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 β 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 β 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 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.

Long Time Behavior of the Volume of the Wiener Sausage on Dirichlet Spaces

“…Chavel-Feldman-Rosen [CFR91] obtained a second order radial asymptotic result for 2-dimensional Riemannian manifold, extending Le Gall's expansion [Le88, Theorem 2.1] in R 2 . Gibson-Pivarski [GP15] obtained a time asymptotic result similar to [DV75] for diffusions on local Dirichlet spaces satisfying a sub-Gaussian heat kernel estimate. Bass-Kumagai [BK00] showed the law of the iterated logarithms (LILs) of the ranges of a class of symmetric diffusion processes on metric measure spaces.…”

“…Here the range is the volume of the trace of a sample path up to a fixed time. The results of [BK00, KKW17,GP15] are based on heat kernel estimates.…”