Heat-content and diffusive leakage from material sets in the low-diffusivity limit^*

The imposed boundary condition type in the definition of the semigroup $\mathrm{exp}(\varepsilon t\overline{{\Delta}})$ corresponds to the one (homogeneous Dirichlet/Neumann) imposed in equation (4), see also [20].

To see this, we first observe that the Markov property of SDEs [1, theorem 9.2.3] yields a time-1 transition function p_ɛ satisfying ${p}_{\varepsilon }(x,A)={E}_{x,0}[{\mathbb{1}}_{A}({X}_{1}^{\varepsilon })]$ for $x\in {\mathbb{R}}^{n}$ and measurable $A\subset {\mathbb{R}}^{n}$. The definition of the inner product ${\left\langle \cdot ,\cdot \right\rangle }_{0}$ yields

$$\begin{equation*}{\left\langle x{\mapsto}{E}_{0,x}[{\mathbb{1}}_{A}\left.({X}_{1}^{\varepsilon })\right)],h\right\rangle }_{0}={\int }_{{\mathbb{R}}^{n}}{p}_{\varepsilon }(\cdot ,A)h(\cdot )\omega ={E}_{h}\left[{\mathbb{1}}_{A}({X}_{1}^{\varepsilon })\right].\end{equation*}$$

Observe that t ↦ ψ(t, ɛ) is monotone (and hence measurable) and finite (cf [8, chapter 5, corollary 1.2]). By the monotone convergence theorem, if t_n → t from below, then ψ(t_n , ɛ) → ψ(t, ɛ). In particular, the almost-everywhere (in t) bound from the integral form of Grönwall's lemma (see [6, appendix B.2]) holds everywhere.