@article{Pilipenko_Kulik_2020, title={On regularization by a small noise of multidimensional ODEs with non-Lipschitz coefficients}, volume={72}, url={https://umj.imath.kiev.ua/index.php/umj/article/view/6292}, DOI={10.37863/umzh.v72i9.6292}, abstractNote={<p><span style="font-weight: 400;">UDC 519.21</span></p> <p>In this paper we solve a selection problem for multidimensional SDE <br>$d X^{\epsilon}(t)=a(X^{\epsilon}(t))\, d t + \epsilon\sigma(X^{\epsilon}(t))\, d W(t),$ <br>where the drift and diffusion are locally Lipschitz continuous outside of a fixed hyperplane $H.$<br>It is assumed that $X^{\epsilon}(0)=x^0\in H,$ 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.<br>We show that if the drift pushes the solution away from $H,$ then the limit process with certain probabilities selects some extremal solutions to the limit ODE. <br>If the drift attracts the solution to $H,$ then the limit process satisfies an ODE with some averaged coefficients. <br>To prove the last result we formulate an averaging principle, which is quite general and new.</p>}, number={9}, journal={Ukrainsâ€™kyi Matematychnyi Zhurnal}, author={Pilipenko, A. and Kulik, A.}, year={2020}, month={Sep.}, pages={1254-1285} }