On regularization by a small noise of multidimensional ODEs with non-Lipschitz coefficients - Pilipenko, A.; Kulik, A. (Ukrains’kyi Matematychnyi Zhurnal, 2020)
UDC 519.21In this paper we solve a selection problem for multidimensional SDE $d X^{ε}(t)=a(X^{ε}(t))\, d t + εσ(X^{ε}(t))\, d W(t),$ where the drift and diffusion are locally Lipschitz continuous outside of a fixed hyperplane $H.$It is assumed that $X^{ε}(0)=x^0\in H,$ 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.