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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.
Highlights
Introduction and main resultsThe Theorem of Archimedes [ArcBC] states that the projection π1 from the unit sphere S2 ⊂ R3 to any coordinate preserves the uniform distribution; see [AM13] and the references therein for a very modern account or [PR12] for one with more probabilistic insight
If U (2) is a uniform random vector on S2, π1(U (2)) is uniform on [−1,1]. This holds in any dimension d ≥ 3: for a uniform random vector U (d−1) on the Euclidean unit sphere Sd−1 := z ∈ Rd ; |z| = 1, its projection πd−2(U (d−1)) onto any d − 2 coordinates is uniform on the unit ball Bd−2 := z ∈ Rd−2 ; |z| ≤ 1
Since the uniform distribution on Sd−1 is the invariant measure for Brownian motion on the sphere, it is natural to investigate the process obtained by projecting it to the ball Bd−2
Summary
The Theorem of Archimedes [ArcBC] states that the projection π1 from the unit sphere S2 ⊂ R3 to any coordinate (in R3) preserves the uniform distribution; see [AM13] and the references therein for a very modern account or [PR12] for one with more probabilistic insight. If U (2) is a uniform random vector on S2, π1(U (2)) is uniform on [−1,1] This holds in any dimension d ≥ 3: for a uniform random vector U (d−1) on the Euclidean unit sphere Sd−1 := z ∈ Rd ; |z| = 1 , its projection πd−2(U (d−1)) onto any d − 2 coordinates is uniform on the unit ball Bd−2 := z ∈ Rd−2 ; |z| ≤ 1. Since the uniform distribution on Sd−1 is the invariant measure for Brownian motion on the sphere, it is natural to investigate the process obtained by projecting it to the ball Bd−2. Such a process ought to have a uniform distribution on Bd−2 as its invariant measure. Since σ(x) is the unique non-negative definite square root of the matrix I − xx⊤ for x ∈ Bn, the SDE for the projected process X implies that its infinitesimal generator equals
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