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We consider Kolmogorov-Fokker-Planck operators of the form $$ \mathcal{L}u=\sum_{i,j=1}^{q}a_{ij}(x,t)u_{x_{i}x_{j}}+\sum_{k,j=1}^{N} b_{jk}x_{k}u_{x_{j}}-\partial_{t}u, $$ with $\left( x,t\right) \in\mathbb{R}^{N+1},N\geq q\geq1$. We assume that $a_{ij}\in L^{\infty}\left( \mathbb{R}^{N+1}\right) $, the matrix $\left\{ a_{ij}\right\} $ is symmetric and uniformly positive on $\mathbb{R}^{q}$, and the drift \[ Y=\sum_{k,j=1}^{N}b_{jk}x_{k}\partial_{x_{j}}-\partial_{t} \] has a structure which makes the model operator with constant $a_{ij}$ hypoelliptic, translation invariant w.r.t. a suitable Lie group operation, and $2$-homogeneus w.r.t. a suitable family of dilations. We also assume that the coefficients $a_{ij}$ are $VMO$ w.r.t. the space variable, and only bounded measurable in $t$. We prove, for every $p\in\left( 1,\infty\right) $, global Sobolev estimates of the kind: \begin{align*} \Vert u\Vert _{W_{X}^{2,p}(S_{T})} \equiv & \sum_{i,j=1}^{q}\Vert u_{x_{i}x_{j}}\Vert_{L^{p}(S_{T})} +\Vert Yu\Vert _{L^{p}(S_{T})} +\sum_{i=1}^{q}\Vert u_{x_{i}}\Vert _{L^{p}(S_{T})} +\Vert u\Vert _{L^{p}(S_{T})} \\ & \leq c\big\{ \Vert \mathcal{L}u\Vert _{L^{p}(S_{T})}+\Vert u\Vert_{L^{p}(S_{T})}\big\} \end{align*} with $S_{T}=\mathbb{R}^{N}\times\left( -\infty,T\right) $ for any $T\in(-\infty,+\infty]$. Also, the well-posedness in $W_{X}^{2,p}(\Omega_{T})$, with $\Omega_{T}=\mathbb{R}^{N}\times(0,T) $ and $T\in\mathbb{R}$, of the Cauchy problem% $$ \begin{cases} \mathcal{L}u=f & \text{in $\Omega_{T}$} \\ u(\cdot,0) =g & \text{in $\mathbb{R}^{N}$} \end{cases} $$ is proved, for $f\in L^{p}(\Omega_{T}), g\in W_{X}^{2,p}(\mathbb{R}^{N})$.

We consider a class of nonvariational degenerate elliptic operators of the kind \[ Lu=\sum_{i,j=1}^{m}a_{ij}\left( x\right) X_{i}X_{j}u \] where $\left\{ a_{ij}\left( x\right) \right\} _{i,j=1}^{m}$ is a symmetric uniformly positive matrix of bounded measurable functions defined in the whole $\mathbb{R}^{n}$ ($n>m$), possibly discontinuos but satisfying a $VMO$ assumption, and $X_{1},...,X_{m}$ are real smooth vector fields satisfying H\"{o}rmander rank condition in the whole $\mathbb{R}^{n}$ and $1$-homogeneous w.r.t. a family of nonisotropic dilations. We do not assume that the vector fields are left invariant w.r.t. an underlying Lie group of translations. We prove global $W_{X}^{2,p}$ a-priori estimates, for every $p\in\left( 1,\infty\right) $, of the kind: \[ \Vert u\Vert_{W_{X}^{2,p}(\mathbb{R}^{n})}\leq c\left\{ \left\Vert Lu\right\Vert _{L^{p}\left( \mathbb{R}^{n}\right) }+\left\Vert u\right\Vert _{L^{p}\left( \mathbb{R}^{n}\right) }\right\} \] for every $u\in W_{X}^{2,p}\left( \mathbb{R}^{n}\right) .$ We also prove higher order estimates and corresponding regularity results: if $a_{ij}\in W_{X}^{k,\infty}\left( \mathbb{R}^{n}\right) $, $u\in W_{X}^{2,p}\left( \mathbb{R}^{n}\right) $, $Lu\in W_{X}^{k,p}\left( \mathbb{R}^{n}\right) $, then $u\in W_{X}^{k+2,p}\left( \mathbb{R}^{n}\right) $ and \[ \Vert u\Vert_{W_{X}^{k+2,p}(\mathbb{R}^{n})}\leq c\left\{ \Vert Lu\Vert_{W_{X}^{k,p}(\mathbb{R}^{n})}+\Vert u\Vert_{L^{p}(\mathbb{R}^{n} )}\right\} . \]

