## Electronic Communications in Probability

### Projections of spherical Brownian motion

#### Abstract

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

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

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

https://projecteuclid.org/euclid.ecp/1535767263

Digital Object Identifier
doi:10.1214/18-ECP162

Mathematical Reviews number (MathSciNet)
MR3852266

Zentralblatt MATH identifier
06964395

#### Citation

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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