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Journal of Theoretical Probability

, Volume 28, Issue 4, pp 1253–1270 | Cite as

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

  • Lee R. Gibson
  • Melanie Pivarski
Article

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.

Keywords

Local Dirichlet space Log Asymptotic behavior Wiener sausage Negative exponential moments 

Mathematics Subject Classification (2010)

60G17 (Primary) 60F10 (Secondary) 

Notes

Acknowledgments

The authors would like to thank Laurent Saloff-Coste for helpful discussions and inspiration. Thanks also to the referee for a thorough review.

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.The Infinite ActuaryPelhamUSA
  2. 2.Department of Mathematics and Actuarial ScienceRoosevelt UniversityChicagoUSA

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