Side Boundary potentials for a Kolmogorov-type PDE

Abstract

We solve a Kolmogorov-type hypoelliptic parabolic partial differential equation with a "side" boundary condition (in the direction of the weak H\"ormander condition). We construct an approximate boundary potential which captures the effect of the boundary condition. Integrals against this approximate boundary potential have a novel discontinuity at the boundary. We introduce some polynomial corrections to this approximate boundary potential and then construct a boundary-domain Volterra equation to solve the original partial differential equation. This Volterra integral equation is iteratively solved, and the bounds contain a periodic behavior resulting from the boundary effects.