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 $(F\_j){j \geq 1}$ be a sequence of normalized homeomorphic solutions to the planar Beltrami equation $\partial{\overline z} F\_j (z)=\mu\_j(z,\omega) \partial\_{z} F\_j(z)$, where the random dilatation satisfies $|\mu\_j|\leq k<1$ and has locally periodic statistics, for example of the type $$ \mu\_j (z,\omega)=\phi(z) \sum\_{n\in\mathbb{Z}^2} g(2^j z-n,X\_{n}(\omega)), $$ where $g(z,\omega)$ decays rapidly in $z$, the random variables $X\_{n}$ are i.i.d., and $\phi \in C^\infty\_0$. 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 $T\_1,\ldots , T\_{m}$ be translation and dilation invariant singular integrals on $\mathbb{R}^d$, and consider a $d$-dimensional version of $\mu\_j$, e.g., as defined above or within a more general setting, see Definition 3.4 below. We then prove that there is a deterministic function $f$ such that almost surely,$\mu\_j T\_{m}\mu\_j\ldots T\_1\mu\_j\to f$ as $j\to\infty$ weakly in $L^p$, for $1 < p < \infty$.