Singular SPDEs on Homogeneous Lie Groups
Fuente:
arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866912244624785408 |
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| author | Mayorcas, Avi Singh, Harprit |
| author_facet | Mayorcas, Avi Singh, Harprit |
| contents | The aim of this article is to extend the scope of the theory of regularity structures in order to deal with a large class of singular SPDEs of the form $$\partial_t u = \mathfrak{L} u+ F(u, ξ)\ ,$$ where the differential operator $\mathfrak{L}$ fails to be elliptic. This is achieved by interpreting the base space $\mathbb{R}^{d}$ as a non-trivial homogeneous Lie group $\mathbb{G}$ such that the differential operator $\partial_t -\mathfrak{L}$ becomes a translation invariant hypoelliptic operator on $\mathbb{G}$. Prime examples are the kinetic Fokker-Planck operator $\partial_t -Δ_v - v\cdot \nabla_x$ and heat-type operators associated to sub-Laplacians. As an application of the developed framework, we solve a class of parabolic Anderson type equations $$\partial_t u = \sum_{i} X^2_i u + u (ξ-c)$$ on the compact quotient of an arbitrary Carnot group. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_05121 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Singular SPDEs on Homogeneous Lie Groups Mayorcas, Avi Singh, Harprit Probability Analysis of PDEs 60L30 (Primary), 60H17, 35H10, 35K70 (Secondary) The aim of this article is to extend the scope of the theory of regularity structures in order to deal with a large class of singular SPDEs of the form $$\partial_t u = \mathfrak{L} u+ F(u, ξ)\ ,$$ where the differential operator $\mathfrak{L}$ fails to be elliptic. This is achieved by interpreting the base space $\mathbb{R}^{d}$ as a non-trivial homogeneous Lie group $\mathbb{G}$ such that the differential operator $\partial_t -\mathfrak{L}$ becomes a translation invariant hypoelliptic operator on $\mathbb{G}$. Prime examples are the kinetic Fokker-Planck operator $\partial_t -Δ_v - v\cdot \nabla_x$ and heat-type operators associated to sub-Laplacians. As an application of the developed framework, we solve a class of parabolic Anderson type equations $$\partial_t u = \sum_{i} X^2_i u + u (ξ-c)$$ on the compact quotient of an arbitrary Carnot group. |
| title | Singular SPDEs on Homogeneous Lie Groups |
| topic | Probability Analysis of PDEs 60L30 (Primary), 60H17, 35H10, 35K70 (Secondary) |
| url | https://arxiv.org/abs/2301.05121 |