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In this paper we study a self-consistent Vlasov-Fokker-Planck equations which describes the longitudinal dynamics of an electron bunch in the storage ring of a synchrotron particle accelerator. We show existence and uniqueness of global classical solutions under physical hypotheses on the initial data. The proof relies on a mild formulation of the equation and hypoelliptic regularization estimates. We also address the problem of the long-time behavior of solutions. We prove the existence of steady states, called Haissinski solutions, given implicitly by a nonlinear integral equation. When the beam current (i.e. the nonlinearity) is small enough, we show uniqueness of steady state and local asymptotic nonlinear stability of solutions in appropriate weighted Lebesgue spaces. The proof is based on hypocoercivity estimates. Finally, we discuss the physical derivation of the equation and its particular asymmetric interaction potential. [ABSTRACT FROM AUTHOR]

In the first part we analyze space G ∗ (R + n) and its dual through Laguerre expansions when these spaces correspond to a general sequence { M p } p ∈ N 0 , where * is a common notation for the Beurling and Roumieu cases of spaces. In the second part we are solving equation of the form L u = f , L = ∑ j = 1 k a j A j h j + c E y d + b P (x , D x) , where f belongs to the tensor product of ultradistribution spaces over compact manifolds without boundaries as well as ultradistribution spaces on R + n and R m ; A j , j = 1 ,... , k , E y and P (x , D x) are operators whose eigenfunctions form orthonormal basis of corresponding L 2 - space. The sequence space representation of solutions enable us to study the solvability and the hypoellipticity in the specified spaces of ultradistributions. [ABSTRACT FROM AUTHOR]

This is the third paper in a series of three dealing with sums of squares and hypoellipticity in the infinitely degenerate regime.We establish a C2,d generalization of M. Christ's smooth sum of squares theorem, and then use a bootstrap argument with the sum of squares decomposition for matrix functions, obtained in our second paper of this series, to prove a hypoellipticity theorem that generalizes some cases of the results of Christ, Hoshiro, Koike, Kusuoka and Stroock and Morimoto for sums of squares, and of Fedıi and Kohn for degeneracies not necessarily a sum of squares. [ABSTRACT FROM AUTHOR]

We prove Cameron-Martin type quasi-invariance for the heat kernel measure of infinite-dimensional Kolmogorov and similar degenerate diffusions. We first study quantitative functional inequalities, particularly Wang-type Harnack inequalities, for appropriate finite-dimensional approximations of these diffusions, and we prove that these inequalities hold with dimension-independent constants. Applying an approach developed in [7, 12-13], these uniform bounds may then be used to prove that the heat kernel measure for these infinite-dimensional diffusions is quasi-invariant under changes of the initial state. [ABSTRACT FROM AUTHOR]

In this paper, we prove the hypoellipticity and parabolic hypoellipticity of nonlocal operators under Hörmander's condition. One key step in the proof is to obtain an off-diagonal small time estimate of the densities of the processes associated to these operators. To this end, we use a new version of Malliavin calculus on the Poisson space, namely the lent particle method, recently developed by Bouleau and Denis. [ABSTRACT FROM AUTHOR]

We prove that the approximate null-controllability with uniform cost of the hypoelliptic Ornstein-Uhlenbeck equations posed on Rn is characterized by an integral thickness geometric condition on the control supports. We also provide associated quantitative weak observability estimates. This result for the hypoelliptic Ornstein-Uhlenbeck equations is deduced from the same study for a large class of nonautonomous elliptic equations from moving control supports. We generalize in particular results known for parabolic equations posed on Rn, for which the approximate null-controllability with uniform cost is ensured by the notion of thickness, which is stronger than the integral thickness condition considered in the present paper. Examples of those parabolic equations are the fractional heat equations associated with the operator (-Δ)s in the regime s ≥ 1/2. Our strategy also allows us to characterize the approximate null-controllability with uniform cost from moving control supports for this class of fractional heat equations. [ABSTRACT FROM AUTHOR]

