Hausdorff Dimension of Caloric Measure

We examine caloric measures ω 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 ω is at least n and ωⁿ. On the other hand, we prove that the upper parabolic Hausdorff dimension of ω is at most n+2-βₙ, where βₙ>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<εₙ δ<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 ρ-dimensional parabolic Hausdorff content for every n<ρ≤ 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+δ,1/2-δ]^n×[-ε^2,0),\\[2pt] and Q_*=[-1/2+δ,1/2-δ]^n×[-3ε^2,-2ε^2]. {gather*} Further, we supply a version of the strong Markov property for caloric measures.