On the extinction of measure-valued critical branching Brownian motion (Q909343)

Let \((X_ t)\) be the measure-valued branching motion on \(R^ d\) satisfying \[ E^{\mu}[\exp \{-\int \phi (x)dX_ t(x)\}]=\exp \{-\int u_{\phi}(t,x)\mu (dx)\}, \] where \(X_ 0\equiv \mu\) and \(u_{\phi}\) satisfies the initial value problem \((\partial /\partial t)u=\Delta u- u^ 2\), \(u(0,\cdot)=\phi\). Let \(\rho_ t\) be the diameter of the support of \(X_ t\), and let \(\xi =\inf \{t: X_ t(R^ d)=0\}\) be the extinction time of the process. The authors give an elementary, straightforward proof that if \(\mu\) has compact support, then \(\lim_{t\uparrow \xi}\rho_ t=0\) \(P^{\mu}\)-almost surely.