Smooth density field of catalytic super-Brownian motion
The authors consider a continuous super-Brownian motion (SBM) \((\rho_t)_{t\geq 0}\) in \(\mathbb{R}^d\) of dimensions \(d= 2,3\) with constant branching rate, and a catalytic SBM \((X^\rho_t)_{t\geq 0}\), that is, a continuous SBM in \(\mathbb{R}^d\) with ``local branching rate'' \(\rho_s(dx)\). The main result is that \(X^\rho_t\) is absolutely continuous with density function \(\xi^\rho_t\), and that there exists a version of the map \((t,z)\to \xi^\rho_t(z)\) which is \({\mathcal C}^\infty\) and solves the heat equation of the catalyst \(\rho\). This result and self-similarity are applied to show that in dimension \(d= 2\), if \(\rho\) and \(X^\rho\) are both started with a Lebesgue measure, then \(X^\rho_T\) converges in distribution as \(T\to\infty\) towards a random multiple of Lebesgue measure, the random multiple being \(\xi^\rho_1(0)\); thus persistent convergence holds.
scientific article; zbMATH DE number 1346297
scientific article