TY - JOUR
AU - A. Pilipenko
AU - A. Kulik
PY - 2020/09/22
Y2 - 2020/10/30
TI - On regularization by a small noise of multidimensional ODEs with non-Lipschitz coefficients
JF - Ukrainsâ€™kyi Matematychnyi Zhurnal
JA - Ukr. Mat. Zhurn.
VL - 72
IS - 9
SE - Research articles
DO - 10.37863/umzh.v72i9.6292
UR - http://umj.imath.kiev.ua/index.php/umj/article/view/6292
AB - UDC 519.21In this paper we solve a selection problem for multidimensional SDE $d X^{\epsilon}(t)=a(X^{\epsilon}(t))\, d t + \epsilon\sigma(X^{\epsilon}(t))\, d W(t),$ where the drift and diffusion are locally Lipschitz continuous outside of a fixed hyperplane $H.$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.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. 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.
ER -