Characterisation of homogenisation for nonlocal diffusion by local topologies

We consider fractional variants of divergence form problems with highly oscillatory local coefficients. We characterise the convergence of these coefficients by means of classical $H$-convergence covering the local behaviour of the fractional divergence form problem and weak-$\ast$ convergence on the complement caused by the nonlocality of the differential operators. The results are further described in the light of nonlocal $H$-convergence as introduced in [Waurick, Calc Var PDEs, 57, 2018] and certain Schur topologies. Applications to symmetric coefficients and a homogenisation problem for a fractional heat type equation are provided.