Extinction properties of super-Brownian motions with additional spatially dependent mass production.
Extinction and local extinction of the finite measure-valued continuous super-Brownian motion \(X\) on \(\mathbb R^d\) corresponding to the log-Laplace equation \(\frac {\partial }{\partial t} u =\frac {1}{2}\Delta u +\beta u -u^2\) are studied, where the additional mass production coefficient \(\beta \) is Hölder continuous and bounded from above. A threshold decay rate \(k_d| x| ^{-2}\) as \(| x| \to \infty \) is found such that the extinction does not occur if \(\beta \) is above the threshold, whereas it does occur below the threshold, where \(k_d=\frac {1}{8} (d-2)^2\) if \(d\leq 6\) and \(k_d=d-4\) if \(d>6\). An analogous criterion has been proved for the local extinction. It is also shown that in dimension 1 if \(\beta \) is not bounded from above, \(X\) does not exhibit local extinction and degenerates to a single point source \(\delta _0\).
scientific article; zbMATH DE number 2098322