A Note on the formulation of the Neumann boundary condition for a nonlocal problem

The nonlocal diffusion equation with continuous kernel $K(x,y$, with ${\int }_{R}K(y,x)dy=1$ has been proposed as a model for some evolution process with diffusion, including population models. However, in general, we don’t have ${\int }_{\mathrm{\Omega }}K(y,x)dy=1$, as expected from its interpretation as a probability density. In this note, we propose a modification of the kernel, based on the idea of `reflection’ at the boundary, familiar in one dimensional problems. We show that a similar construction is possible in higher dimensions, with the new kernel satisfying the above integral equality and being also symmetric in some special cases.