Seminar on Stochastic Processes, 1988 pp 87-98 | Cite as

# Gauge Theorem for Unbounded Domains

Chapter

## Abstract

Let {x and (1) where E

_{t}, t⩾0} be the Brownian motion process in R^{d}, d⩾1; D a domain (nonempty, open and connected set) in R^{d}; q a Borel function on D. Put$${\tau_D} = \inf \left\{ {t > 0:{X_t} \notin D} \right\}, $$

$$u(x) = {E^X}\left\{ {{\tau_D} < \infty; \;\exp \left[ {\int\limits_0^\tau {{}^Dq\left( {{X_t}} \right)dt} } \right]} \right\} $$

(1)

^{x}(P^{x}) denotes the expectation (probability) under X_{0}= x. The function u is called the gauge for (D,q), provided it is well-defined, namely when the integral involved exists. A result of the following form is called gauge theorem: (2) either u ≡ +∞ in D, or u is bounded in D. Let D̄ denote the closure of D in R^{đ}(no point at infinity). It is easy to show that if it is bounded in D, then the same upper bound serves for u in D̄, so that u is in fact bounded in Rd since it is equal to one in R^{đ}- D̄. In this case we say that (D,q) is gaugeable.### Keywords

Hunt Summing## Preview

Unable to display preview. Download preview PDF.

### References

- [1]K. L. Chung and K. M. Rao, Feynman-Kac functional and the Schrödinger equation, Seminar in Stochastic Processes 1(1981), 1–29.MathSciNetCrossRefGoogle Scholar
- [2]M. Aizenman and B. Simon, Brownian motion and Harnack inequality for Schrödinger operators, Comm. Pure Appl. Math. 35(1982), 209–273.MathSciNetMATHCrossRefGoogle Scholar
- [3]Z. Zhao, Conditional gauge with unbounded potential, Z. Wahrscheinlichkeitstheorie Verw. Geb. 65(1983), 13–18.MATHCrossRefGoogle Scholar
- [4]G. A. Hunt, Some theorems concerning Brownian motion, Trans. Amer. Math. Soc. 81(1956), 294–319.MathSciNetMATHCrossRefGoogle Scholar

## Copyright information

© Birkhäuser Boston 1989