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 $\overline{\partial} F_j (z)=\mu_j(z,\omega) \partial F_j(z),$ where the random dilatation satisfies $|\mu_j|\leq k<1$ and has locally periodic statistics, for example of the type $$\mu_j (z,\omega)=\phi(z)\sum_{n\in \mathbf{Z}^2}g(2^j z-n,X_{n}(\omega)), $$ where $g(z,\omega)$ decays rapidly in $z$, the random variables $X_{n}$ are i.i.d., and $\phi\in C^\infty_0$. 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 ${\bf 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 $j\to\infty$, $$ \mu_j T_{m}\mu_j\ldots T_1\mu_j\to f \quad \textrm{weakly in } L^p,\quad 1 < p < \infty\ . $$