Accurate stationary densities with partitioned numerical methods for stochastic partial differential equations

We consider the numerical solution, by finite differences, of second-order-in-time stochastic partial differential equations (SPDEs) in one space dimension. New timestepping methods are introduced by generalising recently-introduced methods for second-order-in-time stochastic differential equations to multidimensional systems. These stochastic methods, based on leapfrog and Runge–Kutta methods, are designed to give good approximations to the stationary variances and the correlations in the position and velocity variables. In particular, we introduce the reverse leapfrog method and stochastic Runge–Kutta Leapfrog methods, analyse their performance applied to linear SPDEs and perform numerical experiments to examine their accuracy applied to a type of nonlinear SPDE

ISL1 Directly Regulates FGF10 Transcription during Human Cardiac Outflow Formation

The LIM homeodomain gene Islet-1 (ISL1) encodes a transcription factor that has been associated with the multipotency of human cardiac progenitors, and in mice enables the correct deployment of second heart field (SHF) cells to become the myocardium of atria, right ventricle and outflow tract. Other markers have been identified that characterize subdomains of the SHF, such as the fibroblast growth factor Fgf10 in its anterior region. While functional evidence of its essential contribution has been demonstrated in many vertebrate species, SHF expression of Isl1 has been shown in only some models. We examined the relationship between human ISL1 and FGF10 within the embryonic time window during which the linear heart tube remodels into four chambers. ISL1 transcription demarcated an anatomical region supporting the conserved existence of a SHF in humans, and transcription factors of the GATA family were co-expressed therein. In conjunction, we identified a novel enhancer containing a highly conserved ISL1 consensus binding site within the FGF10 first intron. ChIP and EMSA demonstrated its direct occupation by ISL1. Transcription mediated by ISL1 from this FGF10 intronic element was enhanced by the presence of GATA4 and TBX20 cardiac transcription factors. Finally, transgenic mice confirmed that endogenous factors bound the human FGF10 intronic enhancer to drive reporter expression in the developing cardiac outflow tract. These findings highlight the interest of examining developmental regulatory networks directly in human tissues, when possible, to assess candidate non-coding regions that may be responsible for congenital malformations

Irreducibility and Asymptotics of Stochastic Burgers Equation Driven by α-stable Processes

The irreducibility, moderate deviation principle and $\psi$-uniformly exponential ergodicity with $\psi(x):=1+\|x\|_0$ are proved for stochastic Burgers equation driven by the $\aa$-stable processes for $\aa\in (1,2),$ where the first two are new for the present model, and the last strengthens the exponential ergodicity under total variational norm derived in \cite{Do-Xu-Zh-14}