Properties of noninteger moments in a first passage time problem

We calculate the moments 〈t^q〉, where q is not necessarily an integer, of the first passage time to trapping for a simple diffusion problem in one dimension. If a characteristic length of the system is L and 〈t_q〉 ~L_τ(q) as L→∞, then we show that there is a phase transition at q=q_csuch that when q<q_c,τ(g)=0, and for q>q_c, τ(q) is a linear function of q. These analytical results can be used to explain results for large moments for diffusion on a hierarchic structure. We also show how to calculate noninteger moments in terms of characteristic functions.