  In the context of a heat kernel diffusion which admits a Gaussian type
estimate with parameter beta on a local Dirichlet space, we consider the log
asymptotic behavior of the negative exponential moments of the Wiener sausage.
We show that the log asymptotic behavior up to time t^{beta}V(x,t) is V(x,t),
which is analogous to the Euclidean result. Here V(x,t) represents the mass of
the ball of radius t about a point x of the local Dirichlet space. The proof
uses a known coarse graining technique to obtain the upper asymptotic, but must
be adapted to for use without translation invariance in this setting. This
result provides the first such asymptotics for several other contexts,
including diffusions on complete Riemannian manifolds with non-negative Ricci
curvature.
