Date: 12 Sep 2013

Conditional gauge with unbounded potential

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Summary

Let B be a ball in R d and q be a Borel function on it. We prove that if \(\mathop {\sup }\limits_{x \in \bar B} \int\limits_B {\frac{{|q(y)|}}{{|x - y|^{d - 2} }} dy}\) is sma11 enough, then $$\mathop {\sup }\limits_{\mathop {x \in B}\limits_{z \in \partial B} } E_z^x \left[ {\exp \int\limits_0^{t_B } {q(x_t )dt} } \right] < + \infty$$

This paper gives a new proof of one of the two main results by Aizenman and Simon in [1] by a simple and elementary method. A basic theorem in Chung and Rao [2] is extended to the class of q treated in [1].