This paper is the second in a series of three papers devoted to sums of squares and hypoellipticity of infinitely degenerate operators. In the first paper we established a sharp ω -monotonicity criterion for writing a smooth nonnegative function f that is flat at, and positive away from, the origin, as a finite sum of squares of C 2 , δ functions for some δ > 0 , namely that f is ω -monotone for some Hölder modulus of continuity ω. Counterexamples were provided for any larger modulus of continuity. In this paper we consider the analogous sum of squares problem for smooth nonnegative matrix functions M that are flat at, and positive away from, the origin. We show that such a matrix function M = [ a k j ] k , j = 1 n can be written as a finite sum of squares of C 2 , δ vector fields if the diagonal entries a k k are ω -monotone for some Hölder modulus of continuity ω , and if the off diagonal entries satisfy certain differential bounds in terms of powers of the diagonal entries. Examples are given to show that in some cases at least, these differential inequalities cannot be relaxed. Various refinements of these results are also given in which one or more of the diagonal entries need not be assumed to have any monotonicity properties at all. These sum of squares decompositions will be applied to hypoellipticity in the infinitely degenerate regime in the third paper in this series. [ABSTRACT FROM AUTHOR]

This paper is part of a continuing examination into the geometric and analytic properties of the Kohn Laplacian and its inverse on general quadric submanifolds of \mathbb {C}^n\times \mathbb {C}^m. The goal of this article is explore the complex Green operator in the case that the eigenvalues of the directional Levi forms are nonvanishing. We (1) investigate the geometric conditions on M which the eigenvalue condition forces, (2) establish optimal pointwise upper bounds on complex Green operator and its derivatives, (3) explore the L^p and L^p-Sobolev mapping properties of the associated kernels, and (4) provide examples. [ABSTRACT FROM AUTHOR]

We study the link between Ψdos and Wick operators via the Bargmann transform. We deduce a formula for the symbol of the Wick operator in terms of the short-time Fourier transform of the Weyl symbol. This gives characterizations of Wick symbols of Ψdos of Shubin type and of infinite order, and results on composition. We prove a series expansion of Wick operators in terms of anti-Wick operators which leads to a sharp Gårding inequality and transition of hypoellipticity between Wick and Shubin symbols. Finally we show continuity results for anti-Wick operators, and estimates for the Wick symbols of anti-Wick operators. [ABSTRACT FROM AUTHOR]

We deal with the problem of parameter estimation in stochastic differential equations (SDEs) in a partially observed framework. We aim to design a method working for both elliptic and hypoelliptic SDEs, the latters being characterized by degenerate diffusion coefficients. This feature often causes the failure of constrast estimator based on Euler Maruyama discretization scheme and dramatically impairs classic stochastic filtering methods used to reconstruct the unobserved states. All of theses issues make the estimation problem in hypoelliptic SDEs difficult to solve. To overcome this, we construct a well-defined cost function no matter the elliptic nature of the SDEs. We also bypass the filtering step by considering a control theory perspective. The unobserved states are estimated by solving deterministic optimal control problems using numerical methods which do not need strong assumptions on the diffusion coefficient conditioning. Numerical simulations made on different partially observed hypoelliptic SDEs reveal our method produces accurate estimate while dramatically reducing the computational price comparing to other estimation procedures. [ABSTRACT FROM AUTHOR]

In this article, we construct and analyse an explicit numerical splitting method for a class of semi-linear stochastic differential equations (SDEs) with additive noise, where the drift is allowed to grow polynomially and satisfies a global one-sided Lipschitz condition. The method is proved to be mean-square convergent of order 1 and to preserve important structural properties of the SDE. First, it is hypoelliptic in every iteration step. Second, it is geometrically ergodic and has an asymptotically bounded second moment. Third, it preserves oscillatory dynamics, such as amplitudes, frequencies and phases of oscillations, even for large time steps. Our results are illustrated on the stochastic FitzHugh-Nagumo model and compared with known mean-square convergent tamed/truncated variants of the Euler-Maruyama method. The capability of the proposed splitting method to preserve the aforementioned properties may make it applicable within different statistical inference procedures. In contrast, known Euler-Maruyama type methods commonly fail in preserving such properties, yielding ill-conditioned likelihood-based estimation tools or computationally infeasible simulation-based inference algorithms. [ABSTRACT FROM AUTHOR]

We study the change of quantization for a class of global pseudodifferential operators of infinite order in the setting of ultradifferentiable functions of Beurling type. The composition of different quantizations as well as the transpose of a quantization are also analysed, with applications to the Weyl calculus. We also compare global ω -hypoellipticity and global ω -regularity of these classes of pseudodifferential operators. [ABSTRACT FROM AUTHOR]

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.

