Conditional convergence of the diagonal-diffusion scheme under comparison

Apply [A1] to the upper subsolution and lower supersolution supplied by [F1]. Their common initial trace gives the required ordering, hence $\bar{U}\le \underset{‾}{U}$. Together with the reverse inequality in [F1], this proves equality.

The common function is both upper and lower semicontinuous by [F1], hence continuous. It is both a viscosity subsolution and supersolution, so is a viscosity solution with the prescribed initial data. Comparison in [A1] implies uniqueness.

Fix $T^{\mathrm{\prime }}<T$ and let $K=[0,1{]}^d\times [0,T^{\mathrm{\prime }}]$ in the periodic lift; this box is compact by Heine--Borel. Write $u$ for the common limit from step 2.1 and fix $\epsilon >0$. For $z\in K$ and rational $q>0$, include $B(z,q\mathrm{/}2)$ in a family when for every integer $k$ with $1\mathrm{/}k<q$ and every $y$ with $\mathrm{\mid }y-z\mathrm{\mid }<q$, $\mathrm{\mid }{V}_k(y)-u(z)\mathrm{\mid }<\epsilon \mathrm{/}3$, and continuity gives $\mathrm{\mid }u(y)-u(z)\mathrm{\mid }<\epsilon \mathrm{/}3$. The set definitions in [F2] guarantee that these balls cover $K$: each point has a positive real radius, and a smaller rational radius works. The family consists of all qualifying pairs, so no radius is selected point by point. By [F2], a finite subcover exists; take the least of its finitely many rational radii. The triangle inequality then gives $\mathrm{\mid }{V}_k(y)-u(y)\mathrm{\mid }<2\epsilon \mathrm{/}3<\epsilon$ uniformly on $K$ for all sufficiently large $k$. This proves uniform convergence on the fundamental box through $T^{\mathrm{\prime }}$, hence on the torus and every compact subset of ${\mathbb{T}}^d\times [0,T)$. Thus the extensions converge locally uniformly.