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2001 A 2-Dimensional SDE Whose Solutions are Not Unique
Jan Swart
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Electron. Commun. Probab. 6: 67-71 (2001). DOI: 10.1214/ECP.v6-1035

Abstract

In 1971, Yamada and Watanabe showed that pathwise uniqueness holds for the SDE $dX= \sigma (X)dB$ when sigma takes values in the n-by-m matrices and satisfies $|\sigma (x)- \sigma (y)| < |x-y|\log(1/|x-y|)^{1/2}$. When $n=m=2$ and $\sigma$ is of the form $\sigma _{ij}(x)= \delta_{ij}s(x)$, they showed that this condition can be relaxed to $| \sigma(x)-\sigma(y)| < |x-y|\log(1/|x-y|)$, leaving open the question whether this is true for general $ 2\times m$ matrices. We construct a $2\times 1$ matrix-valued function which negatively answers this question. The construction demonstrates an unexpected effect, namely, that fluctuations in the radial direction may stabilize a particle in the origin.

Citation

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Jan Swart. "A 2-Dimensional SDE Whose Solutions are Not Unique." Electron. Commun. Probab. 6 67 - 71, 2001. https://doi.org/10.1214/ECP.v6-1035

Information

Accepted: 12 July 2001; Published: 2001
First available in Project Euclid: 19 April 2016

zbMATH: 0988.60059
MathSciNet: MR1846542
Digital Object Identifier: 10.1214/ECP.v6-1035

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