We consider a Kolmogorov-Fokker-Planck operator of the kind: \[ \mathcal{L}u=\sum_{i,j=1}^{q}a_{ij}\left( t\right) \partial_{x_{i}x_{j}} ^{2}u+\sum_{k,j=1}^{N}b_{jk}x_{k}\partial_{x_{j}}u-\partial_{t}u,\qquad (x,t)\in\mathbb{R}^{N+1} \] where $\left\{ a_{ij}\left(t\right) \right\} _{i,j=1}^{q}$ is a symmetric uniformly positive matrix on $\mathbb{R}^{q}$, $q\leq N$, of bounded measurable coefficients defined for $t\in\mathbb{R}$ and the matrix $B=\left\{ b_{ij}\right\} _{i,j=1}^{N}$ satisfies a structural assumption which makes the corresponding operator with constant $a_{ij}$ hypoelliptic. We construct an explicit fundamental solution $\Gamma$ for $\mathcal{L}$, study its properties, show a comparison result between $\Gamma$ and the fundamental solution of some model operators with constant $a_{ij}$, and show the unique solvability of the Cauchy problem for $\mathcal{L}$ under various assumptions on the initial datum.

In this paper we show the Hörmander hypoelliptic theorem for nonlocal operators by a purely probabilistic method: the Malliavin calculus. Roughly speaking, under general Hörmander's Lie bracket conditions, we show the regularization effect of discontinuous Lévy noises for possibly degenerate stochastic differential equations with jumps. To treat the large jumps, we use the perturbation argument together with interpolation techniques and some short time asymptotic estimates of the semigroup. As an application, we show the existence of fundamental solutions for operator ∂ t - K , where K is the following nonlocal kinetic operator: K f (x , v) = p. v. ∫ R d (f (x , v + w) - f (x , v)) κ (x , v , w) | w | d + α d w + v · ∇ x f (x , v) + b (x , v) · ∇ v f (x , v). Here κ 0 - 1 ⩽ κ (x , v , w) ⩽ κ 0 belongs to C b ∞ (R 3 d) and is symmetric in w, p.v. stands for the Cauchy principal value, and b ∈ C b ∞ (R 2 d ; R d) . [ABSTRACT FROM AUTHOR]

We obtain asymptotic expansions for the large deviation principle (LDP) for continuous time stochastic processes with weakly-dependent increments. As a key example, we show that additive functionals of solutions of stochastic differential equations (SDEs) satisfying Hörmander condition on a d -dimensional compact manifold admit these asymptotic expansions of all orders. [ABSTRACT FROM AUTHOR]

This article addresses the local boundedness and Hölder continuity of weak solutions to kinetic Fokker--Planck equations with general transport operators and rough coefficients. These results are due to the mixing effect of diffusion and transport. Although the equation is parabolic only in the velocity variable, it has a hypoelliptic structure provided that the transport part ∂ t + b(v). ∇x is nondegenerate in some sense. We achieve the results by revisiting the method, proposed by Golse, Imbert, Mouhot, and Vasseur in the case b(v) = v, that combines the elliptic De Giorgi--Nash--Moser theory with velocity averaging lemmas. [ABSTRACT FROM AUTHOR]

We show that any generalized smooth distribution on a smooth manifold, possibly of non-constant rank, admits a Riemannian metric. Using such a metric, we attach a Laplace operator to any smooth distribution as such. When the underlying manifold is compact, we show that it is essentially self-adjoint. Viewing this Laplacian in the longitudinal pseudodifferential calculus of the smallest singular foliation which includes the distribution, we prove hypoellipticity. [ABSTRACT FROM AUTHOR]

