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

Lee R. Gibson () and Melanie Pivarski ()
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Lee R. Gibson: The Infinite Actuary
Melanie Pivarski: Roosevelt University

Journal of Theoretical Probability, 2015, vol. 28, issue 4, 1253-1270

Abstract: Abstract In the context of a heat kernel diffusion which admits a Gaussian type estimate with parameter $$\beta $$ 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 $$t^{\beta }V(x,t)$$ t β V ( x , t ) is $$-V(x,t)$$ - V ( x , t ) , which is analogous to the Euclidean result. Here, $$V(x,t)$$ V ( x , t ) represents the mass of the ball of radius $$t$$ t about a point $$x$$ 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.

Keywords: Local Dirichlet space; Log Asymptotic behavior; Wiener sausage; Negative exponential moments; 60G17 (Primary); 60F10 (Secondary) (search for similar items in EconPapers)
Date: 2015
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DOI: 10.1007/s10959-014-0553-0

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