A Concentration Inequality for Harmonic Measures on the Sphere

Logarithmic Sobolev inequalities for Moebius measures on spheres

“…In fact, this probability is the image of μ on under the Moebius transformation. The factor ( − |x| ) n− ( − (x, y)) n− is known as the invariant Poisson kernel P(x, y): as a function of x, it is not harmonic but satisfies the equatioñ ∆P( ⋅ , y) = , where∆ denotes the invariant Laplacian (the reader is referred to [14] for further information on this measure).…”

“…In [14], G. Schechtman and M. Schmuckenschläger proved that both Moebius measure μ n x and harmonic measure with |x| < have a uniform Gaussian concentration.…”

Harmonic Measures on the Sphere via Curvature-Dimension

“…obtained by Schechtman-Schmuckenschläger [31]. However, as soon as an α bn for some 0 < a < b < 3, observe that Finally, we apply the Lichnerowicz spectral-gap estimate from our previous work with A. Kolesnikov [14] (proved for weighted manifolds with convex boundaries, and also independently obtained by S.-I.…”

“…In the case α = 1 we have c n,1 x = 1 − |x| 2 , and it is well known (e.g. [31]) that µ n,1 x is precisely the harmonic measure on S n , characterized as the hitting distribution of S n by standard Brownian motion started at x, or equivalently, as the measure whose density is the Poisson kernel for the Laplace equation in B n+1 = x ∈ R n+1 ; |x| < 1 with Dirichlet boundary conditions on S n = ∂B n+1 . Denoting by g the canonical Riemannian metric on S n , the triplet (S n , g, µ n,α x ) constitutes a weighted Riemannian manifold.…”

“…Clearly, given n, α and |x|, the density of such measures on y ∈ S n only depends on the angle θ(y) ∈ [0, π] given by cos(θ(y)) = y, x/|x| , and so various functional and concentration properties of these measures may be reduced to the study of the one-dimensional measure θ * (µ This is essentially the approach taken by G. Schechtman and M. Schmuckenschläger [31] and F. Barthe, Y. Ma and Z. Zhang [5] in their study of the harmonic case α = 1 (further details to be provided momentarily).…”