The Gaussian free-field as a stream function: continuum version of the scale-by-scale homogenization result
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2024
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| _version_ | 1866915630078230528 |
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| author | Morfe, Peter Otto, Felix Wagner, Christian |
| author_facet | Morfe, Peter Otto, Felix Wagner, Christian |
| contents | This note is about a drift-diffusion process $X$ with a time-independent, divergence-free drift $b$, where $b$ is a smooth Gaussian field that decorrelates over large scales. In two space dimensions, this just fails to fall into the standard theory of stochastic homogenization, and leads to a borderline super-diffusive behavior. In a previous paper by Chatzigeorgiou, Morfe, Otto, and Wang (2022), precise asymptotics of the annealed second moments of $X$ were derived by characterizing the asymptotics of the effective diffusivity $λ_L$ in terms of an artificially introduced large-scale cut-off $L$. The latter was carried out by a scale-by-scale homogenization, and implemented by monitoring the corrector $ϕ_L$ for geometrically increasing cut-off scales $L^+=ML$. In fact, proxies $(\tildeϕ_L,\tildeσ_L)$ for the corrector and flux corrector were introduced incrementally and the residuum $f_L$ estimated.
In this short supplementary note, we reproduce the arguments of the above paper in the continuum setting of $M\downarrow 1$. This has the advantage that the definition of the proxies $(\tildeϕ_L,\tildeσ_L)$ becomes more transparent -- it is given by a simple Itô SDE with $\ln L$ acting as a time variable. It also has the advantage that the residuum $f_L$, which is a martingale, can be efficiently and precisely estimated by Itô calculus. This relies on the characterization of the quadratic variation of the (infinite-dimensional) Gaussian driver. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_00709 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Gaussian free-field as a stream function: continuum version of the scale-by-scale homogenization result Morfe, Peter Otto, Felix Wagner, Christian Probability Analysis of PDEs This note is about a drift-diffusion process $X$ with a time-independent, divergence-free drift $b$, where $b$ is a smooth Gaussian field that decorrelates over large scales. In two space dimensions, this just fails to fall into the standard theory of stochastic homogenization, and leads to a borderline super-diffusive behavior. In a previous paper by Chatzigeorgiou, Morfe, Otto, and Wang (2022), precise asymptotics of the annealed second moments of $X$ were derived by characterizing the asymptotics of the effective diffusivity $λ_L$ in terms of an artificially introduced large-scale cut-off $L$. The latter was carried out by a scale-by-scale homogenization, and implemented by monitoring the corrector $ϕ_L$ for geometrically increasing cut-off scales $L^+=ML$. In fact, proxies $(\tildeϕ_L,\tildeσ_L)$ for the corrector and flux corrector were introduced incrementally and the residuum $f_L$ estimated. In this short supplementary note, we reproduce the arguments of the above paper in the continuum setting of $M\downarrow 1$. This has the advantage that the definition of the proxies $(\tildeϕ_L,\tildeσ_L)$ becomes more transparent -- it is given by a simple Itô SDE with $\ln L$ acting as a time variable. It also has the advantage that the residuum $f_L$, which is a martingale, can be efficiently and precisely estimated by Itô calculus. This relies on the characterization of the quadratic variation of the (infinite-dimensional) Gaussian driver. |
| title | The Gaussian free-field as a stream function: continuum version of the scale-by-scale homogenization result |
| topic | Probability Analysis of PDEs |
| url | https://arxiv.org/abs/2404.00709 |