We consider degenerate KFP operators \[ Lu=\sum_{i,j=1}^{m_{0}}a_{ij}(x,t)\partial_{x_{i}x_{j}}^{2}u+\sum_{k,j=1}^{N}b_{jk}x_{k}\partial_{x_{j}}u-\partial_{t}u\equiv\sum_{i,j=1}^{m_{0}}a_{ij}(x,t)\partial_{x_{i}x_{j}}^{2}u+Yu \] ($(x,t)\in\mathbb{R}^{N+1}$, $1\leq m_{0}\leq N$) s.t. the model operator having constant $a_{ij}$ is hypoelliptic, translation invariant w.r.t. a Lie group in $\mathbb{R}^{N+1}$ and $2$-homogeneous w.r.t. a family of dilations; $(a_{ij})_{i,j=1}^{m_{0}}$ is symmetric and uniformly positive on $\mathbb{R}^{m_{0}}$; $a_{ij}$ are bounded and Dini continuous in space, bounded measurable in time, i.e.: setting \[ S_{T}=\mathbb{R}^{N}\times\left( -\infty,T\right) , \] \[ \omega_{f,S_{T}}(r)=\sup_{\substack{(x,t),(y,t)\in S_{T}\\\Vert x-y\Vert\leq r}}|f(x,t)-f(y,t)| \] \[ \Vert f\Vert_{\mathcal{D}(S_{T})}=\int_{0}^{1}\frac{\omega_{f,S_{T}}(r)}% {r}dr+\Vert f\Vert_{L^{\infty}\left( S_{T}\right) } \] we ask $\Vert a_{ij}\Vert_{\mathcal{D}(S_{T})}<\infty$. We bound $\omega_{u_{x_{i}x_{j}},S_{T}}$, $\left\Vert u_{x_{i}x_{j}}\right\Vert _{L^{\infty}(S_{T})}$ ($i,j=1,2,...,m_{0}$), $\omega_{Yu,S_{T}}$, $\Vert Yu\Vert_{L^{\infty}(S_{T})}$ in terms of $\omega_{\mathcal{L}u,S_{T}}$, $\Vert Lu\Vert_{L^{\infty}(S_{T})}$ and $\Vert u\Vert_{L^{\infty}\left( S_{T}\right) }$, getting a control on the uniform continuity in space of $u_{x_{i}x_{j}},Yu$ if $Lu$ is bounded and Dini-continuous in space. Under the additional assumption that $a_{ij}$ and $\mathcal{L}u$ are log-Dini continuous, meaning the finiteness of the quantity% \[ \int_{0}^{1}\frac{\omega_{f,S_{T}}\left( r\right) }{r}\left\vert \log r\right\vert dr, \] we prove that $u_{x_{i}x_{j}}$ and $Yu$ are Dini continuous; moreover, in this case, the derivatives $u_{x_{i}x_{j}}$ are locally uniformly continuous in space and time.

We consider degenerate Kolmogorov-Fokker-Planck operators $$ \mathcal{L}u=\sum_{i,j=1}^{q}a_{ij}(x,t)\partial_{x_{i}x_{j}}^{2}u+\sum_{k,j=1}^{N}b_{jk}x_{k}\partial_{x_{j}}u-\partial_{t}u,\qquad (x,t)\in\mathbb{R}^{N+1},N\geq q\geq1 $$ such that the corresponding model operator having constant $a_{ij}$ is hypoelliptic, translation invariant w.r.t. a Lie group operation in $\mathbb{R}^{N+1}$ and $2$-homogeneous w.r.t. a family of nonisotropic dilations. The coefficients $a_{ij}$ are bounded and H\"{o}lder continuous in space (w.r.t. some distance induced by $\mathcal{L}$ in $\mathbb{R}^{N}$) and only bounded measurable in time; the matrix $\{ a_{ij}\}_{i,j=1}^{q}$ is symmetric and uniformly positive on $\mathbb{R}^{q}$. We prove "partial Schauder a priori estimates" the kind $$ \sum_{i,j=1}^{q}\Vert\partial_{x_{i}x_{j}}^{2}u\Vert_{C_{x}^{\alpha}(S_{T})}+\Vert Yu\Vert_{C_{x}^{\alpha}(S_{T})}\leq c\left\{ \Vert\mathcal{L}u\Vert _{C_{x}^{\alpha}(S_{T})}+\Vert u\Vert_{C^{0}(S_{T})}\right\} $$ for suitable functions $u$, where $$ \Vert f\Vert_{C_{x}^{\alpha}(S_{T})}=\sup_{t\leq T}\sup_{x_{1},x_{2}\in\mathbb{R}^{N},x_{1}\neq x_{2}}\frac{\left\vert f\left( x_{1},t\right) -f\left( x_{2},t\right) \right\vert }{\left\Vert x_{1}-x_{2}\right\Vert ^{\alpha}}. $$ We also prove that the derivatives $\partial_{x_{i}x_{j}}^{2}u$ are locally H\"{o}lder continuous in space and time while $\partial_{x_{i}}u$ and $u$ are globally H\"{o}lder continuous in space and time.

