  This paper studies strongly local symmetric Dirichlet forms on general
measure spaces. The underlying space is equipped with the intrinsic metric
induced by the Dirichlet form, with respect to which the metric measure space
does not necessarily satisfy volume-doubling property. Assuming Nash-type
inequality, it is proved in this paper that outside a properly exceptional set,
given a pointwise on-diagonal heat kernel upper bound in terms of the volume
function, the comparable heat kernel lower bound also holds. The only
assumption made on the volume growth rate is that it can be bounded by a
continuous function satisfying doubling property, in other words, is not
exponential.