In this paper, we consider periodic boundary value problems for a differential equation whose coefficients are trigonometric polynomials. The spaces of generalized functions are constructed, in which the problems considered have solutions, in particular, the solvability space of a periodic analogue of the Mizohata equation is constructed. A periodic analogue and a generalization of the construction of a nonstandard analysis are constructed, containing not only functions, but also functional spaces. As an illustration of the statement that not all constructions on a torus lead to simplification compared to a plane, a periodic analogue of the concept of a hypoelliptic differential operator is considered, where number-theoretic properties are significant. In particular, it turns out that if a polynomial with integer coefficients is irreducible in the rational field, then the corresponding differential operator is hypoelliptic on the torus.

We study a 1-dimensional chain of N weakly anharmonic classical oscillators coupled at its ends to heat baths at different temperatures. Each oscillator is subject to pinning potential and it also interacts with its nearest neighbors. In our set up both potentials are homogeneous and bounded (with N dependent bounds) perturbations of the harmonic ones. We show how a generalised version of Bakry–Emery theory can be adapted to this case of a hypoelliptic generator which is inspired by Baudoin (J Funct Anal 273(7):2275-2291, 2017). By that we prove exponential convergence to non-equilibrium steady state in Wasserstein–Kantorovich distance and in relative entropy with quantitative rates. We estimate the constants in the rate by solving a Lyapunov-type matrix equation and we obtain that the exponential rate, for the homogeneous chain, has order bigger than N - 3 . For the purely harmonic chain the order of the rate is in [ N - 3 , N - 1 ] . This shows that, in this set up, the spectral gap decays at most polynomially with N. [ABSTRACT FROM AUTHOR]

Multi-dimensional stochastic differential equations (SDEs) are a powerful tool to describe dynamics of phenomena that change over time. We focus on the parametric estimation of such SDEs based on partial observations when only a one-dimensional component of the system is observable. We consider two families of SDE, the elliptic family with a full-rank diffusion coefficient and the hypoelliptic family with a degenerate diffusion coefficient. Estimation for the second class is much more difficult and only few estimation methods have been proposed. Here, we adopt the framework of the optimal control theory to derive a 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 drift and diffusion coefficient parameters. Numerical simulations made on different examples (Harmonic Oscillator, FitzHugh–Nagumo, Lotka–Volterra) reveal our method produces good pointwise estimate for an acceptable computational price with, interestingly, no performance drop for hypoelliptic systems. [ABSTRACT FROM AUTHOR]

To a system of n closed one-forms on the torus T m + n , we associate a differential complex and compute the induced cohomology groups in the s-Gevrey category, provided that a related matrix of periods satisfies a Diophantine condition. Also, we present a complete characterization for the s-global hypoellipticity at the level of q-forms for q ≥ 0 . [ABSTRACT FROM AUTHOR]

We prove hypoellipticity of second order linear operators on R n + m of the form L (x , y , D x , D y) = L 1 (x , D x) + g (x) L 2 (y , D y) , where L j , j = 1 , 2 , satisfy Morimoto's super-logarithmic estimates | | log ξ 2 u ^ (ξ) | | 2 ≤ ε (L j u , u) + C ε , K | | u | | 2 , and g is smooth, nonnegative, and vanishes only at the origin in R n to any arbitrary order. We also show examples in which our hypotheses are necessary for hypoellipticity. [ABSTRACT FROM AUTHOR]

The classical Bakry-Émery calculus is extended to study, for degenerated (non-elliptic, non-reversible, or non-diffusive) Markov processes, questions such as hypoellipticity, hypocoercivity, functional inequalities or Wasserstein contraction. In particular we obtain the optimal speed of convergence to equilibrium for any ergodic Ornstein-Uhlenbeck process, which is given by the spectral gap of the drift matrix and the size of the corresponding Jordan blocks. We also study chains of N interacting overdamped particles and establish for their invariant measures log-Sobolev inequalities with constants of order N2, which is optimal. [ABSTRACT FROM AUTHOR]

