Homogenization of iterated singular integrals with applications to random quasiconformal maps

Abstract

We study homogenization of iterated randomized singular integrals and homeomorphic solutions to the Beltrami differential equation with a random Beltrami coefficient. More precisely, let $(F_j{)}_{j\ge 1}$ be a sequence of normalized homeomorphic solutions to the planar Beltrami equation $\overline{\partial }{F}_j(z)={\mu }_j(z,\omega )\partial F_j(z),$ where the random dilatation satisfies $|{\mu }_j|\le k<1$ and has locally periodic statistics, for example of the type $${\mu }_j(z,\omega )=\varphi (z)\sum _{n\in {\mathbf{Z}}^2}g(2^{j}z-n,X_n(\omega )),$$ where $g(z,\omega )$ decays rapidly in $z$, the random variables $X_n$ are i.i.d., and $\varphi \in C_0^{\infty }$. We establish the almost sure and local uniform convergence as $j\to \infty$ of the maps $F_j$ to a deterministic quasiconformal limit $F_{\infty }$. This result is obtained as an application of our main theorem, which deals with homogenization of iterated randomized singular integrals. As a special case of our theorem, let $T_1,\dots ,T_m$ be translation and dilation invariant singular integrals on ${\mathbf{R}}^d,$ and consider a $d$-dimensional version of ${\mu }_j$, e.g., as defined above or within a more general setting. We then prove that there is a deterministic function $f$ such that almost surely as $j\to \infty$, $${\mu }_j{T}_m{\mu }_j\dots T_1{\mu }_j\to f\text{weakly in }{L}^p,1<p<\infty .$$

56 pages