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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2511.20556 |
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| _version_ | 1866912729000837120 |
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| author | Mayorcas, Avi Mądry, Łukasz |
| author_facet | Mayorcas, Avi Mądry, Łukasz |
| contents | We show existence and uniqueness of invariant measures for SDE of the form \[
dX_t = g(X_t)dt + u(X_t)dt + dW^H_t \]
where $W^H$ is a fractional Brownian motion (fBm) with Hurst parameter $H\in (0,\frac{1}{2})$, $u$ is a linearly dispersive term and $g$ is any $B^α_{\infty,\infty}(\mathbb{R}^d)$ distribution in the class treated by Catellier--Gubinelli `16, i.e. $α>1-\frac{1}{2H}$. The significant challenge is to combine the regularizing effect of the fBm with an ergodic theory suited to non-Markovian SDE. Concerning the latter our first main contribution is to construct a bona fide stochastic dynamical system (SDS) (Hairer `05 and Hairer--Ohashi `07) associated to the equation above. Since the solution map is only continuous in the support of the stationary noise process we weaken the definitions introduced by Hairer `05 and Hairer--Ohashi `07 but manage to retain the Doob--K'hashminksii provided by Hairer--Ohashi `07. Our second innovation is to introduce a family of flexible local-global stochastic sewing lemmas, in the vein of Lê `20, which allows us to efficiently treat small and large scales simultaneously. By tuning the local scale as a function of $\|g\|_{B^α_{\infty,\infty}}$ we are able to obtain the necessary continuity of the semi-group and stability estimates to show unique ergodicity for all $g\in B^α_{\infty,\infty}(\mathbb{R}^d)$. We believe that these local-global sewing lemmas may be of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_20556 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Ergodic Theory for Fractional SDE with Singular Coefficients Mayorcas, Avi Mądry, Łukasz Probability We show existence and uniqueness of invariant measures for SDE of the form \[ dX_t = g(X_t)dt + u(X_t)dt + dW^H_t \] where $W^H$ is a fractional Brownian motion (fBm) with Hurst parameter $H\in (0,\frac{1}{2})$, $u$ is a linearly dispersive term and $g$ is any $B^α_{\infty,\infty}(\mathbb{R}^d)$ distribution in the class treated by Catellier--Gubinelli `16, i.e. $α>1-\frac{1}{2H}$. The significant challenge is to combine the regularizing effect of the fBm with an ergodic theory suited to non-Markovian SDE. Concerning the latter our first main contribution is to construct a bona fide stochastic dynamical system (SDS) (Hairer `05 and Hairer--Ohashi `07) associated to the equation above. Since the solution map is only continuous in the support of the stationary noise process we weaken the definitions introduced by Hairer `05 and Hairer--Ohashi `07 but manage to retain the Doob--K'hashminksii provided by Hairer--Ohashi `07. Our second innovation is to introduce a family of flexible local-global stochastic sewing lemmas, in the vein of Lê `20, which allows us to efficiently treat small and large scales simultaneously. By tuning the local scale as a function of $\|g\|_{B^α_{\infty,\infty}}$ we are able to obtain the necessary continuity of the semi-group and stability estimates to show unique ergodicity for all $g\in B^α_{\infty,\infty}(\mathbb{R}^d)$. We believe that these local-global sewing lemmas may be of independent interest. |
| title | Ergodic Theory for Fractional SDE with Singular Coefficients |
| topic | Probability |
| url | https://arxiv.org/abs/2511.20556 |