Magnetohydrodynamics system consists of a coupling of the Navier-Stokes and Maxwell's equations and is most useful in studying the motion of electrically conducting fluids. We prove the existence of a unique invariant, and consequently ergodic, measure for the Galerkin approximation system of the three-dimensional magnetohydrodynamics system. The proof is inspired by those of [E. Weinan and J.C. Mattingly, Ergodicity for the Navier-Stokes equation with degenerate random forcing: Finitedimensional approximation, Comm. Pure Appl. Math. LIV (2001), pp. 1386-1402; M. Romito, Ergodicity of the finite dimensional approximation of the 3D Navier-Stokes equations forced by a degenerate noise, J. Stat. Phys. 114 (2004), pp. 155-177] on the Navier-Stokes equations; however, computations involve significantly more complications due to the coupling of the velocity field equations with those of magnetic field that consists of four non-linear terms. [ABSTRACT FROM AUTHOR]

The Gevrey hypoellipticity of a class of “sums of squares” with real analytic coefficients is studied in detail. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 L2 space, and where dependencies on the mean-free path τ and the Debye length δ are made explicit. In our analysis the mean free path covers the full range of possible values: from the regime of evanescent collisions τ→∞ to the strongly collisional regime τ→0. 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 τ-dependent constraint on δ ensuring exponential decay with explicit τ-dependent rates towards the stationary solution. In the strongly collisional limit τ→0, 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 L2 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. [ABSTRACT FROM AUTHOR]

We consider a diffusion process under a local weak Hörmander condition on the coefficients.We find Gaussian estimates for the density in short time and exponential lower and upper bounds for the probability that the diffusion remains in a small tube around a deterministic trajectory (skeleton path). These bounds depend explicitly on the radius of the tube and on the energy of the skeleton path. We use a norm which reflects the non-isotropic structure of the problem, meaning that the diffusion propagates in R2 with different speeds in the directions σ and [σ, b]. We establish a connection between this norm and the standard control distance. [ABSTRACT FROM AUTHOR]

We consider the three-dimensional magnetohydrodynamics (MHD) equations in the presence of a spatially degenerate stochastic forcing as a model for magnetostrophic turbulence in the Earth's uid core. We examine the multiparameter singular limit of vanishing Rossby number ε and magnetic Reynold's number δ, and establish that (i) the formal limit (with ε = δ = 0), a stochastically driven active scalar equation, possesses a unique ergodic invariant measure, and (ii) any suitable sequence of statistically invariant states of the full MHD system converges weakly, as ε δ → 0, to the unique invariant measure of the limit equation. This latter convergence result does not require any conditions on the relative rates at which ε, δ decay. Our analysis of the limit equation relies on a recently developed theory of hypoellipticity for infinite-dimensional stochastic dynamical systems. We carry out a detailed study of the interactions between the nonlinear and stochastic terms to demonstrate that a Hörmander bracket condition is satisfied for the limit equation, which yields a contraction property in a suitable Wasserstein metric. This contraction property reduces the convergence of invariant states in the multiparameter limit to the convergence of solutions at finite times. However, in view of the phase space mismatch between the small parameter system and the limit equation, and due to the multiparameter nature of the problem, further analysis is required to establish the singular limit. In particular, we develop methods to lift the contraction for the limit equation to the extended phase space, including the velocity and magnetic fields. Moreover, for the convergence of solutions at finite times we make use of a probabilistic modification of the Grönwall inequality, relying on a delicate stopping time argument. [ABSTRACT FROM AUTHOR]

Our aim is to give a result on Gevrey regularity for Gevrey vectors (in-space) for Hörmander’s operators and a very short survey of some works in the area, particularly recent ones on systems of vector fields or the partial differential operators of second-order associated to these systems. We also recall the regularity results in the case of elliptic operators or of principal type. [ABSTRACT FROM PUBLISHER]

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. [ABSTRACT FROM AUTHOR]

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 ℝd+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 dominant. [ABSTRACT FROM AUTHOR]