Canonical solutions to non-translation invariant singular SPDEs
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866915198719229952 |
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| author | Singh, Harprit |
| author_facet | Singh, Harprit |
| contents | We exhibit a canonical, finite dimensional solution family to certain singular SPDEs of the form \begin{equation} \left(\partial_t- \sum_{i,j=1}^d a_{i,j}(x,t) \partial_i \partial_j - \sum_{i=1}^d b_i(x,t) \partial_i - c(x,t)\right) u = F(u, \partial u, ξ) \ , \end{equation} where $a_{i,j}, b_i, c: \mathbb{T}^d\times \mathbb{R} \to \mathbb{R}$ and $A=\{a_{i,j}\}_{i,j=1}^d$ is uniformly elliptic. More specifically, we solve the non-translation invariant g-PAM, $ϕ^4_2$, $ϕ^4_3$ and KPZ-equation and show that the diverging renormalisation-functions are local functions of $A$. We also establish a continuity result for the solution map with respect to the differential operator for these equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_01085 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Canonical solutions to non-translation invariant singular SPDEs Singh, Harprit Analysis of PDEs Probability 60H15, 60L30 We exhibit a canonical, finite dimensional solution family to certain singular SPDEs of the form \begin{equation} \left(\partial_t- \sum_{i,j=1}^d a_{i,j}(x,t) \partial_i \partial_j - \sum_{i=1}^d b_i(x,t) \partial_i - c(x,t)\right) u = F(u, \partial u, ξ) \ , \end{equation} where $a_{i,j}, b_i, c: \mathbb{T}^d\times \mathbb{R} \to \mathbb{R}$ and $A=\{a_{i,j}\}_{i,j=1}^d$ is uniformly elliptic. More specifically, we solve the non-translation invariant g-PAM, $ϕ^4_2$, $ϕ^4_3$ and KPZ-equation and show that the diverging renormalisation-functions are local functions of $A$. We also establish a continuity result for the solution map with respect to the differential operator for these equations. |
| title | Canonical solutions to non-translation invariant singular SPDEs |
| topic | Analysis of PDEs Probability 60H15, 60L30 |
| url | https://arxiv.org/abs/2310.01085 |