Further study of a simple {PDE}

Abstract

This essentially self-contained paper continues the study of a simple PDE in which an unorthodox sign in the spacial boundary condition destroys the usual Minimum Principle. The long-term behavior of solutions with time-parameter set $(0,\infty)$ is established, and this clarifies in analytic terms the characterization of non-negative solutions which had been obtained previously by probabilistic methods. The paper then studies by direct methods bounded 'ancient' solutions in which the time-parameter set is $(-\infty,0)$. In the final section, Martin-boundary theory is used to describe all non-negative ancient solutions in the most interesting case. The relevant Green kernel density behaves rather strangely, exhibiting two types of behavior in relation to scaling of its arguments. The Martin kernel density, a ratio of Green kernel densities, behaves more sensibly. Doob $h$-transforms illuminate the structure. As a somewhat surprising consequence of our Martin-boundary analysis, we find that non-negative solutions to our parabolic-looking equation satisfy an elliptic-type Harnack principle.