Let $X_{1},...,X_{m}$ be a family of real smooth vector fields defined in $\mathbb{R}^{n}$, $1$-homogeneous with respect to a nonisotropic family of dilations and satisfying H\"{o}rmander's rank condition at $0$ (and therefore at every point of $\mathbb{R}^{n}$). The vector fields are not assumed to be translation invariant with respect to any Lie group structure. Let us consider the nonvariational evolution operator $$ \mathcal{H}:=\sum_{i,j=1}^{m}a_{i,j}(t,x)X_{i}X_{j}-\partial_{t}% $$ where $(a_{i,j}(t,x))_{i,j=1}^{m}$ is a symmetric uniformly positive $m\times m$ matrix and the entries $a_{ij}$ are bounded H\"{o}lder continuous functions on $\mathbb{R}^{1+n}$, with respect to the "parabolic" distance induced by the vector fields. We prove the existence of a global heat kernel $\Gamma(\cdot;s,y)\in C_{X,\mathrm{loc}}^{2,\alpha}(\mathbb{R}^{1+n}\setminus\{(s,y)\})$ for $\mathcal{H}$, such that $\Gamma$ satisfies two-sided Gaussian bounds and $\partial_{t}\Gamma, X_{i}\Gamma,X_{i}X_{j}\Gamma$ satisfy upper Gaussian bounds on every strip $[0,T]\times\mathbb{R}^n$. We also prove a scale-invariant parabolic Harnack inequality for $\mathcal{H}$, and a standard Harnack inequality for the corresponding stationary operator $$ \mathcal{L}:=\sum_{i,j=1}^{m}a_{i,j}(x)X_{i}X_{j}. $$ with H\"{o}lder continuos coefficients.

The emergence of the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) disease (COVID-19) has posed a serious threat to global health. As no specific therapeutics are yet available to control disease evolution, more in-depth understanding of the pathogenic mechanisms induced by SARS-CoV-2 will help to characterize new targets for the management of COVID-19. The present study identified a specific set of biological pathways altered in primary human lung epithelium upon SARS-CoV-2 infection, and a comparison with SARS-CoV from the 2003 pandemic was studied. The transcriptomic profiles were also exploited as possible novel therapeutic targets, and anti-signature perturbation analysis predicted potential drugs to control disease progression. Among them, Mitogen-activated protein kinase kinase (MEK), serine-threonine kinase (AKT), mammalian target of rapamycin (mTOR) and I kappa B Kinase (IKK) inhibitors emerged as candidate drugs. Finally, sex-specific differences that may underlie the higher COVID-19 mortality in men are proposed.

Let $\mathcal{H}=\sum_{j=1}^{m}X_{j}^{2}-\partial_{t}$ be a heat-type operator in $\mathbb{R}^{n+1}$, where $X=\{X_{1},\ldots,X_{m}\}$ is a system of smooth H\"{o}rmander's vector fields in $\mathbb{R}^{n}$, and every $X_{j}$ is homogeneous of degree $1$ with respect to a family of non-isotropic dilations in $\mathbb{R}^{n}$, while no underlying group structure is assumed. In this paper we prove global (in space and time) upper and lower Gaussian estimates for the heat kernel $\Gamma(t,x;s,y)$ of $\mathcal{H}$, in terms of the Carnot-Carath\'{e}odory distance induced by $X$ on $\mathbb{R}^{n}$, as well as global upper Gaussian estimates for the $t$- or $X$-derivatives of any order of $\Gamma$. From the Gaussian bounds we derive the unique solvability of the Cauchy problem for a possibly unbounded continuous initial datum satisfying exponential growth at infinity. Also, we study the solvability of the H-Dirichlet problem on an arbitrary bounded domain. Finally, we establish a global scale-invariant Harnack inequality for non-negative solutions of $\mathcal{H}u=0$.

