A mild rough Gronwall Lemma with applications to non-autonomous evolution equations
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866912674488516608 |
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| author | Blessing, Alexandra Varzaneh, Mazyar Ghani Seitz, Tim |
| author_facet | Blessing, Alexandra Varzaneh, Mazyar Ghani Seitz, Tim |
| contents | We derive a Gronwall type inequality for mild solutions of non-autonomous parabolic rough partial differential equations (RPDEs). This inequality together with an analysis of the Cameron-Martin space associated to the noise, allows us to obtain the existence of moments of all order for the solution of the corresponding RPDE and its Jacobian when the random input is given by a Gaussian Volterra process. Applying further the multiplicative ergodic theorem, these integrable bounds entail the existence of Lyapunov exponents for RPDEs. We illustrate these results for stochastic partial differential equations with multiplicative boundary noise. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_03628 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A mild rough Gronwall Lemma with applications to non-autonomous evolution equations Blessing, Alexandra Varzaneh, Mazyar Ghani Seitz, Tim Probability Analysis of PDEs Dynamical Systems 60G22, 60L20, 60L50, 37H05, 37L55 We derive a Gronwall type inequality for mild solutions of non-autonomous parabolic rough partial differential equations (RPDEs). This inequality together with an analysis of the Cameron-Martin space associated to the noise, allows us to obtain the existence of moments of all order for the solution of the corresponding RPDE and its Jacobian when the random input is given by a Gaussian Volterra process. Applying further the multiplicative ergodic theorem, these integrable bounds entail the existence of Lyapunov exponents for RPDEs. We illustrate these results for stochastic partial differential equations with multiplicative boundary noise. |
| title | A mild rough Gronwall Lemma with applications to non-autonomous evolution equations |
| topic | Probability Analysis of PDEs Dynamical Systems 60G22, 60L20, 60L50, 37H05, 37L55 |
| url | https://arxiv.org/abs/2503.03628 |