Using the method of paracontrolled distributions, we show the local well-posedness of an additive noise approximation to the fluctuating hydrodynamics of the Keller-Segel model on the two-dimensional torus. Our approximation is a non-linear, non-local, parabolic-elliptic stochastic PDE with an irregular, heterogeneous space-time noise. As a consequence of the irregularity and heterogeneity, solutions to this equation must be renormalised by a sequence of diverging fields. Using the symmetry of the elliptic Green’s function, which appears in our non-local term, we establish that the renormalisation diverges at most logarithmically, an improvement over the linear divergence one would expect by power counting. Similar cancellations also serve to reduce the number of diverging counterterms.

The main goal of this work is to construct the killed rough Super Brownian Motion, a superprocess in a white noise environment first introduced by Perkowksi and Rosati. To achieve this, we use a Branching Brownian Motion and a novel intermediate process called the killed mollified Super Brownian Motion. Not only does this simplify the construction of Rosati et al., it may also be more aligned with biological intuition. In order to show the uniqueness of the killed rough Super Brownian Motion, we construct a solution to the associated Evolution Equation, a certain Singular Stochastic Partial Differential Equation with a logistic non-linearity. This construction is carried out by introducing what we shall call Paracontrolled Wild sums, a lightweight tool which also yields the approximability of solutions and differentiability with respect to a small parameter in the initial condition.