# Gauge Theorem for Unbounded Domains

• Kai Lai Chung
Chapter
Part of the Progress in Probability book series (PRPR, volume 17)

## Abstract

Let {xt, t⩾0} be the Brownian motion process in Rd, d⩾1; D a domain (nonempty, open and connected set) in Rd; q a Borel function on D. Put
$${\tau_D} = \inf \left\{ {t > 0:{X_t} \notin D} \right\},$$
and (1)
$$u(x) = {E^X}\left\{ {{\tau_D} < \infty; \;\exp \left[ {\int\limits_0^\tau {{}^Dq\left( {{X_t}} \right)dt} } \right]} \right\}$$
(1)
where Ex (Px) denotes the expectation (probability) under X0 = 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 Rđ (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 Rđ - D̄. In this case we say that (D,q) is gaugeable.

## Keywords

Brownian Motion Green Function Unbounded Domain Markov Property Borel Function
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

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G. A. Hunt, Some theorems concerning Brownian motion, Trans. Amer. Math. Soc. 81(1956), 294–319.