitesse différente dans les directions du coefficient de diffusion et dans les directions des crochets de Lie. Pour cette raison, on introduit une norme qui prend en compte ce comportement anisotrope, qui peut être adaptée aux cas d'Hormander fort et faible. Dans le cas Hormander fort on établit un lien entre cette norme et la distance de contrôle classique. Dans le cas Hormander faible on introduit une distance de contrôle équivalente appropriée. Dans la deuxième partie de la thèse, on travaille avec des modèles à volatilité stochastique avec retour à la moyenne, oú la volatilité est dirigée par un processus de saut. On suppose d'abord que les sauts suivent un processus de Poisson, et on considère la décroissance des corrélations croisées, théoriquement et empiriquement. Ceci nous amène à étudier un algorithme pour la détection de sauts de la volatilité. On considère ensuite un phénomène plus subtil largement observé dans les indices financiers: le "multiscaling" des moments, c'est-à-dire le fait que les moments d'ordre q des log-incréments du prix sur un temps h, ont une amplitude d'ordre h à une certaine puissance, qui est non linéaire dans q. On travaille avec des modèles oú la volatilité suit une EDS avec retour à la moyenne dirigée par un subordinateur de Lévy. On montre que le multiscaling se produit si la mesure caractéristique du Lévy a des queues de loi de puissance et le retour à la moyenne est superlinéaire à l'infini. Dans ce cas l'exposant de scaling est linéaire par morceaux

We study the kinetic Fokker-Planck equation perturbed by a stochastic Vlasov force term. When the noise intensity is not too large, we solve the Cauchy Problem in a class of well-localized (in velocity) functions. We also show that, when the noise intensity is sufficiently small, the system with prescribed mass admits a unique invariant measure which is exponentially mixing. The proof uses hypocoercive decay estimates and hypoelliptic gains of regularity. At last we also exhibit an explicit example showing that some restriction on the noise intensity is indeed required., Comment: Extended version

The celebrated H\"{o}rmander condition is a sufficient (and nearly necessary) condition for a second-order linear Kolmogorov partial differential equation (PDE) with smooth coefficients to be hypoelliptic. As a consequence, the solutions of Kolmogorov PDEs are smooth at all positive times if the coefficients of the PDE are smooth and satisfy H\"{o}rmander's condition even if the initial function is only continuous but not differentiable. First-order linear Kolmogorov PDEs with smooth coefficients do not have this smoothing effect but at least preserve regularity in the sense that solutions are smooth if their initial functions are smooth. In this article, we consider the intermediate regime of nonhypoelliptic second-order Kolmogorov PDEs with smooth coefficients. The main observation of this article is that there exist counterexamples to regularity preservation in that case. More precisely, we give an example of a second-order linear Kolmogorov PDE with globally bounded and smooth coefficients and a smooth initial function with compact support such that the unique globally bounded viscosity solution of the PDE is not even locally H\"{o}lder continuous. From the perspective of probability theory, the existence of this example PDE has the consequence that there exists a stochastic differential equation (SDE) with globally bounded and smooth coefficients and a smooth function with compact support which is mapped by the corresponding transition semigroup to a function which is not locally H\"{o}lder continuous. In other words, degenerate noise can have a roughening effect. A further implication of this loss of regularity phenomenon is that numerical approximations may converge without any arbitrarily small polynomial rate of convergence to the true solution of the SDE. More precisely, we prove for an example SDE with globally bounded and smooth coefficients that the standard Euler approximations converge to the exact solution of the SDE in the strong and numerically weak sense, but at a rate that is slower then any power law., Comment: Published in at http://dx.doi.org/10.1214/13-AOP838 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

With a view to numerical applications we address the following question: given an ergodic Brownian diffusion with a unique invariant distribution, what are the invariant distributions of the duplicated system consisting of two trajectories? We mainly focus on the interesting case where the two trajectories are driven by the same Brownian path. Under this assumption, we first show that uniqueness of the invariant distribution (weak confluence) of the duplicated system is essentially always true in the one-dimensional case. In the multidimensional case, we begin by exhibiting explicit counter-examples. Then, we provide a series of weak confluence criterions (of integral type) and also of a.s. pathwise confluence, depending on the drift and diffusion coefficients through a non-infinitesimal Lyapunov exponent. As examples, we apply our criterions to some non-trivially confluent settings such as classes of gradient systems with non-convex potentials or diffusions where the confluence is generated by the diffusive component. We finally establish that the weak confluence property is connected with an optimal transport problem. As a main application, we apply our results to the optimization of the Richardson-Romberg extrapolation for the numerical approximation of the invariant measure of the initial ergodic Brownian diffusion., Comment: to appear in "Annales de l'Institut Henri Poincar\'e"

