Density estimates on a parabolic spde

We consider a general class of parabolic spde's [formula] with (t, x) [member of] [0, T]×[0, 1] and [epsilon]Wt,x, [epsilon] > 0, a perturbed Gaussian space-time white noise. For (t, x) [member of] (0, T]×(0, 1) we prove the called Davies and Varadhan-Léandre estimates of the density p[epsilon]t,x of the solution u[epsilon]t,x

Asymptotic behaviour of the density in a parabolic SPDE

Consider the density of the solution $X(t,x)$ of a stochastic heat equation with small noise at a fixed $t\in [0,T]$, $x \in [0,1]$. In the paper we study the asymptotics of this density as the noise is vanishing. A kind of Taylor expansion in powers of the noise parameter is obtained. The coefficients and the residue of the expansion are explicitly calculated. In order to obtain this result some type of exponential estimates of tail probabilities of the difference between the approximating process and the limit one is proved. Also a suitable local integration by parts formula is developped.Malliavin Calculus, parabolic SPDE, large deviations, Taylor expansion of a density, exponential estimates of the tail probabilities, stochastic integration by parts formula

Logarithmic asymptotics of the densities of SPDEs driven by spatially correlated noise

We consider the family of stochastic partial differential equations indexed by a parameter \eps\in(0,1], \begin{equation*} Lu^{\eps}(t,x) = \eps\sigma(u^\eps(t,x))\dot{F}(t,x)+b(u^\eps(t,x)), \end{equation*} (t,x)\in(0,T]\times\Rd with suitable initial conditions. In this equation, $L$ is a second-order partial differential operator with constant coefficients, $\sigma$ and $b$ are smooth functions and $\dot{F}$ is a Gaussian noise, white in time and with a stationary correlation in space. Let p^\eps_{t,x} denote the density of the law of u^\eps(t,x) at a fixed point (t,x)\in(0,T]\times\Rd. We study the existence of \lim_{\eps\downarrow 0} \eps^2\log p^\eps_{t,x}(y) for a fixed $y\in\R$. The results apply to a class of stochastic wave equations with $d\in\{1,2,3\}$ and to a class of stochastic heat equations with $d\ge1$.Comment: 39 pages. Will be published in the book " Stochastic Analysis and Applications 2014. A volume in honour of Terry Lyons". Springer Verla

Differential clinical characteristics and prognosis of intraventricular conduction defects in patients with chronic heart failure

Intraventricular conduction defects (IVCDs) can impair prognosis of heart failure (HF), but their specific impact is not well established. This study aimed to analyse the clinical profile and outcomes of HF patients with LBBB, right bundle branch block (RBBB), left anterior fascicular block (LAFB), and no IVCDs. Clinical variables and outcomes after a median follow-up of 21 months were analysed in 1762 patients with chronic HF and LBBB (n = 532), RBBB (n = 134), LAFB (n = 154), and no IVCDs (n = 942). LBBB was associated with more marked LV dilation, depressed LVEF, and mitral valve regurgitation. Patients with RBBB presented overt signs of congestive HF and depressed right ventricular motion. The LAFB group presented intermediate clinical characteristics, and patients with no IVCDs were more often women with less enlarged left ventricles and less depressed LVEF. Death occurred in 332 patients (interannual mortality = 10.8%): cardiovascular in 257, extravascular in 61, and of unknown origin in 14 patients. Cardiac death occurred in 230 (pump failure in 171 and sudden death in 59). An adjusted Cox model showed higher risk of cardiac death and pump failure death in the LBBB and RBBB than in the LAFB and the no IVCD groups. LBBB and RBBB are associated with different clinical profiles and both are independent predictors of increased risk of cardiac death in patients with HF. A more favourable prognosis was observed in patients with LAFB and in those free of IVCDs. Further research in HF patients with RBBB is warranted

Whole-genome sequencing reveals host factors underlying critical COVID-19

Critical COVID-19 is caused by immune-mediated inflammatory lung injury. Host genetic variation influences the development of illness requiring critical care1 or hospitalization2,3,4 after infection with SARS-CoV-2. The GenOMICC (Genetics of Mortality in Critical Care) study enables the comparison of genomes from individuals who are critically ill with those of population controls to find underlying disease mechanisms. Here we use whole-genome sequencing in 7,491 critically ill individuals compared with 48,400 controls to discover and replicate 23 independent variants that significantly predispose to critical COVID-19. We identify 16 new independent associations, including variants within genes that are involved in interferon signalling (IL10RB and PLSCR1), leucocyte differentiation (BCL11A) and blood-type antigen secretor status (FUT2). Using transcriptome-wide association and colocalization to infer the effect of gene expression on disease severity, we find evidence that implicates multiple genes—including reduced expression of a membrane flippase (ATP11A), and increased expression of a mucin (MUC1)—in critical disease. Mendelian randomization provides evidence in support of causal roles for myeloid cell adhesion molecules (SELE, ICAM5 and CD209) and the coagulation factor F8, all of which are potentially druggable targets. Our results are broadly consistent with a multi-component model of COVID-19 pathophysiology, in which at least two distinct mechanisms can predispose to life-threatening disease: failure to control viral replication; or an enhanced tendency towards pulmonary inflammation and intravascular coagulation. We show that comparison between cases of critical illness and population controls is highly efficient for the detection of therapeutically relevant mechanisms of disease

A first update on mapping the human genetic architecture of COVID-19

peer reviewe

Density estimates on a parabolic spde

We consider a general class of parabolic spde's [formula] with (t, x) [member of] [0, T]×[0, 1] and [epsilon]Wt,x, [epsilon] > 0, a perturbed Gaussian space-time white noise. For (t, x) [member of] (0, T]×(0, 1) we prove the called Davies and Varadhan-Léandre estimates of the density p[epsilon]t,x of the solution u[epsilon]t,x

On stochastic partial differential equations with spatially correlated noise: smoothness of the law

AbstractWe deal with the following general kind of stochastic partial differential equations:Lu(t,x)=α(u(t,x))Ḟ(t,x)+β(u(t,x)),t⩾0,x∈Rdwith null initial conditions, L a second-order partial differential operator and F a Gaussian noise, white in time and correlated in space. Firstly, we prove that the solution u(t,x) possesses a smooth density pt,x for every t>0,x∈Rd. We use the tools of Malliavin Calculus. Secondly, we apply this general result to two particular cases: the d-dimensional spatial heat equation, d⩾1, and the wave equation, d∈{1,2}