Smooth density field of catalytic super-Brownian motion

Abstract

Given an (ordinary) super-Brownian motion (SBM) $\varrho$ on $\mathbf{R}^d$ of dimension $d = 2, 3$, we consider a (catalytic) SBM $X^{\varrho}$ on $\mathbf{R}^d$ with "local branching rates" $\varrho_s(dx)$. We show that $X_t^{\varrho}$ is absolutely continuous with a density function $\xi_t^{\varrho}$, say. Moreover, there exists a version of the map $(t, z) \mapsto \xi_t^{\varrho}(z)$ which is $\mathscr{C}^{\infty}$ and solves the heat equation off the catalyst $\varrho$; more precisely, off the (zero set of) closed support of the time-space measure $ds\varrho_s(dx)$. Using self-similarity, we apply this result to give the following answer to an open problem on the long-term behavior of $X^{\varrho}$ in dimension $d = 2$: If $\varrho$ and $X^{\varrho}$ start with a Lebesgue measure, then does $X_T^{\varrho}$ converge (persistently) as $T \to \infty$ toward a random multiple of Lebesgue measure?