Continuum limit of nonlocal diffusion on inhomogeneous networks (Q6612559)

The paper deals with the following nonlocal reaction-diffusion equation on inhomogeneous networks \N\[\N \frac{d}{dt}u(t,x)=\int_0^1J(x,y)(u(t,y)-u(t,x))dy+Au(t,x)+B,\ \ A<0, \N\]\Nwith conditions \({u(t,0)=u(t,1)}\) and \({0\leq u(0,x)\in C([0,1])}.\) \N\NThe kernel \({J:[0,1]\times[0, 1]\to\mathbb{R}}\) is a nonnegative, symmetric, positive-definite, and continuous function interpreted as the distribution of jumping from location \(x\) to \(y\).\N\NThe author considers some adequate discretizations of \(J\) and constructs a stochastic model using a reproducing kernel Hilbert space to consider the inhomogeneities of the network structure.\N\NAdopting the methods of [\textit{D. Blount}, Ann. Probab. 19, No. 4, 1440--1462 (1991; Zbl 0741.92022)], the law of large numbers and the central limit theorem are proved for the stochastic model.