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Quantitative instability for stochastic scalar reaction-diffusion equations

Alexandra Blessing & Tommaso Rosati

Annals of Applied Probability2026https://doi.org/10.1214/25-aap2255article
ABDC A
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0.50

Abstract

This work studies the instability of stochastic scalar reaction diffusion equations, driven by a multiplicative noise that is white in time and smooth in space, near to zero, which is assumed to be a fixed point for the equation. We prove that if the Lyapunov exponent at zero is positive, then the flow of nonzero solutions admits uniform bounds on small negative moments. The proof builds on ideas from stochastic homogenisation. We require suitable corrector estimates for the solution to a Poisson problem involving an infinite-dimensional projective process, together with a linearisation step that hinges on quantitative parametrix-like arguments. Overall, we are able to construct an appropriate Lyapunov functional for the nonlinear dynamics and address some problems left open in the literature.

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https://doi.org/https://doi.org/10.1214/25-aap2255

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@article{alexandra2026,
  title        = {{Quantitative instability for stochastic scalar reaction-diffusion equations}},
  author       = {Alexandra Blessing & Tommaso Rosati},
  journal      = {Annals of Applied Probability},
  year         = {2026},
  doi          = {https://doi.org/https://doi.org/10.1214/25-aap2255},
}

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Quantitative instability for stochastic scalar reaction-diffusion equations

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