2014/03/07 by Philippe H. A. Charmoy, Charmoy, Philippe H. A.
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Geometric Analysis and Curvature Flows #Mathematical Dynamics and Fractals #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.1403.1811
openalex publication_date 2014/03/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the short time asymptotics of the heat content E of a domain D of ℝd. The novelty of this paper is that we consider the situation where D is a domain whose boundary ∂ D is a random Koch type curve. When ∂ D is spatially homogeneous, we show that we can recover the lower and upper Minkowski dimensions of ∂ D from the short time behaviour of E(s). Furthermore, in some situations where the Minkowski dimension exists, finer geometric fluctuations can be recovered and the heat content is controlled by sαef(log(1/s)) for small s, for some α∈ (0, ∞) and some regularly varying function f. The function f is not constant is general and carries some geometric information. When ∂ D is statistically self-similar, then the Minkowski dimension and content of ∂ D typically exist and can be recovered from E(s). Furthermore, the heat content has an almost sure expansion E(s) = c sα N_∞ + o(sα) for small s, for some c and α∈ (0, ∞) and some positive random variable N_∞ with unit expectation arising as the limit of some martingale.