Local Phase Tracking and Metastability of Planar Waves in Stochastic Reaction-Diffusion Systems
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| Format: | Preprint |
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2025
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| _version_ | 1866909578571022336 |
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| author | Bosch, Mark van den Hupkes, Hermen Jan |
| author_facet | Bosch, Mark van den Hupkes, Hermen Jan |
| contents | Planar travelling waves on $\mathbb R^d,$ with $ d\geq 2,$ are shown to persist in systems of reaction-diffusion equations with multiplicative noise on significantly long timescales with high probability, provided that the wave is orbitally stable in dimension one ($d=1$). While a global phase tracking mechanism is required to determine the location of the stochastically perturbed wave in one dimension, or on a cylindrical domain, we show that the travelling wave on the full unbounded space can be controlled by keeping track of local deviations only. In particular, the energy infinitesimally added to or withdrawn from the system by noise dissipates almost fully into the transverse direction, leaving behind small localised phase shifts. The noise process considered is white in time and coloured in space, possibly weighted, and either translation invariant or trace class. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_09350 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Local Phase Tracking and Metastability of Planar Waves in Stochastic Reaction-Diffusion Systems Bosch, Mark van den Hupkes, Hermen Jan Analysis of PDEs Dynamical Systems Probability Planar travelling waves on $\mathbb R^d,$ with $ d\geq 2,$ are shown to persist in systems of reaction-diffusion equations with multiplicative noise on significantly long timescales with high probability, provided that the wave is orbitally stable in dimension one ($d=1$). While a global phase tracking mechanism is required to determine the location of the stochastically perturbed wave in one dimension, or on a cylindrical domain, we show that the travelling wave on the full unbounded space can be controlled by keeping track of local deviations only. In particular, the energy infinitesimally added to or withdrawn from the system by noise dissipates almost fully into the transverse direction, leaving behind small localised phase shifts. The noise process considered is white in time and coloured in space, possibly weighted, and either translation invariant or trace class. |
| title | Local Phase Tracking and Metastability of Planar Waves in Stochastic Reaction-Diffusion Systems |
| topic | Analysis of PDEs Dynamical Systems Probability |
| url | https://arxiv.org/abs/2504.09350 |