Stochastic Stefan Problems Driven By Standard Brownian Sheets

TL;DR

This paper investigates stochastic Stefan problems driven by Brownian sheets, establishing existence, uniqueness, and regularity of solutions, and introduces new methods for analyzing stochastic moving boundary PDEs with space-time randomness.

Contribution

It presents novel techniques for proving existence and regularity of solutions to stochastic Stefan problems driven by Brownian sheets, extending previous results to more complex stochastic boundary PDEs.

Findings

Proved existence and uniqueness of solutions to stochastic Stefan problems.

Derived space and time regularity of solutions using Kolmogorov's Continuity Theorem.

Extended methods to stochastic heat equations driven by space-time Brownian sheets.

Abstract

In this paper we study the effect of stochastic perturbations on a common type of moving boundary value PDE's which endorse Stefan boundary conditions, or Stefan problems, and show the existence and uniqueness of the solutions to a number of stochastic equations of this kind. Moreover we also derive the space and time regularities of the solutions and the associated boundaries via Kolmogorov's Continuity Theorem in an appropriately defined normed space. The paper first conveys our previous results where randomness is smoothly correlated in space and Brownian in time, then introduces a new methodology that enables us to prove the existence and uniqueness of the solution to the standard heat equation driven by a space-time Brownian sheet as well as its boundary regularity, and finally extends it to the stochastic moving boundary partial PDE's driven by the same type of randomness.