论文标题
粗糙的路径理论近似随机动力学系统
Rough Path Theory to approximate Random Dynamical Systems
论文作者
论文摘要
我们认为粗糙的微分方程$ dy = f（y）d \ bm \ om $其中$ \ bm \ om =（ω，\ bbomega）$是由$ \ rr^m $ on Brownian Motion $ω$定义的粗糙路径。在$ f $的通常规律性假设下，即$ f \ in c^3_b（\ rr^d，\ rr^{d \ times m}）$，粗糙的微分方程具有定义随机动力学系统$ ϕ_0 $的独特解决方案。另一方面，我们还考虑了一个普通的随机微分方程$dy_δ= f（y_δ）dΩ_\ de $，其中$ω__\ de $是一个随机过程，具有固定的增量，并且连续可区分的路径近似于$ω$。后一个微分方程也会生成一个随机动力学系统$ ϕ_δ $。我们显示了随机动力学系统的收敛$ ϕ_δ $ to $δ\ to $δ\ to $δ\至0 $的收敛性。
We consider the rough differential equation $dY=f(Y)d\bm \om$ where $\bm \om=(ω,\bbomega)$ is a rough path defined by a Brownian motion $ω$ on $\RR^m$. Under the usual regularity assumption on $f$, namely $f\in C^3_b (\RR^d, \RR^{d\times m})$, the rough differential equation has a unique solution that defines a random dynamical system $ϕ_0$. On the other hand, we also consider an ordinary random differential equation $dY_δ=f(Y_δ)dω_\de$, where $ω_\de$ is a random process with stationary increments and continuously differentiable paths that approximates $ω$. The latter differential equation generates a random dynamical system $ϕ_δ$ as well. We show the convergence of the random dynamical system $ϕ_δ$ to $ϕ_0$ for $δ\to 0$ in Hölder norm.
