Gauge Theorem for the Neumann Problem

  • K. L. Chung
  • Pei Hsu
Part of the Progress in Probability and Statistics book series (PRPR, volume 9)


Let D be a bounded domain in IRd and let (Δ/2 + q)u = 0 be Schrödinger’s equation on D. The Dirichlet problem for the equation was studied first in [2] for bounded q and then in [1] and [4] for q ∈ Kd (see below for definition). The gauge function for the Dirichlet problem is defined in [2] as
$${\text{G(x)}}{\mkern 1mu} {\text{ = }}{\mkern 1mu} {{{\text{E}}}^{{\text{x}}}}[\exp (\int_{{\text{0}}}^{{{{\tau }_{{\text{D}}}}}} {{\text{q(}}{{{\text{B}}}_{{\text{S}}}}{\text{)}}} {\mkern 1mu} {\text{ds}})]$$
, where B = {Bt, t ≥ 0} is the standard Brownian motion on and IRd and τD is the first exit time of D. One striking property of the gauge function proved in [2] and [4] is the following.


Brownian Motion Dirichlet Problem Neumann Problem Exit Time Standard Brownian Motion 
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Copyright information

© Birkhäuser Boston, Inc. 1986

Authors and Affiliations

  • K. L. Chung
    • 1
  • Pei Hsu
    • 2
  1. 1.Department of MathematicsStanford UniversityStanfordUSA
  2. 2.Courant Institute of MathematicsNew York UniversityNew YorkUSA

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