Conditioned Brownian motion and hyperbolic geodesics in simply connected domains (Q1310139)

Let \(D\) be a simply connected plane domain, \(B_ t\) a Brownian motion in \(D\) with life-time \(\tau_ D\), \(P^ D_ t(w,z)\) its transition probability, \(H^ +(D)\) the collection of all positive harmonic functions in \(D\), \(P^ h_ t (w,z)= {1\over {h(w)}} P^ D_ t (w,z) h(z)\) the transition function for the Brownian motion conditioned by \(h\in H^ +(D)\), \(P^ h_ w\) the measure on path space induced by \(P^ h_ t\), and \(E^ h_ w\) the corresponding expectation. The authors extend the Cranston-McConnells inequality and give upper and lower estimates for \(\sup_{w\in D, h\in H^ +(D)} E^ h_ w(\tau_ D)\) in terms of geodesics and hyperbolic distance in \(D\). As a consequence new and shorter proofs of recent results of Xu, Davis and others are obtained. Some sufficient conditions for a domain with infinite area to have finite life-time are also proposed.