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

Chung, Kai Lai

Ann. Inst. Henri Poincaré, Probab. Stat. 20, 349-351 (1984).

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

Keywords:

conditional Brownian motion; lifetime of h-paths; upcrossing lemma

Citations:

Zbl 0506.60071