The Rate of Decay of the Wiener Sausage in Local Dirichlet Space

Abstract

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 of the upper asymptotic uses a known coarse graining technique which must be adapted to the non-transitive setting. This result provides the first such asymptotics for several other contexts, including diffusions on complete Riemannian manifolds with nonnegative Ricci curvature.