Classical Potential Theory and Its Probabilistic Counterpart by J. L. Doob (auth.)

By J. L. Doob (auth.)

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If D is a ball or is IR N , a simple choice of B. is an increasing sequence of balls with closures in D and with union D. Choice of B. as an increasing sequence of open relatively compact subsets of D with union D will be possible for every open set D once TB is defined (in Section VIlLI I ) whenever B is an open relatively compact subset of D. Observation (b). 111. Infima of Families of Superharmonic Functions for superharmonic functions u l , U2 and positive constants c l , c2 • EXAMPLE (a). Let u be superharmonic on D and let B = B(~, b) have closure in D.

Generalization of Theorem 1 Theorem 1 is included in the following theorem but was proved separately because of the importance of its constructive proof. Theorem. If a class r of superharmonic functions on an open subset D of [RN has a subharmonic minorant, then GMr exists and is harmonic. Let robe the class of subharmonic minorants of r. The class r 0 contains U l v U2 with Ul, U2 and is therefore directed upward in the order of pointwise inequality, with limit u', a function majorized by r. We prove the theorem by showing that u' is harmonic.

5. Minimum Theorem for Superharmonic Functions Theorem. Let u be superharmonic on an open subset D of IRN. If D is connected and if u attains its infimum at a point (b) ~, then u is identically constant. The infimum ofu is the limit ofu along a sequence ofpoints approaching (c) If u has a lower semicontinuous extension to D U (a) oD. the extension is attained on oD. oD, the infimum of A typical implication of this theorem is the fact that a super harmonic function on D with a positive inferior limit at every boundary point must be positive.

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