Nonlinear Stochastic Equations in Two and Three Dimensions

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Stochastic partial differential equations (SPDE) describe the behaviour of spatially extended systems under the influence of noise. They arise naturally in various fields of applications as diverse as data mining, mathematical finance, and population dynamics and genetics.The present proposal aims to study a class of stochastic partial differential equations from statistical mechanics. Many particle models exhibit a behaviour called phase transition, where the behaviour of the system changes drastically when one changes a given system parameter beyond a critical point. It is a very exciting question to understand the behaviour of such a system near a critical point. In such a regime one expects the dynamics to be governed by a non-linear SPDE. Analytically the understanding of these equations is very challenging, because of the interaction between the rough noise term and the non-linear evolution. But this is also what gives rise to interesting phenomena. In this proposal, we aim to deepen the understanding of these equations. On the one hand we will study the behaviour under the influence of a small noise term. Then we will establish that two different kinds of particle models can indeed be described by such an SPDE.