On regularization by a small noise of multidimensional ODEs with non-Lipschitz coefficients

In this paper we solve a selection problem for multidimensional SDE $d X^\varepsilon(t)=a(X^\varepsilon(t)) d t+\varepsilon \sigma(X^\varepsilon(t))\, d W(t)$, 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.