Abstract
In Section 5.4, we proved the Gauge Theorem for a bounded domain D: if the gauge u(x) = E x{e q (τD)} ≢ ∞ in D, then it is bounded in \( \bar D \) . In this section, we shall prove the Conditional Gauge Theorem for a bounded Lipschitz domain D: if the conditional gauge u(x, z) = E x z {e q (τD)} ≢ ∞ in D × ∂D, then it is bounded in D × ∂D. For the gauge theorem, no assumption about the boundary is imposed, not even its regularity in the Dirichlet sense. By contrast, the conditional gauge theorem requires a certain smoothness of the boundary. Ad hoc assumptions on D and q may be and have been considered, but we shall settle the case in which D is a bounded Lipschitz domain in ℝd, d≥2 and q ∈ Jloc. For the case d = 1, see Theorem 9.9 and the Appendix to Section 9.2.
Keywords
- Lipschitz Domain
- Harmonic Measure
- Bound Lipschitz Domain
- Strong Markov Property
- Boundary Harnack Principle
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© 1995 Springer-Verlag Berlin Heidelberg
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Chung, K.L., Zhao, Z. (1995). Conditional Gauge and q-Green Function. In: From Brownian Motion to Schrödinger’s Equation. Grundlehren der mathematischen Wissenschaften, vol 312. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-57856-4_7
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DOI: https://doi.org/10.1007/978-3-642-57856-4_7
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-63381-2
Online ISBN: 978-3-642-57856-4
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