The Martin Boundary in the Parabolic Context

Abstract

In discussing the parabolic context Martin boundary of an open subset $\dot{D}$ of ${{\dot{\mathbb{R}}}^{N}}$ for $N1$ we first make the obvious remark that there are necessarily two boundaries, one adapted to the operator $\dot{\Delta }$ and superparabolic potentials, the other adapted to the operator $\mathop{\Delta }\limits^{*}$ and cosuperparabolic potentials. The first is called the exit boundary; the second is called the entrance boundary. These dual contexts are interchanged by a reflection of ${{\dot{\mathbb{R}}}^{N}}$ in the abscissa hyperplane. We shall treat the exit boundary but shall omit the word “exit” unless both boundaries are involved. The following remarks are offered to orient the reader to the new features that arise in parabolic context Martin boundary theory.