Annals of Probability

Lenses in skew Brownian flow

Krzysztof Burdzy and Haya Kaspi

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We consider a stochastic flow in which individual particles follow skew Brownian motions, with each one of these processes driven by the same Brownian motion. One does not have uniqueness for the solutions of the corresponding stochastic differential equation simultaneously for all real initial conditions. Due to this lack of the simultaneous strong uniqueness for the whole system of stochastic differential equations, the flow contains lenses, that is, pairs of skew Brownian motions which start at the same point, bifurcate, and then coalesce in a finite time. The paper contains qualitative and quantitative (distributional) results on the geometry of the flow and lenses.

Article information

Ann. Probab., Volume 32, Number 4 (2004), 3085-3115.

First available in Project Euclid: 8 February 2005

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Zentralblatt MATH identifier

Primary: 60J65: Brownian motion [See also 58J65]
Secondary: 60J55: Local time and additive functionals 60G17: Sample path properties 60H10: Stochastic ordinary differential equations [See also 34F05]

Skew Brownian motion stochastic flow


Burdzy, Krzysztof; Kaspi, Haya. Lenses in skew Brownian flow. Ann. Probab. 32 (2004), no. 4, 3085--3115. doi:10.1214/009117904000000711.

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