Stefan problems for reflected SPDEs driven by space–time white noise |
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Institution: | 1. School of Economics, Nanjing University of Finance and Economics, Nanjing, Jiangsu 210023, China;2. CAS Wu Wen-Tsun Key Laboratory of Mathematics, School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, China;1. School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, People’s Republic of China;2. Department of Mathematics, Jining University, Qufu 273155, Shandong Province, People’s Republic of China |
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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. |
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