We consider a Kolmogorov-Fokker-Planck operator of the kind studied by Lanconelli-Polidoro in [Rend. Sem. Mat. Univ. Politec. Torino 52 (1994)], where the leading coefficients $a_{ij}$, instead of being constant, are bounded measurable functions of t. We construct an explicit fundamental solution for this operator, study its property, show a comparison result between this function and the fundamental solution of some model operators with constant $a_{ij}$, and show the unique solvability of the Cauchy problem under various assumptions on the initial datum.

Let $\mathcal{L}=\sum_{j=1}^m X_j^2$ be a H\"ormander sum of squares of vector fields in space $\mathbb{R}^n$, where any $X_j$ is homogeneous of degree $1$ with respect to a family of non-isotropic dilations in space. In this paper we prove global estimates and regularity properties for $\mathcal{L}$ in the $X$-Sobolev spaces $W^{k,p}_X(\mathbb{R}^n)$, where $X = \{X_1,\ldots,X_m\}$. In our approach, we combine local results for general H\"ormander sums of squares, the homogeneity property of the $X_j$'s, plus a global lifting technique for homogeneous vector fields.

Let $\mathcal{L}=\sum_{j=1}^{m}X_{j}^{2}$ be a H\"{o}rmander sum of squares of vector fields in $\mathbb{R}^{n}$, where any $X_{j}$ is homogeneous of degree $1$ with respect to a family of non-isotropic dilations in $\mathbb{R}^{n}$. Then $\mathcal{L}$ is known to admit a global fundamental solution $\Gamma (x;y)$, that can be represented as the integral of a fundamental solution of a sublaplacian operator on a lifting space $\mathbb{R}^{n}\times \mathbb{R}^{p}$, equipped with a Carnot group structure. The aim of this paper is to prove global pointwise (upper and lower) estimates of $\Gamma $, in terms of the Carnot-Carath\'{e}odory distance induced by $X=\{X_{1},\ldots ,X_{m}\}$ on $\mathbb{R}^{n}$, as well as global pointwise (upper) estimates for the $X$-derivatives of any order of $\Gamma $, together with suitable integral representations of these derivatives. The least dimensional case $n=2$ presents several peculiarities which are also investigated. Applications to the potential theory for $\mathcal{L}$ and to singular-integral estimates for the kernel $X_{i}X_{j}\Gamma $ are also provided. Finally, most of the results about $\Gamma$ are extended to the case of H\"{o}rmander operators with drift $\sum_{j=1}^{m}X_{j}^{2}+X_{0}$, where $X_{0}$ is $2$-homogeneous and $X_{1},...,X_{m}$ are $1$-homogeneous.

We consider a heat-type operator L structured on the left invariant 1-homogeneous vector fields which are generators of a Carnot group, multiplied by a uniformly positive matrix of bounded measurable coefficients depending only on time. We prove that if Lu is smooth with respect to the space variables, the same is true for u, with quantitative regularity estimates in the scale of Sobolev spaces defined by right invariant vector fields. Moreover, the solution and its space derivatives satisfy a 1/2-H\"older continuity estimate with respect to time. The result is proved both for weak solutions and for distributional solutions, in a suitable sense., Comment: 34 pages

We consider a nonvariational degenerate elliptic operator structured on a system of left invariant, 1-homogeneous, H\"ormander's vector fields on a Carnot group in $R^{n}$, where the matrix of coefficients is symmetric, uniformly positive on a bounded domain of $R^{n}$ and the coefficients are bounded, measurable and locally VMO in the domain. We give a new proof of the interior $L^{p}$ estimates on the second order derivatives with respect to the vector fields, first proved by Bramanti-Brandolini in [Rend. Sem. Mat. dell'Univ. e del Politec. di Torino, Vol. 58, 4 (2000), 389-433], extending to this context Krylov' technique, introduced in [Comm. in P.D.E.s, 32 (2007), 453-475], consisting in estimating the sharp maximal function of the second order derivatives., Comment: 26 pages

We prove a local version of Fefferman-Stein inequality for the local sharp maximal function, and a local version of John-Nirenberg inequality for locally BMO functions, in the framework of locally homogeneous spaces, in the sense of Bramanti-Zhu [Manuscripta Math. 138 (2012), no. 3-4, 477-528].

