<p>We solve a Kolmogorov-type parabolic partial differential equation with a “side” boundary condition (in the direction of the weak Hörmander condition). We construct an approximate boundary potential which captures the effect of the boundary condition. Integrals against this approximate boundary potential have a novel jump discontinuity at the boundary which includes a measure discovered by McKean. 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. We discuss some applications to a problem of McKean.</p>