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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2405.05778 |
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| _version_ | 1866911871562416128 |
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| author | Yang, Huanyu Yang, Zhilin |
| author_facet | Yang, Huanyu Yang, Zhilin |
| contents | In this article, we study the weak coupling limit of the following equation in $\mathbb{R}^2$: $$dX_t^\varepsilon=\frac{\hatλ}{\sqrt{\log\frac1\varepsilon}}ω^\varepsilon(X_t^\varepsilon)dt+νdB_t,\quad X_0^\varepsilon=0. $$ Here $ω^\varepsilon=\nabla^{\perp}ρ_\varepsilon*ξ$ with $ξ$ representing the $2d$ Gaussian Free Field (GFF) and $ρ_\varepsilon$ denoting an appropriate identity. $B_t$ denotes a two-dimensional standard Brownian motion, and $\hatλ,ν>0$ are two given constants. We use the approach from \cite{Cannizzaro.2023} to show that the second moment of $X_t^\varepsilon$ under the annealed law converges to $(c(ν)^2+2ν^2)t$ with a precisely determined constant $c(ν)>0$, which implies a non-trivial limit of the drift terms as $\varepsilon$ vanishes. We also prove that in this weak coupling regime, the sequence of solutions converges in distribution to $\left(\sqrt{\frac{c(ν)^2}{2}+ν^2}\right)\widetilde{B}_t$ as $\varepsilon$ vanishes, where $\widetilde{B}_t$ is a two-dimensional standard Brownian motion. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_05778 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Weak coupling limit of a Brownian particle in the curl of the 2D GFF Yang, Huanyu Yang, Zhilin Probability In this article, we study the weak coupling limit of the following equation in $\mathbb{R}^2$: $$dX_t^\varepsilon=\frac{\hatλ}{\sqrt{\log\frac1\varepsilon}}ω^\varepsilon(X_t^\varepsilon)dt+νdB_t,\quad X_0^\varepsilon=0. $$ Here $ω^\varepsilon=\nabla^{\perp}ρ_\varepsilon*ξ$ with $ξ$ representing the $2d$ Gaussian Free Field (GFF) and $ρ_\varepsilon$ denoting an appropriate identity. $B_t$ denotes a two-dimensional standard Brownian motion, and $\hatλ,ν>0$ are two given constants. We use the approach from \cite{Cannizzaro.2023} to show that the second moment of $X_t^\varepsilon$ under the annealed law converges to $(c(ν)^2+2ν^2)t$ with a precisely determined constant $c(ν)>0$, which implies a non-trivial limit of the drift terms as $\varepsilon$ vanishes. We also prove that in this weak coupling regime, the sequence of solutions converges in distribution to $\left(\sqrt{\frac{c(ν)^2}{2}+ν^2}\right)\widetilde{B}_t$ as $\varepsilon$ vanishes, where $\widetilde{B}_t$ is a two-dimensional standard Brownian motion. |
| title | Weak coupling limit of a Brownian particle in the curl of the 2D GFF |
| topic | Probability |
| url | https://arxiv.org/abs/2405.05778 |