On the ultimate value of local time of one-dimensional super-Brownian motion

We study the random field of local time picked up over the entire life of a super-Brownian motion on the real line. The finite-dimensional distributions of the field are characterized via their Laplace transforms by unique solutions of certain boundary-value differential equations. In some cases the one-dimensional distributions can be found explicitly, giving some insight into how super-Brownian motion behaves before extinction or local extinction.