Strong clumping of super-Brownian motion in a stable catalytic medium

Biodiversity of Catalytic Super-Brownian Motion

In this paper we investigate the structure of the equilibrium state of three{dimensional catalytic super-Brownian motion where the catalyst is itself a classical super-Brownian motion. We show that the reactant has an in nite local biodiversity or genetic abundance. This contrasts the nite local biodiversity of the equilibrium of classical super-Brownian motion. Another question we address is t...

Persistence of a Two-dimensional Super-brownian Motion in a Catalytic Medium

The super-Brownian motion X % in a super-Brownian medium % constructed in DF96a] is known to be persistent (no loss of expected mass in the longtime behaviour) in dimensions one ((DF96a]) and three ((DF96b]). Here we ll the gap in showing that persistence holds also in the critical dimension two. The key to this result is that in any dimension (d 3); given the catalyst, the variance of the proc...

On the Hot Spots of a Catalytic Super-brownian Motion

A Super-brownian Motion with a Locally Innnite Catalytic Mass

A super-Brownian motion X in IR with \hyperbolic" branching rate %2(b) = 1=b 2 , b 2 IR; is constructed, which symbolically could be described by the formal stochastic equation dXt = 1 2 Xt dt + p 2%2Xt dWt ; t > 0; (1) (with a space-time white noise dW). Starting at X0 = a ; a 6 = 0; this superprocess X will never hit the catalytic center: There is an increasing sequence of Brownian stopping t...

Smooth Density Eld of Catalytic Super-brownian Motion

Given an (ordinary) super-Brownian motion (SBM) % on R d of dimension d = 2; 3; we consider a (catalytic) SBM X % on R d with \local branching rates" %s(dx). We show that X % t is absolutely continuous with a density function % t ; say. Moreover, there exists a version of the map (t; z) 7 ! % t (z) which is C 1 and solves the heat equation oo the catalyst %, more precisely, oo the (zero set of)...