Zero Noise Selections of Multidimensional Peano Phenomena

Liangquan Zhang

Published 2012-02-19, updated 2024-03-28Version 6

For the ordinary differential equation (ODE in short) \begin{equation*} \left\{ \begin{array}{crl} \xi ^{\prime }\left( t\right) & = & b\left( \xi \left( t\right) \right) ,% \text{\qquad }t\geq 0, \\ \xi \left( 0\right) & = & x\in \mathbb{R}^{d},% \end{array}% \right. \end{equation*}% where $b:\mathbb{R}^{d}\rightarrow \mathbb{R}^{d},$ there is a general local existence theory if $b$ is only supposed to be continuous (Peano's theorem), even though uniqueness may fail in this case. However, the perturbed stochastic differential equation (SDE in short) \begin{equation*} \left\{ \begin{array}{crl} \mbox{\rm d}X^{x,\varepsilon }\left( t\right) & = & b\left( X^{x,\varepsilon }\left( t\right) \right) \mbox{\rm d}t+\varepsilon \mbox{\rm d}W\left( t\right) ,\qquad t\geq 0, \\ \text{ }X^{x,\varepsilon }\left( 0\right) & = & x\in \mathbb{R}^{d},% \end{array}% \right. \end{equation*}% where $W$ is a $d$-dimensional standard Brownian motion, has a unique strong solution when $b$ is assumed to be continuous and bounded. Moreover, when $% \varepsilon \rightarrow 0^{+},$ the solutions to the perturbed SDEs converge, in a suitable sense, to the solutions of the ODE. This phenomenon has been extensively studied for one-dimensional case in literature. The goal of present paper is to analyze some multi-dimensional cases. When $b$ has an isolated zero and is non Lipschitz continuous at zero, the ODE may have infinitely many solutions. Our main result shows which solutions of the ODE can be the limits of the solutions of the SDEs (as $\varepsilon \rightarrow 0^{+}$) by stopping time technique which can be thought of as responses to the questions imposed by Bafico and Baldi (\emph{Stochastics}, \cite{BB}, 1982, page 292). The main novelty consists in the treatment of multi-dimensional case in a simple manner.