Fluctuations in continuum and conservative stochastic partial differential equations

Introduction

Fluctuations are ubiquitous in real-world contexts and in key technological challenges, ranging from thermal fluctuations in physical systems to algorithmic stochasticity in machine learning, and fluctuations caused by small-scale weather patterns in climate dynamics.

At the same time, such complex systems are subject to an abundance of influences and depend on a large variety of parameters and interactions. A systematic understanding of the interplay of stochasticity and complex dynamical behavior aims at unveiling universal properties, irrespective of the many details of the concrete systems at hand. Its development relies on the derivation and analysis of universal concepts for their scaling limits, capturing not only their average behavior but also their fluctuations.

Proposed Analysis

We propose to analyze the class of conservative stochastic partial differential equations (SPDE) as such a universal fluctuating continuum model, and unveil its mathematical analysis as a fruitful field for the discovery of new mathematical structures and methods.

Key Challenges

The key challenges targeted in this proposal are:

1. Well-posedness of singular conservative SPDEs
2. Singular limits for supercritical conservative SPDEs
3. Stochastic dynamics for conservative SPDEs

Approach

We aim to approach these challenges by a novel combination of recent scientific breakthroughs in the fields of strongly nonlinear, conservative SPDEs and singular SPDEs.

We thereby intend to advance the highly active and productive field of (singular) SPDEs, which has inspired striking mathematical progress in the last decade.

Interdisciplinary Connections

The analysis of conservative SPDEs conjoins several contemporary fields of analysis and probability: singular SPDEs, nonlinear PDEs, kinetic theory, supercriticality, and stochastic dynamical systems.

We are, therefore, confronted with an interplay of stochasticity, irregularity, and nonlinearity, posing new challenges and going far beyond established methods.