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 th...