Electronic Communications in Probability

Projections of spherical Brownian motion

Aleksandar Mijatović, Veno Mramor, and Gerónimo Uribe Bravo

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We obtain a stochastic differential equation (SDE) satisfied by the first $n$ coordinates of a Brownian motion on the unit sphere in $\mathbb{R} ^{n+\ell }$. The SDE has non-Lipschitz coefficients but we are able to provide an analysis of existence and pathwise uniqueness and show that they always hold. The square of the radial component is a Wright-Fisher diffusion with mutation and it features in a skew-product decomposition of the projected spherical Brownian motion. A more general SDE on the unit ball in $\mathbb{R} ^{n+\ell }$ allows us to geometrically realize the Wright-Fisher diffusion with general non-negative parameters as the radial component of its solution.

Article information

Electron. Commun. Probab., Volume 23 (2018), paper no. 52, 12 pp.

Received: 1 June 2018
Accepted: 8 August 2018
First available in Project Euclid: 1 September 2018

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Zentralblatt MATH identifier

Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 58J65: Diffusion processes and stochastic analysis on manifolds [See also 35R60, 60H10, 60J60]

non-Lipschitz stochastic differential equation skew-product decomposition pathwise uniqueness Wright-Fisher diffusion

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Mijatović, Aleksandar; Mramor, Veno; Uribe Bravo, Gerónimo. Projections of spherical Brownian motion. Electron. Commun. Probab. 23 (2018), paper no. 52, 12 pp. doi:10.1214/18-ECP162. https://projecteuclid.org/euclid.ecp/1535767263

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