Approximation of the ergodic measure of SDEs with singular drift by Euler-Maruyama scheme

We study the approximation of the ergodic measure of the following stochastic differential equation (SDE) on $\mathbb{R}^d$: \begin{eqnarray}\label{e:SDEE} d X_t &=& (b_1(X_t)+b_2(X_t)) d t+\sigma(X_t) d W_t, \end{eqnarray} where $W_t$ is a $d$-dimensional standard Brownian motion, and $b_1: \mathbb{R}^d \mapsto \mathbb{R}^d$, $b_2: \mathbb{R}^d \mapsto \mathbb{R}^d$ and $\sigma: \mathbb{R}^d \mapsto \mathbb{R}^{d\times d}$ are the functions to be specified in Assumption 2.1 below. In particular, $b_1$ satisfies $b_1\in \mathbb{L}^\infty(\mathbb{R}^d)\cap \mathbb{L}^1(\mathbb{R}^d)$ or $b_1 \in \mathcal{C}_b^{\alpha}(\mathbb{R}^d)$ with $\alpha\in (0,1)$, which makes the standard numerical schemes not work or fail to give a good convergence rate. In order to overcome these two difficulties, we first apply a Zvonkin's transform to SDE and obtain a new SDE which has coefficients with nice properties and admits a unique ergodic measure $\widehat \mu$, then discretize the new equation by Euler-Maruyama scheme to approximate $\widehat \mu$, and finally use the inverse Zvonkin's transform to get an approximation of the ergodic measure of SDE, denoted by $\mu$. Our approximation method is inspired by Xie and Zhang [22]. The proof of our main result is based on the method of introducing a stationary Markov chain, a key ingredient in this method is establishing the regularity of a Poisson equation, which is done by combining the classical PDE local regularity and a nice extension trick introduced by Gurvich [10]

Thermodynamics and Agglomeration Behavior on Spinel Inclusion in Al-Deoxidized Steel Coupling with Mg Treatment

There are many types of non-metallic MgAl2O4 inclusions observed in Al-deoxidized steel coupling with Mg treatment, including single-particle MgAl2O4, agglomerated MgAl2O4, and MgAl2O4-MnS. Thermodynamic calculation shows that MgAl2O4 precipitates in the liquid phase. The phase transformation follows liquid + Al2O3 + MgAl2O4 → liquid + MgAl2O4 → liquid + MgO + MgAl2O4 → liquid + MgO with the Mg content increasing when the Al content is a constant in molten steel, and it is in agreement with the experimental results for the formation of MgAl2O4 in molten steel. The calculation results of various attractive forces between two particles show that the cavity bridge force plays a dominant role in the agglomeration process and results in the agglomerated MgAl2O4. The lattice mismatch calculation result shows that MgAl2O4 can provide effective sites for MnS nucleating in steel

Transcriptomic responses to RGNNV and Vibrio alginolyticus infection in the spleen of hybrid grouper (Epinephelus lanceolatus♂ × Epinephelus fuscoguttatus♀)

The hybrid grouper (Epinephelus lanceolatus♂ × E. fuscoguttatus♀) is a novel and commercially important fish for mariculture in China. Recent years, the outbreak of viral and bacterial diseases have caused huge losses of hybrid grouper culture. However, the immune response of hybrid grouper against viral and bacterial infection remains unexplored. In this study, a total of 9 splenic transcriptomic libraries of hybrid grouper including untreated control, Red-spotted grouper nervous necrosis virus (RGNNV)-challenged, Vibrio alginolyticus-challenged were constructed. A sum of 6024 differentially expressed genes (DEGs) with significant differences were identified, of which the overlapped DEGs between RGNNV and V. alginolyticus challenged were significantly enriched in multiple immune-related pathway including interleukin-17 NOD-like receptor, C-type lectin receptor, chemokine signaling pathway and natural killer cell mediated cytotoxicity, indicating their important roles in immune defense of hybrid grouper against (both) viral and bacterial infection. To the best of our knowledge, this is the first report that investigated the immune response of hybrid grouper following viral and bacterial pathogen stimuli using transcriptomic analysis. These results provide comprehensive understandings of the innate immunity of hybrid grouper and valuable references to develop disease control strategy in grouper aquaculture

Additional file 1 of Evidence-based expert consensus on the management of primary central nervous system lymphoma in China

Additional file 1. Selection of clinical questions