The Annals of Probability

Lenses in skew Brownian flow

Krzysztof Burdzy and Haya Kaspi

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Abstract

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

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

Dates
First available in Project Euclid: 8 February 2005

Permanent link to this document
https://projecteuclid.org/euclid.aop/1107883347

Digital Object Identifier
doi:10.1214/009117904000000711

Mathematical Reviews number (MathSciNet)
MR2094439

Zentralblatt MATH identifier
1071.60073

Subjects
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]

Keywords
Skew Brownian motion stochastic flow

Citation

Burdzy, Krzysztof; Kaspi, Haya. Lenses in skew Brownian flow. Ann. Probab. 32 (2004), no. 4, 3085--3115. doi:10.1214/009117904000000711. https://projecteuclid.org/euclid.aop/1107883347


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