Moving boundary problems for reflected SPDEs

The work in this thesis is based on the study of reflected SPDE (stochastic partial differential equation) moving boundary problems. These are systems consisting of two competing profiles which each evolve according to a reflected stochastic heat equation in one spatial dimension, and share a common boundary point. The reflection here minimally pushes the profiles upwards in order to maintain positivity. The evolution of the shared boundary depends on the state of the profiles, and so is coupled with the dynamics of the two competing sides. Such equations are suited to modelling competition between two types. An example of this is the limit order book. We can think of the competing profiles as being order volumes to buy/sell an asset at different prices, with positivity of these ensured by the reflection terms. The shared boundary then represents the current midprice. Key results include scaling laws, which show how we can connect particle systems to particular reflected SPDE moving boundary problems, and existence and uniqueness results for different classes of reflected SPDE moving boundary models. In the latter case, we first study the situation where the boundary mechanism is driven by the competing profiles considered as continuous functions. Following this we impose stronger assumptions and examine the case where the boundary mechanism is driven by the derivatives of the competing profiles at the shared interface, as in the classical Stefan problem. We also study other properties of our systems, including their Hölder regularity and the existence of invariant measures.