Multidimensional Stability of Planar Travelling Waves for Stochastically Perturbed Reaction-Diffusion Systems
arxiv(2024)
摘要
We consider reaction-diffusion systems with multiplicative noise on a spatial
domain of dimension two or higher. The noise process is white in time, coloured
in space, and invariant under translations. In the deterministic setting,
multidimensional stability of planar waves on the whole space ℝ^d has
been studied by many. Inspired by previous works on the real line, we establish
the multidimensional stability of planar waves on a cylindrical domain on time
scales 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 time scales. 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.
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