Gauge Theorem for Unbounded Domains

Let {x_t, 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) (1)$$u(x) = {E^X}\left\{ {{\tau_D} < \infty; \;\exp \left[ {\int\limits_0^\tau {{}^Dq\left( {{X_t}} \right)dt} } \right]} \right\} $$ 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.