An Additive-Noise Approximation to Keller-Segel-Dean-Kawasaki Dynamics: Local Well-Posedness of Paracontrolled Solutions
Adrian Martini, Avi Mayorcas
TL;DR
This work addresses local well-posedness for a singular SPDE that serves as an additive-noise approximation to fluctuating Keller–Segel–Dean–Kawasaki dynamics on the 2D torus. The authors employ paracontrolled distributions to meaningfully define and renormalise ill-defined nonlinearities arising from rough noise, introducing a renormalised enhancement of the noise (and a canonical version) to capture divergent terms. A key technical insight is that symmetry of the elliptic Green's function reduces potential divergences from linear to logarithmic, decreasing the number of required counterterms. The paper proves local existence and uniqueness of renormalised, paracontrolled solutions and sets up a framework that connects to macroscopic limit results (LLN/CLT/LDP) in subsequent work.
Abstract
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.