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
and (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.
Research supported in part by AFOSR Grant 85–0330.
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References
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© 1989 Birkhäuser Boston
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Chung, K.L. (1989). Gauge Theorem for Unbounded Domains. In: Çinlar, E., Chung, K.L., Getoor, R.K., Glover, J. (eds) Seminar on Stochastic Processes, 1988. Progress in Probability, vol 17. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-3698-6_4
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DOI: https://doi.org/10.1007/978-1-4612-3698-6_4
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