{"title":"Multidimensional Stability of Planar Traveling Waves for Stochastically Perturbed Reaction–Diffusion Systems","authors":"M. van den Bosch, H. J. Hupkes","doi":"10.1111/sapm.70114","DOIUrl":null,"url":null,"abstract":"<p>We consider reaction–diffusion systems with multiplicative noise on a spatial domain of dimension two or higher. The noise process is white in time, colored in space, and invariant under translations. Inspired by previous works on the real line, we establish the multidimensional stability of planar waves on a cylindrical domain on timescales that are exponentially long with respect to the noise strength. This is achieved by means of a stochastic phase-tracking mechanism that can be maintained over such long timescales. The corresponding mild formulation of our problem features stochastic integrals with respect to anticipating integrands, which hence cannot be understood within the well-established setting of Itô-integrals. To circumvent this problem, we exploit and extend recently developed theory concerning forward integrals.</p>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"155 3","pages":""},"PeriodicalIF":2.3000,"publicationDate":"2025-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1111/sapm.70114","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Studies in Applied Mathematics","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1111/sapm.70114","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
We consider reaction–diffusion systems with multiplicative noise on a spatial domain of dimension two or higher. The noise process is white in time, colored in space, and invariant under translations. Inspired by previous works on the real line, we establish the multidimensional stability of planar waves on a cylindrical domain on timescales that are exponentially long with respect to the noise strength. This is achieved by means of a stochastic phase-tracking mechanism that can be maintained over such long timescales. The corresponding mild formulation of our problem features stochastic integrals with respect to anticipating integrands, which hence cannot be understood within the well-established setting of Itô-integrals. To circumvent this problem, we exploit and extend recently developed theory concerning forward integrals.
期刊介绍:
Studies in Applied Mathematics explores the interplay between mathematics and the applied disciplines. It publishes papers that advance the understanding of physical processes, or develop new mathematical techniques applicable to physical and real-world problems. Its main themes include (but are not limited to) nonlinear phenomena, mathematical modeling, integrable systems, asymptotic analysis, inverse problems, numerical analysis, dynamical systems, scientific computing and applications to areas such as fluid mechanics, mathematical biology, and optics.