Flow equation approach to singular stochastic PDEs
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866908273256431616 |
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| author | Duch, Paweł |
| author_facet | Duch, Paweł |
| contents | We prove universality of a macroscopic behavior of solutions of a large class of semi-linear parabolic SPDEs on $\mathbb{R}_+\times\mathbb{T}$ with fractional Laplacian $(-Δ)^{σ/2}$, additive noise and polynomial non-linearity, where $\mathbb{T}$ is the $d$-dimensional torus. We consider the weakly non-linear regime and not necessarily Gaussian noises which are stationary, centered, sufficiently regular and satisfy some integrability and mixing conditions. We prove that the macroscopic scaling limit exists and has a universal law characterized by parameters of the relevant perturbations of the linear equation. We develop a new solution theory for singular SPDEs of the above-mentioned form using the Wilsonian renormalization group theory and the Polchinski flow equation. In particular, in the case of $d=4$ and the cubic non-linearity our analysis covers the whole sub-critical regime $σ>2$. Our technique avoids completely all the algebraic and combinatorial problems arising in different approaches. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2109_11380 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Flow equation approach to singular stochastic PDEs Duch, Paweł Probability Mathematical Physics 60H17, 81T17 We prove universality of a macroscopic behavior of solutions of a large class of semi-linear parabolic SPDEs on $\mathbb{R}_+\times\mathbb{T}$ with fractional Laplacian $(-Δ)^{σ/2}$, additive noise and polynomial non-linearity, where $\mathbb{T}$ is the $d$-dimensional torus. We consider the weakly non-linear regime and not necessarily Gaussian noises which are stationary, centered, sufficiently regular and satisfy some integrability and mixing conditions. We prove that the macroscopic scaling limit exists and has a universal law characterized by parameters of the relevant perturbations of the linear equation. We develop a new solution theory for singular SPDEs of the above-mentioned form using the Wilsonian renormalization group theory and the Polchinski flow equation. In particular, in the case of $d=4$ and the cubic non-linearity our analysis covers the whole sub-critical regime $σ>2$. Our technique avoids completely all the algebraic and combinatorial problems arising in different approaches. |
| title | Flow equation approach to singular stochastic PDEs |
| topic | Probability Mathematical Physics 60H17, 81T17 |
| url | https://arxiv.org/abs/2109.11380 |