Homogenization of iterated singular integrals with applications to random quasiconformal maps (Q2692510)

Summary: 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 \(\partial_{\bar{z}} F_j (z) = \mu_j (z, \omega) \partial_z F_j (z)\), where the random dilatation satisfies \(|\mu_j| \leq k < 1\) and has locally periodic statistics, for example of the type \[ \mu_j (z, \omega) = \phi (z) \sum_{n \in \mathbb{Z}^2} g (2^j z - n, X_n (\omega)), \tag{0.1} \] where \(g (z, \omega)\) decays rapidly in \(z\), the random variables \(X_n\) are i.i.d., and \(\phi \in C_0^{\infty}\). 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 \(\mathbb{R}^d\), 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, \(\mu_j T_m \mu_j \dots T_1 \mu_j \to f\) as \(j \to \infty\) weakly in \(L^p\), for \(1 < p < \infty\).