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Stefan problems for reflected SPDEs driven by space–time white noise

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  • Hambly, Ben
  • Kalsi, Jasdeep

Abstract

We prove the existence and uniqueness of solutions to a one-dimensional 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 the evolution of phases driven by competition at an interface, 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; the reflection measures, which maintain positivity for the competing profiles; and a sufficient condition to make sense of the Stefan condition at the boundary. We illustrate the behaviour of the solution numerically to show that this sufficient condition is close to necessary.

Suggested Citation

  • Hambly, Ben & Kalsi, Jasdeep, 2020. "Stefan problems for reflected SPDEs driven by space–time white noise," Stochastic Processes and their Applications, Elsevier, vol. 130(2), pages 924-961.
    • Handle: RePEc:eee:spapps:v:130:y:2020:i:2:p:924-961
    DOI: 10.1016/j.spa.2019.04.003
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    References listed on IDEAS

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    1. Xu, Tiange & Zhang, Tusheng, 2009. "White noise driven SPDEs with reflection: Existence, uniqueness and large deviation principles," Stochastic Processes and their Applications, Elsevier, vol. 119(10), pages 3453-3470, October.
    2. Пигнастый, Олег & Koжевников, Георгий, 2019. "Распределенная Динамическая Pde-Модель Программного Управления Загрузкой Технологического Оборудования Производственной Линии [Distributed dynamic PDE-model of a program control by utilization of t," MPRA Paper 93278, University Library of Munich, Germany, revised 02 Feb 2019.
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