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

  • A. Pilipenko Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv,and Nat. Techn. Univ. Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”
  • A. Kulik Wroclaw Univ. Sci. and Technology, Poland

Abstract

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.

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Published
22.09.2020
How to Cite
PilipenkoA., and KulikA. “On Regularization by a Small Noise of Multidimensional ODEs With Non-Lipschitz Coefficients”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, no. 9, Sept. 2020, pp. 1254-85, doi:10.37863/umzh.v72i9.6292.
Section
Research articles