Convergence Rates for Random Polarizations

Abstract

It is shown by Burchard and Fortier that the expected $L^1$ distance between $f^{\ast }$ and $n$ random polarizations of an essentially bounded function $f$ with support in a ball of radius $L$ is bounded by $2dm(B_{2L})\Vert f{\Vert }_{\infty }{n}^{-1}$. This article expands on these results. We show that the expected $L^1$ distance is bounded by $c_n{n}^{-1}$ with ${\mathrm{limsup}}_{n\to \infty }{c}_n\le 2^{d+1}\Vert \nabla f{\Vert }_1$ for every $f\in W^{1,1}(B_L)\cap L^{\infty }(B_L)$. Furthermore, we establish that the expected $L^1$ distance is $O(n^{-1/q})$ for $f\in L^p(B_L)$ with $1/p+1/q=1$. The rate $n^{-1}$ is shown to be best possible; specifically, $n$ times the measure of the symmetric difference between the random polarizations of a ball and its Schwarz symmetrization converges in distribution to a random variable with explicitly derived moments. We also prove that the expected symmetric difference between the random polarizations of a measurable set and its Schwarz symmetrization is slower than $n^{-r}$ for any $r>2d$, and that if the rate is $n^{-1}$, the normalized symmetric difference converges in distribution. We introduce a new sequence of random polarizations where the transition probability depends on the state of the underlying Markov chain, yielding a convergence rate of $O\left(n^{-\left(2-\frac{1}{d}\right)}(\log n{)}^{1-\frac{1}{d}}\right)$ for $d>1$. Finally, we show that for every compact set $A\subset \mathbb{R}$ with finite perimeter, there exists a sequence of polarizations converging exponentially to its Schwarz symmetrization.

23 pages