发表机构
Delft Institute of Applied Mathematics TU Delft（代尔夫特应用数学研究所 代尔夫特理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明驯化指数欧拉格式对超线性双曲型随机偏微分方程具有1/2阶强收敛率，将结果从全局推广至局部Lipschitz非线性，并应用于多种非线性方程。
AI 中文摘要
本文证明了具有超线性增长非线性和乘性噪声的半线性双曲型随机发展方程的驯化指数欧拉格式的依路径一致收敛率可达$1/2$阶。我们采用“双曲型”这一术语，意指主算子生成一个压缩$C_0$-半群，但不发生抛物型光滑化。在非线性项满足局部Lipschitz、多项式增长、强制性和单调性条件下，我们在Hilbert空间$X$上对$p\in[2,\infty)$建立了如下形式的依路径一致强误差估计：\begin{equation*} \Big(\mathbb{E}\max_{0\le j \le N} \\|U(t_j)-U^j\\|_X^p\Big)^{1/p} \lesssim \sqrt{k} \end{equation*}，其中$U$是温和解，$U^j$是步长为$k>0$、在时刻$t_j=jk$处的驯化指数欧拉近似。这将非抛物型随机偏微分方程的先前收敛结果从全局Lipschitz非线性推广到局部Lipschitz非线性，允许漂移项和扩散项均呈多项式增长。在随机Kato框架下，我们进一步建立了温和解及其近似的局部和全局适定性以及一致先验估计。文中包含了非线性随机输运方程、Airy方程、波动型方程和耗散阻尼非线性薛定谔方程的应用，涵盖了不同的非线性项——截断和分数驯化格式。对于具有三次速度阻尼的Klein-Gordon方程，这补充了先前针对加性噪声获得的结果。
英文摘要
In this paper, we prove pathwise uniform convergence at rates up to $1/2$ for tamed exponential Euler schemes for semilinear hyperbolic stochastic evolution equations with superlinearly growing nonlinearities and multiplicative noise. We take the term hyperbolic to mean that the leading operator generates a contractive $C_0$-semigroup but no parabolic smoothing occurs. Under local Lipschitz, polynomial growth, coercivity, and monotonicity conditions on the nonlinearities, we establish pathwise uniform strong error estimates of the form \begin{equation*} \Big(\mathbb{E}\max_{0\le j \le N} \|U(t_j)-U^j\|_X^p\Big)^{1/p} \lesssim \sqrt{k} \end{equation*} on a Hilbert space $X$ for $p\in [2,\infty)$. Here, $U$ is the mild solution and $U^j$ is the tamed exponential Euler approximation at time $t_j=jk$ with step size $k>0$. This extends previous convergence results for non-parabolic SPDEs from globally to locally Lipschitz nonlinearities, allowing both drift and diffusion to grow polynomially. In a stochastic Kato framework, we further establish local and global well-posedness as well as uniform a priori estimates for the mild solution and its approximation. Applications to nonlinear stochastic transport, Airy, wave-type, and dissipatively damped nonlinear Schrödinger equations are included, covering different nonlinearities-stopped and fractionally tamed schemes. For the Klein-Gordon equation with cubic velocity damping, this complements previous results obtained for additive noise.
Comments50 pages. Comments are welcome!
