Convergence Rates for Random Polarizations

Abstract

It is shown by Burchard and Fortier that the expected $L^1$ distance between $f^*$ and $n$ random polarizations of an essentially bounded function $f$ with support in a ball of radius $L$ is bounded by $2dm(B_{2L})\|f\|_{\infty}n^{-1}$. This article expands on these results. We show that the expected $L^1$ distance is bounded by $c_n n^{-1}$ with $\limsup_{n\rightarrow \infty} c_n \leq 2^{d+1}\|\nabla f\|_1$ 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.