This paper is dedicated to the application of the DeGiorgi-Nash-Moser regularity theory to the kinetic Fokker-Planck equation. This equation is hypoelliptic. It is parabolic only in the velocity variable, while the Liouville transport operator has a mixing effect in the position/velocity phase space. The mixing effect is incorporated in the classical DeGiorgi method via the averaging lemmas. The result can be seen as a Hölder regularity version of the classical averaging lemmas.

We study degenerate hypoelliptic Ornstein-Uhlenbeck operators in $L^2$ spaces with respect to invariant measures. The purpose of this article is to show how recent results on general quadratic operators apply to the study of degenerate hypoelliptic Ornstein-Uhlenbeck operators. We first show that some known results about the spectral and subelliptic properties of Ornstein-Uhlenbeck operators may be directly recovered from the general analysis of quadratic operators with zero singular spaces. We also provide new resolvent estimates for hypoelliptic Ornstein-Uhlenbeck operators. We show in particular that the spectrum of these non-selfadjoint operators may be very unstable under small perturbations and that their resolvents can blow-up in norm far away from their spectra. Furthermore, we establish sharp resolvent estimates in specific regions of the resolvent set which enable us to prove exponential return to equilibrium., Comment: 37 pages, 3 figures

This work is devoted to radiative transfer equations with long-range interactions. Such equations arise in the modeling of high frequency wave propagation in random media with long-range dependence. In the regime we consider, the singular collision operator modeling the interaction between the wave and the medium is conservative, and as a consequence wavenumbers take values on the unit sphere. Our goal is to investigate the regularizing effects of grazing collisions, the diffusion limit, and the peaked-forward limit. As in the case where wavenumbers take values in ${\mathbb R}^{d+1}$, we show that the transport operator is hypoelliptic, which implies in particular that the solutions are infinitely differentiable in all variables. Using probabilistic techniques, we show as well that the diffusion limit can be carried out as in the case of a regular collision operator and that as a consequence, the diffusion coefficient is nonzero and finite. Finally, we consider the regime where grazing collisions are domi...

We prove an optimal embedding result for the domains of Kolmogorov (or degenerate hypoelliptic Ornstein–Uhlenbeck) operators in L 2 spaces with respect to invariant measures. We use an interpolation method together with optimal L 2 estimates for the space derivatives of T ( t ) f near t = 0 , where T ( t ) is the Ornstein–Uhlenbeck semigroup and f is any function in L 2 .

We study the existence of “Lp-type” gradient estimates for the heat kernel of the natural hypoelliptic “Laplacian” on the real three-dimensional Heisenberg Lie group. Using Malliavin calculus methods, we verify that these estimates hold in the case p>1. The gradient estimate for p=2 implies a corresponding Poincaré inequality for the heat kernel. The gradient estimate for p=1 is still open; if proved, this estimate would imply a logarithmic Sobolev inequality for the heat kernel.

We construct parametrices for a class of pseudodifferential operators of infinite order acting on spaces of tempered ultradistributions of Beurling and Roumieu type. As a consequence we obtain a result of hypoellipticity in these spaces., 16 pages

We propose an approach based on methods from microlocal analysis, for characterizing the hypoellipticity in C°° and Gevrey classes of semilinear anisotropic partial differential operators with multiple characteristics, in dimension n > 3., 2002 Mathematics Subject Classification: 35S05

We consider the spatially inhomogeneous non-cutoff Kac's model of the Boltzmann equation. We prove that the Cauchy problem for the fluctuation around the Maxwellian distribution enjoys Gelfand-Shilov regularizing properties with respect to the velocity variable and Gevrey regularizing properties with respect to the position variable., Comment: 47 pages

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