Sparse Beltrami coefficients, integral means of conformal mappings and   the Feynman-Kac formula
  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. 
Oleg Ivrii
math.CV math.PR