Stefan Problems for Reflected SPDEs Driven by Space-Time White Noise

Ben Hambly, Jasdeep Kalsi

Published 2018-06-12Version 1

We prove the existence and uniqueness of solutions to a Stefan Problem for reflected SPDEs which are driven by space-time white noise. The solutions are shown to exist until almost surely positive blow-up times. Such equations can model competition between two types, with the dynamics of the shared boundary depending on the derivatives of two competing profiles at this point. The novel features here are the presence of space-time white noise and the reflection measures, which maintain positivity for the competing profiles. In general, reflected SPDEs driven by space-time white noise are only H\"{o}lder regular up to 1/2 in space. In particular, they do not have spatial derivatives, so we can't make sense of the Stefan moving boundary condition, that $p^{\prime}(t)= h(\frac{\partial u_1}{\partial x}(0), \frac{\partial u_2}{\partial x}(0)).$ By including a volatility coefficient in front of the space-time white noise which decays linearly as $x \rightarrow 0$, and by demonstrating that solutions to parabolic obstacle problems have spatial derivatives at their endpoint when the obstacle does, we are able to overcome these issues. The blow-up time is characterised as the first time that one of the derivatives at the boundary blows up. We conclude the paper with some simple numerical illustrations.