## The lifetime of conditional Brownian motion in the plane.(English)Zbl 0573.60070

This note simplifies the proof of joint work of T. R. McConnell and the reviewer [Z. Wahrscheinlichkeitstheor. Verw. Geb. 65, 1-11 (1983; Zbl 0506.60071)] on the lifetime of h-paths. The simplification is in two directions. The first is eliminating the need to discuss Martin boundaries. The second is the observation that $$(X_ t)/h$$ is a $$P^ x_ h$$-supermartingale and the $$E^ x_ h$$-expected number of crossings of $$\{$$ y:$$2^{n-1}h(x_ 0)<h(y)<2^{n+1}h(x_ 0)\}$$ by $$X_ t$$ is handled easily by the upcrossing lemma: $$(X_ t)/h$$ must cross $$[2^{-(n+1)}h(x_ 0)^{-1}, 2^{-(n-1)}h(x_ 0)^{-1}].$$
There are a couple of misprints. In the statement of the main result, the constant C is independent of D. The estimate (3) holds for $$E^ x_ h$$.
Reviewer: M.C.Cranston

### MSC:

 60J65 Brownian motion 60J45 Probabilistic potential theory

Zbl 0506.60071
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