In this talk, I will present an estimate for the dimension of the image of the unit circle under a quasiconformal mapping whose dilatation is supported on a sparse set, e.g. a union of disjoint horoballs which are at least R apart in the hyperbolic metric. To motivate the proof, I will discuss an analogous estimate for the growth of solutions of second-order parabolic equations given by the Feynman-Kac formula. Time permitting, I will give further examples illustrating the dictionary between the two settings.