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Sharp estimates for capacities and applications to symmetric diffusions
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  • Published: March 1995

Sharp estimates for capacities and applications to symmetric diffusions

  • K. -T. Sturm1 

Probability Theory and Related Fields volume 103, pages 73–89 (1995)Cite this article

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Summary

We give sharp estimates for the capacity of compact setF⊂X whereX is the state space of a strongly regular Dirichlet form of diffusion type. These estimates are in terms of the reference measurem and the Carathéodory metric ρ onX. For instance,

$$Cap_0 F \leqq \left( {\mathop \smallint \limits_0^\infty \frac{{dr}}{{v'(r)}}} \right)^{ - 1} \leqq 2 \left( {\mathop \smallint \limits_0^\infty \frac{{r dr}}{{v(r)}}} \right)^{ - 1} ,$$

wherev(r)=m({0<ρ(x, F)<r}). From these estimates one easily obtains lower estimates for the Green function as well as sharp criteria for polarity and for recurrence. The latter, we apply to the question of hitting the nodal set {ϕ=0} for the process associated with the operator\(\Delta + 2\tfrac{{\nabla _\varphi }}{\varphi }\nabla \) and to recurrence questions for divergence form operators ∇a∇ onD ⊂ ℝn witha=(a ij ) which degenerate at ∂D.

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Authors and Affiliations

  1. Mathematisches Institut, Universität Erlangen-Nürnberg, Bismarckstrasse 1 1/2, D-91054, Erlangen, Germany

    K. -T. Sturm

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Sturm, K.T. Sharp estimates for capacities and applications to symmetric diffusions. Probab. Th. Rel. Fields 103, 73–89 (1995). https://doi.org/10.1007/BF01199032

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  • Received: 01 April 1994

  • Revised: 04 April 1995

  • Issue Date: March 1995

  • DOI: https://doi.org/10.1007/BF01199032

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Mathematics subject classification

  • 60J60
  • 31C25
  • 35J20
  • 58G03
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