We consider operators of the form $L=\sum_{i=1}^{n}X_{i}^{2}+X_{0}$ in a bounded domain of R^p where X_0, X_1,...,X_n are nonsmooth H\"ormander's vector fields of step r such that the highest order commutators are only H\"older continuous. Applying Levi's parametrix method we construct a local fundamental solution \gamma\ for L and provide growth estimates for \gamma\ and its first derivatives with respect to the vector fields. Requiring the existence of one more derivative of the coefficients we prove that \gamma\ also possesses second derivatives, and we deduce the local solvability of L, constructing, by means of \gamma, a solution to Lu=f with H\"older continuous f. We also prove $C_{X,loc}^{2,\alpha}$ estimates on this solution., Comment: 75 pages, LaTeX

For a linear nonvariational operator structured on smooth H\"ormander's vector fields, with H\"older continuous coefficients, we prove a regularity result in the spaces of H\"older functions. We deduce an analogous regularity result for nonvariational degenerate quasilinear equations.

We consider a class of nonvariational linear operators formed by homogeneous left invariant Hormander's vector fields with respect to a structure of Carnot group. The bounded coefficients of the operators belong to "vanishing logarithmic mean oscillation" class with respect to the distance induced by the vector fields (in particular they can be discontinuous). We prove local estimates in "local BMO" spaces intersected with the Lebesgue spaces. Even in the uniformly elliptic case our estimates improve the known results.

We consider a class of degenerate Ornstein-Uhlenbeck operators in $\mathbb{R}^{N}$, of the kind [\mathcal{A}\equiv\sum_{i,j=1}^{p_{0}}a_{ij}(x) \partial_{x_{i}x_{j}}^{2}+\sum_{i,j=1}^{N}b_{ij}x_{i}\partial_{x_{j}}%] where $(a_{ij})$ is symmetric uniformly positive definite on $\mathbb{R}^{p_{0}}$ ($p_{0}\leq N$), with uniformly continuous and bounded entries, and $(b_{ij})$ is a constant matrix such that the frozen operator $\mathcal{A}_{x_{0}}$ corresponding to $a_{ij}(x_{0})$ is hypoelliptic. For this class of operators we prove global $L^{p}$ estimates ($1

Let X_1,...,X_q be the basis of the space of horizontal vector fields on a homogeneous Carnot group in R^n (q

In this paper we provide two different characterizations of sets with finite perimeter and functions of bounded variation in Carnot groups, analogous to those which hold in Euclidean spaces, in terms of the short-time behaviour of the heat semigroup. The second one holds under the hypothesis that the reduced boundary of a set of finite perimeter is rectifiable, a result that presently is known in Step 2 Carnot groups.

We consider linear second order nonvariational partial differential operators of the kind a_{ij}X_{i}X_{j}+X_{0}, on a bounded domain of R^{n}, where the X_{i}'s (i=0,1,2,...,q, n>q+1) are real smooth vector fields satisfying H\"ormander's condition and a_{ij} (i,j=1,2,...,q) are real valued, bounded measurable functions, such that the matrix {a_{ij}} is symmetric and uniformly positive. We prove that if the coefficients a_{ij} are H\"older continuous with respect to the distance induced by the vector fields, then local Schauder estimates on X_{i}X_{j}u, X_{0}u hold; if the coefficients belong to the space VMO with respect to the distance induced by the vector fields, then local L^{p} estimates on X_{i}_{j}u, X_{0}u hold. The main novelty of the result is the presence of the drift term X_{0}, so that our class of operators covers, for instance, Kolmogorov-Fokker-Planck operators.

We introduce the concept of locally homogeneous space, and prove in this context L^p and Holder estimates for singular and fractional integrals, as well as L^p estimates on the commutator of a singular or fractional integral with a BMO or VMO function. These results are motivated by local a-priori estimates for subelliptic equations.

