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A 2-Dimensional SDE Whose Solutions are Not Unique

   
@article{ECP1035,
	author = {Jan Swart},
	title = {A 2-Dimensional SDE Whose Solutions are Not Unique},
	journal = {Electron. Commun. Probab.},
	fjournal = {Electronic Communications in Probability},
	volume = {6},
	year = {2001},
	keywords = {Stochastic differential equation, pathwise uniqueness / strong uniqueness, diffusion process.},
	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.},
	pages = {no. 6, 67-71},
	issn = {1083-589X},
	doi = {10.1214/ECP.v6-1035},    
        url = {http://ecp.ejpecp.org/article/view/1035}}