Sturm comparison for nonlocal equations

In 1836 Sturm published two papers in Liouville’s J. Math. Pures et Appl., giving his famous comparison and oscillation theorems. In its simplest form, Sturm’s comparison result is Theorem Let q1(x)≤q2(x)be not identically equal on [a, b]. Let ube a nontrivial solution to u′′+q1(x)u=0on [a, b], and let vbe a nontrivial solution tov′′+q2(x)v=0on [a, b]. Suppose that u(a)=u(b)=0.Then vhas at least one zero in (a, b). One may ask if there is a discrete version of this. There is, where the second derivative operator is replaced by the usual finite difference approximation, as has been observed by many people. Furthermore, it continues to hold for more general tri-diagonal operators. The second derivative operator may also be approximated by a discretization which includes next nearest neighbor interactions. Does the above theorem hold for such penta-diagonal operators? Not in general, as we show by example. In another direction, being the main point of this paper, one can replace the second derivative operator by an integral operator, bounded or unbounded, resulting in a nonlocal diffusion-like operator. Can one prove a similar comparison theorem? Here we discuss the case where u′′ is replaced by (Formula presented.) where J≥0 is continuous, even, with compact support and J(0)>0. After this, we discuss the case when the nonlocal diffusion operator is replaced by a scaled version (Formula presented.) for ϵ>0 and small, and where Ω⊂Rn is smoothly bounded, and J is radially symmetric. This operator converges, in some sense, to a multiple of the Laplacian, as ϵ→0, and so the question arises as to whether or not a Sturm-like comparison theorem holds with this nonlocal diffusion operator.