Formula omitted]-conformal maps on the triangular lattice

Abstract

In Akahori and Ida (2014), p-conformal (or Parisian-conformal) maps on the triangular lattice were defined. The definition of p-conformality is nonstandard in comparison to ordinary discrete derivatives, but was seen to be natural in connection with a particular type of random walk, the Parisian random walk. In this note, we establish the fact that the only p-conformal polynomials in z and z̄ on the triangle lattice are linear combinations of 1, z and z2−z̄, but that if one extends the notion of p-conformality to functions of two variables (the complex variable z and the time variable t) we obtain a rich class of polynomials which yield martingales when applied to the Parisian walk. These polynomials make use of a particular type of martingale transform, which is defined in the paper.