Smooth densities of the laws of perturbed diffusion processes

Under some regularity conditions on $b$, $\sigma$ and $\alpha$, we prove that the following perturbed stochastic differential equation \begin{equation} X_t=x+\int_0^t b(X_s)ds+\int_0^t \sigma(X_s) dB_s+\alpha \sup_{0 \le s \le t} X_s, \ \ \ \alpha<1 \end{equation} admits smooth densities for all $0 \le t \le t_0$, where $t_0>0$ is some finite number

Non-integrable stable approximation by Stein's method

We develop Stein's method for $\alpha$-stable approximation with $\alpha\in(0,1]$, continuing the recent line of research by Xu \cite{lihu} and Chen, Nourdin and Xu \cite{C-N-X} in the case $\alpha\in(1,2).$ The main results include an intrinsic upper bound for the error of the approximation in a variant of Wasserstein distance that involves the characterizing differential operators for stable distributions, and an application to the generalized central limit theorem. Due to the lack of first moment for the approximating sequence in the latter result, we appeal to an additional truncation procedure and investigate fine regularity properties of the solution to Stein's equation

Distribution Estimation of Contaminated Data via DNN-based MoM-GANs

This paper studies the distribution estimation of contaminated data by the MoM-GAN method, which combines generative adversarial net (GAN) and median-of-mean (MoM) estimation. We use a deep neural network (DNN) with a ReLU activation function to model the generator and discriminator of the GAN. Theoretically, we derive a non-asymptotic error bound for the DNN-based MoM-GAN estimator measured by integral probability metrics with the $b$-smoothness H\"{o}lder class. The error bound decreases essentially as $n^{-b/p}\vee n^{-1/2}$, where $n$ and $p$ are the sample size and the dimension of input data. We give an algorithm for the MoM-GAN method and implement it through two real applications. The numerical results show that the MoM-GAN outperforms other competitive methods when dealing with contaminated data

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]

CiteSpace Visual Analysis of Community Pharmacy Services in China: Bridging Local Insights and Global Trends

Purpose: This research traces the history of pharmacy services research in Chinese pharmacies over the past twenty years and attempts to identify key trends and hotspots consistent with global practices. Methodology: From January 2002 to December 2022, we leveraged databases such as CNKI, VIP, and Wanfang and deployed CiteSpace 5.8.R3 to visualize publication trends, authorship, and institutional contributions within the field. Results: Our synthesis highlights critical developments in drug safety monitoring, prescription management, pharmacoeconomics, and public health education and highlights the leadership of respected researchers and institutions. Conclusion: The “Internet+” paradigm is identified as an important catalyst for pharmacy services innovation, with implications that extend beyond China and suggest models for international adaptation and strategy development

The dynamics of the stochastic shadow Gierer-Meinhardt system

We consider the dynamics of the stochastic shadow Gierer-Meinhardt system with one-dimensional standard Brownian motion. We establish the global existence and uniqueness of solutions. We also prove a large deviation result

Exponential Ergodicity of stochastic Burgers equations driven by α\alpha-stable processes

In this work, we prove the strong Feller property and the exponential ergodicity of stochastic Burgers equations driven by $\alpha/2$-subordinated cylindrical Brownian motions with $\alpha\in(1,2)$. To prove the results, we truncate the nonlinearity and use the derivative formula for SDEs driven by $\alpha$-stable noises established in Zhang (arXiv:1204.2630v2).Comment: 17p

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}