Let S be a Sobolev or Orlicz-Sobolev space of functions not necessarily vanishing at the boundary of the domain. We give sufficient conditions on a nonnegative function in S in order that its spherical rearrangement ("Schwartz symmetrization") still belongs to S. These results are obtained via relative isoperimetric inequalities and somewhat generalize a well-known Polya-Szego's theorem. We also prove that the rearrangement of any function in S is locally in S., Comment: 11 pages

We prove a version of Rothschild-Stein's theorem of lifting and approximation and some related results in the context of nonsmooth Hormander's vector fields for which the highest order commutators are only Holder continuous. The theory explicitly covers the case of one vector field having weight two while the others have weight one., Comment: 46 pages, LaTeX. Minor changes in Section 3

We consider a family of vector fields defined in some bounded domain of R^p, and we assume that they satisfy Hormander's rank condition of some step r, and that their coefficients have r-1 continuous derivatives. We extend to this nonsmooth context some results which are well-known for smooth Hormander's vector fields, namely: some basic properties of the distance induced by the vector fields, the doubling condition, Chow's connectivity theorem, and, under the stronger assumption that the coefficients belong to C^{r-1,1}, Poincare's inequality. By known results, these facts also imply a Sobolev embedding. All these tools allow to draw some consequences about second order differential operators modeled on these nonsmooth Hormander's vector fields., Comment: 60 pages, LaTeX; Section 6 added and Section 7 (6 in the previous version) changed. Some references added

We consider a class of degenerate Ornstein-Uhlenbeck operators in $\mathbb{R}^{N}$, of the kind \[ \mathcal{A}\equiv\sum_{i,j=1}^{p_{0}}a_{ij}\partial_{x_{i}x_{j}}^{2} +\sum_{i,j=1}^{N}b_{ij}x_{i}\partial_{x_{j}}% \] where $(a_{ij}) ,(b_{ij}) $ are constant matrices, $(a_{ij}) $ is symmetric positive definite on $\mathbb{R} ^{p_{0}}$ ($p_{0}\leq N$), and $(b_{ij}) $ is such that $\mathcal{A}$ is hypoelliptic. For this class of operators we prove global $L^{p}$ estimates ($1

Recent theoretical work suggests that violation of the Equivalence Principle might be revealed in a measurement of the fractional differential acceleration $\eta$ between two test bodies -of different composition, falling in the gravitational field of a source mass- if the measurement is made to the level of $\eta\simeq 10^{-13}$ or better. This being within the reach of ground based experiments, gives them a new impetus. However, while slowly rotating torsion balances in ground laboratories are close to reaching this level, only an experiment performed in low orbit around the Earth is likely to provide a much better accuracy. We report on the progress made with the "Galileo Galilei on the Ground" (GGG) experiment, which aims to compete with torsion balances using an instrument design also capable of being converted into a much higher sensitivity space test. In the present and following paper (Part I and Part II), we demonstrate that the dynamical response of the GGG differential accelerometer set into supercritical rotation -in particular its normal modes (Part I) and rejection of common mode effects (Part II)- can be predicted by means of a simple but effective model that embodies all the relevant physics. Analytical solutions are obtained under special limits, which provide the theoretical understanding. A simulation environment is set up, obtaining quantitative agreement with the available experimental data on the frequencies of the normal modes, and on the whirling behavior. This is a needed and reliable tool for controlling and separating perturbative effects from the expected signal, as well as for planning the optimization of the apparatus., Comment: Accepted for publication by "Review of Scientific Instruments" on Jan 16, 2006. 16 2-column pages, 9 figures

We show that the dynamic approach to L\'{e}vy statistics is characterized by aging and multifractality, induced by an ultra-slow transition to anomalous scaling. We argue that these aspects make it a protoptype of complex systems., Comment: 12 pages, including 2 figure pages

We prove uniqueness in law for possibly degenerate SDEs having a linear part in the drift term. Diffusion coefficients corresponding to non-degenerate directions of the noise are assumed to be continuous. When the diffusion part is constant we recover the classical degenerate Ornstein-Uhlenbeck process which only has to satisfy the H\"ormander hypoellipticity condition. In the proof we use global $L^p$-estimates for hypoelliptic Ornstein-Uhlenbeck operators recently proved in Bramanti-Cupini-Lanconelli-Priola (Math. Z. 266 (2010)) and adapt the localization procedure introduced by Stroock and Varadhan. Appendix contains a quite general localization principle for martingale problems.