Wiener's criterion at ∞ for the heat equation and its measure-theoretical counterpart

This paper introduces a notion of regularity (or irregularity) of the point at infinity (infinity) for an unbounded open set Omega subset of RN+1 with regard to the heat equation, according to whether the parabolic measure of 1 is zero or positive. A necessary and sufficient condition for the existence of a unique bounded solution to the parabolic Dirichlet problem in an arbitrary unbounded open subset of RN+1 is established. It is expressed in terms of Wiener's criterion for the regularity of infinity.