论文标题
通过少量多维ODE和非lipschitz系数正规化
On regularization by a small noise of multidimensional ODEs with non-Lipschitz coefficients
论文作者
论文摘要
在本文中，我们解决了一个 多维SDE的选择问题 $ d x^\ varepsilon（t）= a（x^\ varepsilon（t））d t+\ varepsilonσ（x^\ varepsilon（t））\，d w（t）$，漂移和扩散是本地的lipschitz continally Lipschitz的连续固定型超级Pleplane $ h $。 假定$ x^\ varepsilon（0）= x^0 \在h $中，漂移$ a（x）$具有hoelder渐近肌，$ x $接近$ h $，而极限$ d x（t）= a（x（x（t））\，d t $没有独特的解决方案。 我们表明，如果漂移将解决方案推开$ h $，则具有某些概率的限制过程将选择一些极限的解决方案。如果漂移将解决方案吸引到$ h $，则极限过程可以通过一些平均系数满足颂歌。为了证明最后的结果，我们制定了一个平均原理，这是一般和新的。
In this paper we solve a selection problem for multidimensional SDE $d X^\varepsilon(t)=a(X^\varepsilon(t)) d t+\varepsilon σ(X^\varepsilon(t))\, d W(t)$, where the drift and diffusion are locally Lipschitz continuous outside of a fixed hyperplane $H$. It is assumed that $X^\varepsilon(0)=x^0\in H$, the drift $a(x)$ has a Hoelder asymptotics as $x$ approaches $H$, and the limit ODE $d X(t)=a(X(t))\, d t$ does not have a unique solution. We show that if the drift pushes the solution away of $H$, then the limit process with certain probabilities selects some extreme solutions to the limit ODE. If the drift attracts the solution to $H$, then the limit process satisfies an ODE with some averaged coefficients. To prove the last result we formulate an averaging principle, which is quite general and new.
