Recurrence and Transience

We have the following dichotomy result. Theorem 2. The following are equivalent: (1) X is transient; (2) X is not recurrent; (3) U0(B(0 � �)) < ∞ for all � > 0. Theorem 2 has the following important dichotomous implication: Either X is transient, or X is recurrent [and not both]. The proof relies on a convenient series of equivalences. Proposition 3. The following are equivalent: (1) sup�∈R� U0(B(� � �)) < ∞ for all � > 0; (2) U0(B(0 � �)) < ∞ for all � > 0; (3) U0(B(0 � �)) < ∞ for some � > 0; (4) �∞ 0 1lB(0��)(X�) d� < ∞ a.s. for all � > 0; (5) lim�↑∞ sup�∈B(0�2�) P{ �∞ 0 1lB(���)(X�) d� > �} < 1 for all � > 0.