Homogenization of iterated singular integrals with applications to random quasiconformal maps

We study homogenization of iterated randomized singular integrals and homeomorphic solutions to the Beltrami differential equation with a random coefficient. More precisely, let $(F\_j){j \geq 1}$ be sequence normalized planar $\partial{\overline z} F\_j (z)=\mu\_j(z,\omega) \partial\_{z} F\_j(z)$, where dilatation satisfies $|\mu\_j|\leq k<1$ has locally periodic statistics, for example type $$ \mu\_j (z,\omega)=\phi(z) \sum\_{n\in\mathbb{Z}^2} g(2^j z-n,X\_{n}(\omega)), $g(z,\omega)$ decays rapidly in $z$, variables $X\_{n}$ are i.i.d., $\phi \in C^\infty\_0$. establish almost sure local uniform convergence as $j \to \infty$ maps $F\_j$ deterministic quasiconformal limit $F\_\infty$. This result is obtained an application our main theorem, which deals integrals. As special case $T\_1,\ldots , T\_{m}$ translation dilation invariant on $\mathbb{R}^d$, consider $d$-dimensional version $\mu\_j$, e.g., defined above or within more general setting, see Definition 3.4 below. then prove that there function $f$ such surely,$\mu\_j T\_{m}\mu\_j\ldots T\_1\mu\_j\to f$ $j\to\infty$ weakly $L^p$, $1 < p \infty$.