Stochastic Partial Differential Equation for a Measure-valued processes

Abstract: 

The theory of Stochastic Partial Differential Equations (SPDEs) lies at the intersection of probability theory, measure theory, functional analysis and the theory of partial differential equations. Informally, an SPDE is a Partial Differential Equation with a random forcing term described by the derivative of Brownian Motion. 

In statistical physics, the so-called Mean Field Method explores the limiting behaviour of a large system of interacting particles. We will consider a large system of dependent measure-valued processes $\mu^n$, each describing the dynamics of an N-particle system. In this thesis we assume that each particle follows an Ornstein-Uhlenbeck process that admits an invariant measure. As the number of processes tends to infinity, the average of these processes converges weakly to a limit measure-valued process due to a generalised Strong Law of Large Numbers. We will show that the limit measure-valued process has a density which satisfies a certain SPDE. Therefore, we can study the evolution of the limit random measure and use it as an approximate description of a very large system of interacting particles. In order to solve this limit SPDE, we turn to the theory of strongly continuous semigroups. 

This system can be used to model various social and natural science phenomena which exhibit large scale equilibrium behaviour. In particular, it is also useful in financial mathematics for modelling the default rate of companies.