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

• Datum: 2020
• Hauptverfasser: Pilipenko, A., Kulik, A.
• Format: Artikel
• Sprache: Ukrainisch
• Veröffentlicht: Institute of Mathematics, NAS of Ukraine 2020
• Online Zugang: https://umj.imath.kiev.ua/index.php/umj/article/view/6292
• Назва журналу: Ukrains’kyi Matematychnyi Zhurnal

UDC 519.21 In 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.