Forward-Invariance and Wong-Zakai Approximation for Stochastic Moving Boundary Problems

TL;DR

This paper studies stochastic moving boundary problems, proving phase separation and developing a Wong-Zakai approximation to extend deterministic results to stochastic PDEs with moving interfaces.

Contribution

It introduces a Wong-Zakai approximation for stochastic moving boundary problems and analyzes phase separation using stochastic evolution equations.

Findings

Proper phase separation is achieved with suitable initial conditions.

Wong-Zakai approximation effectively extends deterministic results to stochastic cases.

Reformulation in terms of stochastic evolution equations enables analysis of complex boundary dynamics.

Abstract

We discuss a class of stochastic second-order PDEs in one space-dimension with an inner boundary moving according to a possibly non-linear, Stefan-type condition. We show that proper separation of phases is attained, i.e., the solution remains negative on one side and positive on the other side of the moving interface, when started with the appropriate initial conditions. To extend results from deterministic settings to the stochastic case, we establish a Wong-Zakai type approximation. After a coordinate transformation the problems are reformulated and analysed in terms of stochastic evolution equations on domains of fractional powers of linear operators.