Hausdorff Dimension of Caloric Measure

abstract: We examine caloric measures on general domains in ^n+1=ⁿ (space time) from the perspective of geometric measure theory. On one hand, we give a direct proof of a consequence of a theorem of Taylor and Watson (1985) that the lower parabolic Hausdorff dimension of is at least n and ⁿ. On the other hand, we prove that the upper parabolic Hausdorff dimension of is at most n+2-ₙ, where ₙ>0 depends only on n. Analogous bounds for harmonic measures were first shown by Nevanlinna (1934) and Bourgain (1987). Heuristically, we show that the density of obstacles in a cube needed to make it unlikely that a Brownian motion started outside of the cube exits a domain near the center of the cube must be chosen according to the ambient dimension. In the course of the proof, we give a caloric measure analogue of Bourgain's alternative: for any constants 0<ₙ <1/2 and closed set E^n+1, either (i) E Q has relatively large caloric measure in Q E for every pole in F or (ii) E Q_* has relatively small -dimensional parabolic Hausdorff content for every n< n+2, where Q is a cube, F is a subcube of Q aligned at the center of the top time-face, and Q_* is a subcube of Q that is close to, but separated backwards-in-time from F: gather* Q= (-1/2, 1/2) ⁿ (-1, 0), F=-1/2+, 1/2-ⁿ-², 0), \\[2pt and Q_*=-1/2+, 1/2-ⁿ-3², -2². gather* Further, we supply a version of the strong Markov property for caloric measures.