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Sharp estimates for capacities and applications to symmetric diffusions. (English) Zbl 0828.60062
Summary: We give sharp estimates for the capacity of compact set \(F \subset X\) where \(X\) is the state space of a strongly regular Dirichlet form of diffusion type. These estimates are in terms of the reference measure \(m\) and the Carathéodory metric \(\rho\) on \(X\). For instance, \[ \text{Cap}_0 F \leq \left( \int^\infty_0 {dr \over v'(r)} \right)^{- 1} \leq 2 \left( \int^\infty_0 {r dr \over v(r)} \right)^{-1}, \] where \(v(r) = m(\{0 < \rho (x,F) < r\})\). 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 \(\{\varphi = 0\}\) for the process associated with the operator \(\Delta + 2 {\nabla \varphi \over \varphi} \nabla\) and to recurrence questions for divergence form operators \(\nabla a \nabla\) on \(D \subset \mathbb{R}^n\) with \(a = (a_{ij})\) which degenerate at \(\partial D\).

MSC:
60J60 Diffusion processes
31C25 Dirichlet forms
35J20 Variational methods for second-order elliptic equations
58J05 Elliptic equations on manifolds, general theory
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