We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
Let 
$(X_t)_{t \geqslant 0}$ be the solution of the stochastic differential equation where 
\[\mathrm{d} X_t = b(X_t) \,\mathrm{d} t+A\,\mathrm{d} Z_t, \quad X_{0}=x,\]
$b\,:\, \mathbb{R}^d \rightarrow \mathbb{R}^d$ is a Lipschitz-continuous function, 
$A \in \mathbb{R}^{d \times d}$ is a positive-definite matrix, 
$(Z_t)_{t\geqslant 0}$ is a d-dimensional rotationally symmetric 
$\alpha$-stable Lévy process with 
$\alpha \in (1,2)$ and 
$x\in\mathbb{R}^{d}$. We use two Euler–Maruyama schemes with decreasing step sizes 
$\Gamma = (\gamma_n)_{n\in \mathbb{N}}$ to approximate the invariant measure of 
$(X_t)_{t \geqslant 0}$: one uses independent and identically distributed 
$\alpha$-stable random variables as innovations, and the other employs independent and identically distributed Pareto random variables. We study the convergence rates of these two approximation schemes in the Wasserstein-1 distance. For the first scheme, under the assumption that the function b is Lipschitz and satisfies a certain dissipation condition, we demonstrate a convergence rate of 
$\gamma^{\frac{1}{\alpha}}_n$. This convergence rate can be improved to 
$\gamma^{1+\frac {1}{\alpha}-\frac{1}{\kappa}}_n$ for any 
$\kappa \in [1,\alpha)$, provided b has the additional regularity of bounded second-order directional derivatives. For the second scheme, where the function b is assumed to be twice continuously differentiable, we establish a convergence rate of 
$\gamma^{\frac{2-\alpha}{\alpha}}_n$; moreover, we show that this rate is optimal for the one-dimensional stable Ornstein–Uhlenbeck process. Our theorems indicate that the recent significant result of [34] concerning the unadjusted Langevin algorithm with additive innovations can be extended to stochastic differential equations driven by an 
$\alpha$-stable Lévy process and that the corresponding convergence rate exhibits similar behaviour. Compared with the result in [6], our assumptions have relaxed the second-order differentiability condition, requiring only a Lipschitz condition for the first scheme, which broadens the applicability of our approach.
Electro-osmotic flow (EOF) in nanochannels exhibits a puzzling non-monotonic dependence on salt concentration, which contrasts with observations in microchannels and remains not fully understood. In this work, we address this phenomenon through a theoretical investigation of EOF in 
$\mathrm{pH}$-regulated channels. New analytical approximations for electrostatic potential, EOF profile and electro-osmotic mobility beyond the Debye–Hückel limit are derived through asymptotic analysis. Our findings reveal that the surface electrostatic potential is independent of the channel size only when the half-channel size exceeds the Gouy–Chapman length. In contrast, surface ionization and net charge distribution play more crucial roles in EOF at the nanoscale, as they govern both the magnitude and the spatial distribution of the Coulomb driving force. As salt concentration increases, EOF velocity initially rises due to enhanced surface ionization, followed by a decline attributed to increased wall shear stress. This work provides key insights for EOF applications in nanofluidics and biomedical devices, and deepens the understanding of electrokinetic phenomena influenced by 
$\mathrm{pH}$-regulation effects.
The propagation of multiple ultraintense femtosecond lasers in underdense plasmas is investigated theoretically and numerically. We find that the energy merging effect between two in-phase seed lasers can be improved by using two obliquely incident guiding lasers whose initial phase is 
$\pi$ and 
$\pi /2$ ahead of the seed laser. Particle-in-cell simulations show that due to the repulsion and energy transfer of the guiding laser, the peak intensity of the merged light is amplified by more than five times compared to the seed laser. The energy conversion efficiency from all incident lasers to the merged light is up to approximately 60
$\%$. The results are useful for many applications, including plasma-based optical amplification, charged particle acceleration and extremely intense magnetic field generation.
