Stability of the essential spectrum for 2D--transport models with Maxwell boundary conditions

We discuss the spectral properties of collisional semigroups associated to various models from transport theory by exploiting the links between the so-called resolvent approach and the semigroup approach. Precisely, we show that the essential spectrum of the full transport semigroup coincides with that of the collisionless transport semigroup in any $L^p$--spaces $(1 <p < \infty)$ for three 2D--transport models with Maxwell--boundary conditions.Comment: 23 page

Invariant density and time asymptotics for collisionless kinetic equations with partly diffuse boundary operators

International audienceThis paper deals with collisionless transport equationsin bounded open domains $\Omega \subset \R^{d}$ $(d\geq 2)$ with $\mathcal{C}^{1}$ boundary $\partial \Omega$, orthogonallyinvariant velocity measure \bm{m}(\d v) with support $V\subset \R^{d}$ and stochastic partly diffuse boundary operators $\mathsf{H}$ relating the outgoing andincoming fluxes. Under very general conditions, such equations are governedby stochastic $C_{0}$-semigroups $\left( U_{\mathsf{H}}(t)\right) _{t\geq 0}$ on $$ We give a general criterion of irreducibility of $$ and we show that, under very natural assumptions, if an invariant densityexists then $\left( U_{\mathsf{H}}(t)\right) _{t\geq 0}$ converges strongly (notsimply in Cesar\`o means) to its ergodic projection. We show also that if noinvariant density exists then $\left( U_{\mathsf{H}}(t)\right) _{t\geq 0}$ is\emph{sweeping} in the sense that, for any density $\varphi$, the total mass of $$ concentrates near suitable sets of zero measure as $$ We show also a general weak compactness theoremwhich provides a basis for a general theory on existence of invariantdensities. This theorem is based on a series of results on smoothness andtransversality of the dynamical flow associated to $\left( U_{\mathsf{H}}(t)\right) _{t\geq0}.

The dissipative linear Boltzmann equation for hard spheres

We prove the existence and uniqueness of an equilibrium state with unit mass to the dissipative linear Boltzmann equation with hard--spheres collision kernel describing inelastic interactions of a gas particles with a fixed background. The equilibrium state is a universal Maxwellian distribution function with the same velocity as field particles and with a non--zero temperature lower than the background one, which depends on the details of the binary collision. Thanks to the H--theorem we then prove strong convergence of the solution to the Boltzmann equation towards the equilibrium.Comment: 17 pages, submitted to Journal of Statistical Physic

A Class of Non-Parametric Statistical Manifolds modelled on Sobolev Space

We construct a family of non-parametric (infinite-dimensional) manifolds of finite measures on Rd. The manifolds are modelled on a variety of weighted Sobolev spaces, including Hilbert-Sobolev spaces and mixed-norm spaces. Each supports the Fisher-Rao metric as a weak Riemannian metric. Densities are expressed in terms of a deformed exponential function having linear growth. Unusually for the Sobolev context, and as a consequence of its linear growth, this "lifts" to a nonlinear superposition (Nemytskii) operator that acts continuously on a particular class of mixed-norm model spaces, and on the fixed norm space W²'¹ i.e. it maps each of these spaces continuously into itself. It also maps continuously between other fixed-norm spaces with a loss of Lebesgue exponent that increases with the number of derivatives. Some of the results make essential use of a log-Sobolev embedding theorem. Each manifold contains a smoothly embedded submanifold of probability measures. Applications to the stochastic partial differential equations of nonlinear filtering (and hence to the Fokker-Planck equation) are outlined

Matrix metalloproteinases (MMP-2,9) and their tissue inhibitors (TIMP-1,2) as novel markers of stress response and atherogenesis in children with chronic kidney disease (CKD) on conservative treatment

The system of matrix metalloproteinases (MMPs) and their tissue inhibitors (TIMPs) may play a key role in atherogenesis of chronic kidney disease (CKD) patients by its impact on matrix accumulation. Connections with inflammation, stress, or endothelial dysfunction are also probable. However, the data on correlations between these parameters in CKD patients are scarce in adults and absent in children. The aim of our study was to evaluate serum concentrations of MMP-2, MMP-9, TIMP-1, and TIMP-2, as well as their correlations with markers of stress response (Hsp90-α, anti-Hsp60), endothelial dysfunction (sE-selectin), and inflammation (high-sensitivity C-reactive protein) in CKD children treated conservatively. Thirty-seven patients were divided into two groups according to the CKD stage (gr.CKDI, 19 children with CKD stages 2–3; gr.CKDII, 18 subjects with CKD stages 4–5). Twenty-four age-matched healthy subjects served as controls. Serum concentrations of MMP-2, MMP-9, TIMP-1, TIMP-2, Hsp90-α, anti-Hsp60, and sE-selectin were assessed by ELISA. Median values of MMP-2, MMP-9, TIMP-1, and TIMP-2 were significantly higher in all CKD children vs. controls and were increased in patients with CKD stages 4–5 vs. CKD stages 2–3. Hsp90-α, anti-Hsp60, sE-selectin, and glomerular filtration rate predicted the values of MMPs and TIMPs. Chronic kidney disease in children is characterized by MMP/TIMP system dysfunction, aggravated by the progression of renal failure. Correlations between examined parameters, heat shock proteins, and markers of endothelial damage suggest the possibility of MMP/TIMP application as indicators of stress response and atherogenesis in children with CKD on conservative treatment

Metalloprotease Meprinβ in Rat Kidney: Glomerular Localization and Differential Expression in Glomerulonephritis

Meprin (EC 3.4.24.18) is an oligomeric metalloendopeptidase found in microvillar membranes of kidney proximal tubular epithelial cells. Here, we present the first report on the expression of meprinβ in rat glomerular epithelial cells and suggest a potential involvement in experimental glomerular disease. We detected meprinβ in glomeruli of immunostained rat kidney sections on the protein level and by quantitative RT-PCR of laser-capture microdissected glomeruli on the mRNA level. Using immuno-gold staining we identified the membrane of podocyte foot processes as the main site of meprinβ expression. The glomerular meprinβ expression pattern was altered in anti-Thy 1.1 and passive Heymann nephritis (PHN). In addition, the meprinβ staining pattern in the latter was reminiscent of immunostaining with the sheep anti-Fx1A antiserum, commonly used in PHN induction. Using Western blot and immunoprecipitation assays we demonstrated that meprinβ is recognized by Fx1A antiserum and may therefore represent an auto-antigen in PHN. In anti-Thy 1.1 glomerulonephritis we observed a striking redistribution of meprinβ in tubular epithelial cells from the apical to the basolateral side and the cytosol. This might point to an involvement of meprinβ in this form of glomerulonephritis