Side boundary conditions for a Kolmogorov Diffusion
We consider a Kolmogorov-type PDE corresponding to a particle under white noise force. We are interested in stopping the process at a fixed position, i.e., imposing Dirichlet conditions at a side boundary. We construct a simple Gaussian heat kernel inside the domain, and investigate a boundary-layer kernel, connected to some work by McKean. We show that this boundary layer heat kernel has a novel jump condition. We outline a polynomial expansion of for the heat kernels, and then construct a Volterra equation for solving the original problem. The novel jump leads to a periodic structure of the Volterra equation.