May 16th, 2021, 8:54 pm

For 2, express the hitting time density as the negative t' derivative of a complementary distribution func. Since, by assumption in that case, the expected hitting time is finite, the hitting time density must decay at large t' like [$] 1/(t')^{2 + \epsilon}[$] or faster, where [$]\epsilon > 0[$]. Use that fact to convince yourself the (new) parts term is zero. Then, revert in the remaining integral from the complementary dist func. to the regular dist func.

For 1, I looked at the text and it seems we are talking about 1D diffusions where [$]\Omega[$] is a two-sided spatial interval. I haven't really checked it, but would check drifting Brownian motion, with a positive drift, starting from the origin, and the interval [$]\Omega = (a,\infty)[$], where [$]a < 0[$]. Is the expected exit time finite? Is having one boundary at infinity ruled out?

Even if having one boundary always at infinity is ruled out, suspect one could cook up some two-sided curving boundaries where the curving was such that the expected exit time was still +infinity.