Super-Brownian Motions in Catalytic Media

Abstract

Some recent results on (critical continuous) super-Brownian motions X in catalytic media, i.e. where the branching rate ϱ is assumed to be a generalized function, will be reviewed. An extremely simplified one-dimensional example is ϱ = δ_c. Here branching occurs only at a single point catalyst with position c ∈ R and with infinite rate; outside c only the heat flow acts. This single point-catalytic super-Brownian motion X has remarkable properties discussed in some detail. For instance, jointly continuous super-Brownian local times $y:= \{y_t(a); t > 0, a \in R\}$ exist, but $\{y_t(c); t > 0\} =: y(c)$ is only singularly continuous. The intuitive reason behind this is that the catalyst normally kills off the mass, by the infinite rate of branching, but “occasionally” (at exceptional times of “full” dimension) branching occurs. The super-Brownian local time y(c) at c is a basic object in this model. In fact, it can be alternatively constructed as the total occupation time measure of a one-sided super-½-stable motion on R ₊. Using y(c), the mass density field $x:= \{x_t(a); t> 0, a \neq c \}$ of X can then be defined by an excursion type formula, so that it solves the heat equation and is C ^∞. Another problem is the construction of higher dimensional catalytic super-Brownian motions with absolutely continuous states (in contrast to the constant medium case). Some nonlinear reaction diffusion equations in which δ-functions enter in various ways are a main analytical tool.