Discrete initial value problems and discrete parabolic potential theory (Q810680)

The authors introduce an infinite network \(N=\{X,Y,K,r\}\), where X is a countably infinite set of nodes, Y is a countably infinite set of arcs, K is the node-arc incidence function and r is a strictly positive real function on Y. Let T be the set of all integers. The discrete parabolic operator \(\Pi u(\xi)=\Delta u(\xi)-\partial u(\xi)\), \(\xi =(x,t)\in X\times T\) is studied. The initial value problem is considered for \(\Pi\) on \(\{N,T_ s\}\), where \(T_ s=\{t\in T\), \(t\geq s\}\). In particular, the existence and uniqueness of the parabolic Green function \(G_{\alpha}\) of \(\{\) N,T\(\}\) with a pole at \(\alpha\in X\times T\) is proved. The following aspects of the potential theory for \(\Pi\) are discussed: (a) a discrete analogue of the Riesz decomposition theorem for nonnegative superparabolic functions, (b) the duality of parabolic and coparabolic potentials.