On- and off-diagonal heat kernel behaviors on certain infinite dimensional local Dirichlet spaces

This paper studies on-diagonal and off-diagonal bounds for symmetric diffusion semi-groups that admit a continuous kernel, in the case where the underlying space is typically infinite dimensional. For invariant diffusions on the infinite dimensional torus, the equivalence between a certain on-diagonal estimate and a certain off-diagonal behavior is proved. This gives a necessary and sufficient condition, in terms of the associated infinite symmetric matrix, for an elliptic Harnack inequality to be satisfied. As a consequence, elliptic and parabolic Harnack inequalities are in fact equivalent properties in this setting. In terms of potential theory, this gives a necessary and sufficient condition for the associated harmonic sheaf to be a Brelot harmonic sheaf. A new distance associated to certain Dirichlet spaces is also introduced. It plays a crucial role in relating on-diagonal behavior to off-diagonal behavior in cases where the intrinsic distance is infinite almost everywhere.