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 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    M. AIZENMAN and B. SIMON. Brownian Motion and Harnack’s inequality for Schrödinger Operators. Comm. Pure Appl. Math. 35(1982), 209–271.MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    K.L. CHUNG and K.M. RAO. Feynman-Kac Functional and the Schrödinger Equation. Seminar on Stochastic Process 1981, pp. 1–29. Birkhäuser, Boston, 1981.CrossRefGoogle Scholar
  3. 3.
    PEI HSU. Reflecting Brownian Motion, Boundary Local Time and the Neumann Problem. Doctoral Dissertation, Stanford University, June 1984.Google Scholar
  4. 4.
    Z.X. ZHAO. Conditional Gauge with Unbounded Potentials. Z. Wahrscheinlichkeitstheorie verw. Geb. 35(1983), 13–18.CrossRefGoogle Scholar

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

Personalised recommendations