Space-time boundaries for random walks obtained from diffuse measures

We define a “space-time” boundary (referring to space-time harmonic functions) to encompass random walks obtained from compactly supported diffuse measures on Euclidean space, and then prove that in many cases, a qualitative analogue of the Ney-Spitzer theorem (1966) holds, namely that the space-time boundary admits a natural identification with the convex hull of the support of the measure. This can also be interpreted as a generalization to the diffuse case of the weighted moment mapping of algebraic geometry. In many more cases, a weaker analogue holds, identifying the faithful extreme space-time harmonic functions with the interior of the convex body. Partially supported by an operating grant from NSERC (Canada).