Conditional gauge with unbounded potential

  • Zhongxin Zhao


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].


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    Aizenman, M., Simon, B.: Brownian motion and Harnack inequality for Schrödinger operators. Comm. Pure Appl. Math. 35, 209–273 (1982)MathSciNetCrossRefMATHGoogle Scholar
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    Chung, K.L., Rao, K.M.: Feynman-Kac functional and Schrödinger equation. In Seminar on Stochastic Process, Boston (1981)Google Scholar
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    Chung, K.L.: An inequality for boundary value problems, preprint (1982)Google Scholar
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    Chung, K.L.: Conditional gauges, preprint (1982)Google Scholar
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    Port, S.C., Stone, C.J.: Brownian motion and classical potential theory. New York: Academic Press 1978MATHGoogle Scholar

Copyright information

© Springer-Verlag 1983

Authors and Affiliations

  • Zhongxin Zhao
    • 1
    • 2
  1. 1.Institute of Systems Science and Mathematical SciencesAcademia SinicaBeijingChina
  2. 2.Visiting Department of MathematicsStanford UniversityStanfordUSA

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