AI 中文总结
研究基于强李雅普诺夫函数的驯服数值格式逼近连续系统解，推导解范数估计证其\(L^1\)收敛，还证明了具有负梯度条件系统的随机动力系统存在数值回溯吸引子。
AI 中文摘要
基于新引入的粗糙微分方程的“强李雅普诺夫函数”概念，我们研究一种驯服数值格式来逼近连续系统的解。推导了驯服系统解范数的显式估计，其与连续系统的类似。结果证明了驯服格式在\(L^1\)意义下的收敛性。对于具有负梯度条件的系统，证明了由驯服数值格式生成的随机动力系统存在数值回溯吸引子，且关于格式步长是可积且上半连续的。
英文摘要
Based on the newly introduced concept of {\it strong Lyapunov functions} for rough differential equations \cite{ducjost25}, we study a tamed numerical scheme to approximate the solutions of the continuous system. We derive explicit estimates of solution norms of the tamed system which look similar to those of the continuous system. As a result, we prove the convergence of the tamed scheme in the $L^1$ sense. For systems with the negative gradient condition, we prove the existence of a numerical pullback attractor for the generated random dynamical system from the tamed numerical scheme which is integrable and upper semi-continuous w.r.t. the scheme step size.
