2021/08/27 by Matthew Badger, Badger, Matthew, Alyssa Genschaw +1 · 2 citations
Computer Science · Mathematics · #28A78 #35K05 #42B37 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Dynamics and Fractals #Meromorphic and Entire Functions #Primary 31B15 #Secondary 28A75
paper · pdf · doi:10.48550/arxiv.2108.12340
openalex publication_date 2021/08/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We examine caloric measures ω on general domains in ℝn+1 = ℝn×ℝ (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 ω≪ Hn. 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 < ε≪n δ< 1/2 and closed set E ⊂ ℝn+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: Q = (-1/2,1/2)n × (-1,0), F = [-1/2+δ,1/2-δ]n×[-ε2,0), and Q_* = [-1/2+δ,1/2-δ]n×[-3ε2,-2ε2]. Further, we supply a version of the strong Markov property for caloric measures.