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

2020/07/21 by Kulik, Alexei, Pilipenko, Andrey
#34F05 #60F17 #60H10 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2007.10911

Abstract

In this paper we solve a selection problem for multidimensional SDE d X^ε(t)=a(X^ε(t)) d t+ε σ(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)=x⁰∈ 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 of H, then the limit process with certain probabilities selects some extreme 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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