Harnack inequalities in infinite dimensions

We consider the Harnack inequality for harmonic functions with respect to three types of infinite-dimensional operators. For the infinite-dimensional Laplacian, we show no Harnack inequality is possible. We also show that the Harnack inequality fails for a large class of Ornstein–Uhlenbeck processes, although functions that are harmonic with respect to these processes do satisfy an a priori modulus of continuity. Many of these processes also have a coupling property. The third type of operator considered is the infinite-dimensional analog of operators in Hörmanderʼs form. In this case a Harnack inequality does hold.