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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2406.04651 |
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| _version_ | 1866916279471833088 |
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| author | Blessing, Alexandra Rosati, Tommaso |
| author_facet | Blessing, Alexandra Rosati, Tommaso |
| contents | 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 non-zero 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_04651 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quantitative instability for stochastic scalar reaction-diffusion equations Blessing, Alexandra Rosati, Tommaso Probability Analysis of PDEs 60H15 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 non-zero 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. |
| title | Quantitative instability for stochastic scalar reaction-diffusion equations |
| topic | Probability Analysis of PDEs 60H15 |
| url | https://arxiv.org/abs/2406.04651 |