Sparse Beltrami coefficients, integral means of conformal mappings and the Feynman-Kac formula

TL;DR

This paper establishes estimates for the image of the unit circle under certain quasiconformal maps and for solutions of parabolic equations with sparse potentials, linking these two areas through a novel dictionary.

Contribution

It introduces a new connection between quasiconformal mappings with sparse dilatation support and solutions of parabolic equations via the Feynman-Kac formula, providing new estimates.

Findings

Estimate for the dimension of the image of the circle under quasiconformal maps with small support

Estimate for the growth rate of solutions to parabolic equations with sparse potentials

Establishment of a dictionary linking the two mathematical settings

Abstract

In this note, we give an estimate for the dimension of the image of the unit circle under a quasiconformal mapping whose dilatation has small support. We also prove an analogous estimate for the rate of growth of a solution of a second-order parabolic equation given by the Feynman-Kac formula (with a sparsely supported potential) and introduce a dictionary between the two settings.