We present some limit theorems for the normalized laws (with respect to functionals involving last passage times at a given level $a$ up to time $t$) of a large class of null recurrent diffusions. Our results rely on hypotheses on the Lévy measure of the diffusion inverse local time at 0. As a special case, we recover some of the penalization results obtained by Najnudel, Roynette and Yor in the (reflected) Brownian setting.
"Penalizing null recurrent diffusions." Electron. J. Probab. 17 1 - 23, 2012. https://doi.org/10.1214/EJP.v17-2267