On the geometric and analytic properties of some random fractals

Thesis

The heat content of a domain D of ℝ^d is defined as

E(s) = ∫_D u(s,x)dx,

where u is the solution to the heat equation with zero initial condition and unit Dirichlet boundary condition. This thesis studies the behaviour of E(s) for small s with a particular emphasis on the case where $D$ is a planar domain whose boundary is a random Koch curve.

When ∂D is spatially homogeneous, we show that we can recover the upper and lower Minkowski dimensions of ∂D from E(s). Furthermore, in some cases where the Minkowski dimension does exist, finer fluctuations can be recovered and the heat content is controlled by s^α exp{f (log(1/s)} for small s, for some positive α and some regularly varying function f.

When ∂D is statistically self-similar, the heat content asymptotics are studied using a law of large numbers for the general branching process, and we show that the Minkowski dimension and content of ∂D exist and can be recovered from E(s). More precisely the heat content has an almost sure expansion

E(s) = c₁ s^α N_∞ + o(s^α), a.s.

for small s, for some positive c₁ and α and a positive random variable N_∞ with unit expectation.

To study the fluctuations around these asymptotics, we prove a central limit theorem for the general branching process. The proof follows a standard Taylor expansion argument and relies on the independence built into the general branching process. The limiting distribution established here is reminiscent of those arising in central limit theorems for martingales.

When ∂D is a statistically self-similar Cantor subset of ℝ, we discuss examples where we have and fail to have a central limit theorem for the heat content. We conclude with an open question about the fluctuations of the heat content when ∂D is a statistically self-similar Koch curve.

Authors: Charmoy, PHA

• Publication date: 2014
• DOI: 10.5287/ora-xrzokg7yy
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: spectral analysis; fractals; probability
• Subjects: Probability theory and stochastic processes