Malliavin Calculus for the stochastic Cahn-Hilliard equation driven by fractional noise
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| Format: | Preprint |
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2026
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| author | Dimitriou, Dimitrios Farazakis, Dimitris Karali, Georgia |
| author_facet | Dimitriou, Dimitrios Farazakis, Dimitris Karali, Georgia |
| contents | The stochastic partial differential equation analyzed in this work is the Cahn-Hilliard equation perturbed by an additive fractional white noise (fractional in time and white in space). We work in the case of one spatial dimension and apply Malliavin calculus to investigate the existence of a density for the stochastic solution $u$. In particular, we show that $u$ admits continuous paths almost surely and construct a localizing sequence through which we prove that its Malliavin derivative exists locally, and that its law is absolutely continuous with respect to the Lebesgue measure on $\bf R$, establishing thus that a density exists. A key contribution of this work is the analysis of the stochastic integral appearing in the mild formulation: we derive sharp estimates for the expectation of the $p$-th power ($p \geq 2$) of the $L^{\infty}(D)$-norm of this stochastic integral as well as for the integral involving the $L^{\infty}(D)$-norm of the operator associated with the kernel appearing in the integral representation of the fractional noise, all of which are essential for this study. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_10490 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Malliavin Calculus for the stochastic Cahn-Hilliard equation driven by fractional noise Dimitriou, Dimitrios Farazakis, Dimitris Karali, Georgia Probability Analysis of PDEs 35J60, 60H15 The stochastic partial differential equation analyzed in this work is the Cahn-Hilliard equation perturbed by an additive fractional white noise (fractional in time and white in space). We work in the case of one spatial dimension and apply Malliavin calculus to investigate the existence of a density for the stochastic solution $u$. In particular, we show that $u$ admits continuous paths almost surely and construct a localizing sequence through which we prove that its Malliavin derivative exists locally, and that its law is absolutely continuous with respect to the Lebesgue measure on $\bf R$, establishing thus that a density exists. A key contribution of this work is the analysis of the stochastic integral appearing in the mild formulation: we derive sharp estimates for the expectation of the $p$-th power ($p \geq 2$) of the $L^{\infty}(D)$-norm of this stochastic integral as well as for the integral involving the $L^{\infty}(D)$-norm of the operator associated with the kernel appearing in the integral representation of the fractional noise, all of which are essential for this study. |
| title | Malliavin Calculus for the stochastic Cahn-Hilliard equation driven by fractional noise |
| topic | Probability Analysis of PDEs 35J60, 60H15 |
| url | https://arxiv.org/abs/2601.10490 |