An additive-noise approximation to Keller–Segel–Dean–Kawasaki dynamics

Thesis

The theory of fluctuating hydrodynamics aims to describe density fluctuations of interacting particle systems as so-called Dean–Kawasaki stochastic partial differential equations (SPDEs). However, those Dean–Kawasaki equations are ill-posed and recent focus has shifted towards finding well-posed approximations that retain the statistical properties of the particle system. In this thesis we consider the fluctuating hydrodynamics of a system in which particles interact with one another through the potential gradient induced by the Green function of the Laplacian (Keller–Segel dynamics).

We propose an additive-noise approximation, which is given by a non-linear, non-local, parabolic-elliptic SPDE with a heterogeneous space-time noise, and show that it retains the same law of large numbers and central limit theorem as (conjectured for) the particle system. We further consider the large deviation principle and show that the approximation error lies in the skeleton equation that drives the rate function.

The results above depend on the relative scaling between two parameters that appear in our equation: the noise intensity, which governs the amplitude of our fluctuations, and the correlation length, which describes their spatial correlations. To loosen the required relationship between the noise intensity and the correlation length, one needs to assume that the approximation lies in a space of lower regularity. Consequently, we consider three different scenarios:

• The regular setting, in which the approximation converges in a Bessel potential space of regularity greater than -1/2, at the expense of a restrictive relative scaling.

• The rough setting, in which the approximation converges in a Hölder--Besov space of regularity less than -1, under a general relative scaling.

• The renormalised setting, in which one can make sense of the approximation even for zero correlation lengths, if one is willing to subtract a formally infinite counterterm.

Depending on the specific setting, we use different tools to analyse the approximation. For example, to deduce a large deviation principle in the regular setting we rely on the weak-convergence approach, whereas in the rough setting we use the paracontrolled decomposition and a generalisation of Freidlin and Wentzell's theory due to Hairer and Weber.

In comparing these different approaches we demonstrate how the range of suitable scaling relations, and the regularity of the approximation, are determined by the asymptotic behaviour of certain stochastic objects for vanishing correlation lengths.

Authors: Martini, A

• DOI: 10.5287/ora-2ryg9q99k
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: law of large numbers; fluctuating hydrodynamics; paracontrolled distributions; large deviation principle; central limit theorem; singular stochastic partial differential equation; parabolic-elliptic Keller–Segel model; Dean–Kawasaki equation
• Subjects: Chemotaxis; Stochastic partial differential equations; Nonequilibrium statistical mechanics