Harmonic functions on the lattice: absolute monotonicity and propagation of smallness (Q902251)

The authors study harmonic functions on the lattice. Let \(u:B_R(0)\to \mathbb R\) be harmonic and define \(Q_u(n) = Eu^2(X_n)\), where \(X_n\) is a simple random walk on the lattice \(\mathbb Z^d\). Then \(Q_u\) is absolutely monotonic on \([0,R]\) in the sense that the iterated differences are all positive. There is a continuous analogue of this result, too. Further results concern convexity properties of discrete harmonic functions as well as smallness results for harmonic functions on the lattice.