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Let τr denote the hitting time of B(0:r) for a multidimensional diffusion process. We give verifiable criteria for finiteness/infiniteness of 
As an application we exhibit classes of diffusion processes which are recurrent but 
is infinite for all p > 0, |x| > r > 0; this includes the two-dimensional Brownian motion and the reflecting Brownian motion in a wedge with a certain parameter α = 0.
