Stochastic Heat Equation and Catalytic Super-Brownian Motion

Abstract

This thesis concerns the existence of certain stochastic processes with infinite dimensional state spaces as well as regularity properties of their sample paths. A main object of interest is the stochastic heat equation in Rd with singular drift and driven by an inhomogeneous time-space white noise. The quadratic variation measure of the white noise is not required to be absolutely continuous w.r.t. the Lebesgue measure, neither in space nor in time. While the homogeneous version of this equation (regular drift, homogeneous white noise) has been studied several times, our general setting has not been investigated so far. We prove the existence of jointly continuous solutions in dimension d = 1, provided the drift and the quadratic variation measure of the white noise are not too singular and the coefficients are continuous. In higher dimensions (d ≥ 2) the disturbance by the white noise is too strong in order to obtain continuous solutions. However, if the noise is skipped, then we can establish that the deterministic heat equation with moderately singular drift possesses jointly continuous solutions in all dimensions d ≥ 1. For both the stochastic and the deterministic equation statements on uniqueness and non-negativity of solutions will be proven under some additional assumptions on the coefficients. Another object of interest is the catalytic super-Brownian motion in Rd which arises as (high-density/short-lifetime) measure-valued diffusion limit of a system of d-dimensional branching Brownian particles. The branching time of a particle is governed by the particle’s collision with a given time-space measure, the so-called catalyst. We introduce a new admissibility condition for the catalyst and present a direct construction (i.e. without referring to any particle system) of the corresponding catalytic super-Brownian motion X = (Xt(dx) : t ≥ 0). For the sake of completeness we also construct the corresponding branching functional (in the sense of Dynkin) for the approximating particle system. An important feature of the catalytic super-Brownian motion X is the characterization as (unique) solution to a certain martingale problem. While it is comparatively easy to verify that X solves the martingale problem, it is more delicate to show uniqueness of solutions. In particular, a general uniqueness result was an open problem. By means of a duality argument we prove that uniqueness holds. Further, it will be shown that X can be assumed to be continuous w.r.t. the weak topology. Such a regularity statement is already known for more special catalysts. However, the question whether X possesses a jointly continuous Lebesgue density field has not been studied so far (except for the case of the classical super-Brownian motion). We show that in dimension d = 1 there is a large class of non-atomic catalysts which induce a jointly continuous density field for X. Moreover, the density field can be characterized as unique solution to the stochastic heat equation (without drift) described above, where the noise coefficient has the shape a(u) = √ u and the quadratic variation measure of the time-space white noise coincides with the catalyst.