On a particle approximation to the Dean-Kawasaki type equation with logarithmic interactions
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866909423733047296 |
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| author | Ding, Hao |
| author_facet | Ding, Hao |
| contents | We consider a class of Dean-Kawasaki type equations on $\mathbb{T}$ with logarithmic repulsive interactions depending on the inverse temperature $β$ and a new spectral approximation to the noise part, which approximately features Otto's metric in $\mathbb{P}(\mathbb{T})$. Following the idea of intrinsic constructions of Brownian motions on the Wasserstein space, we construct a class of particle models whose fluctuating hydrodynamic limits, denoted as $p_t^β$, are solutions to the martingale problems of this class of equations. Specifically, we give a quantitative convergence rate of the particle approximation, which allows us to identify a unique limit distribution depending on $β$.
As the inverse temperature rises, the regularizing effect of repulsive interactions becomes stronger. We prove that there exists three thresholds $0<λ_0\leqλ_1<λ_2$ depending on the noise such that, when $β>λ_0$, $p_t^β$ is a non-atomic measure process in $\mathbb{P}(\mathbb{T})$; when $β>λ_1$, $p_t^β$ is absolutely continuous with respect to Lebesgue measure almost surely; when $β>λ_2$, the expectation of the Rényi entropy of $p_t^β$ satisfies an exponential decay estimate. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2204_11309 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On a particle approximation to the Dean-Kawasaki type equation with logarithmic interactions Ding, Hao Probability 60H15, 60G57, 60K35, 82B31 We consider a class of Dean-Kawasaki type equations on $\mathbb{T}$ with logarithmic repulsive interactions depending on the inverse temperature $β$ and a new spectral approximation to the noise part, which approximately features Otto's metric in $\mathbb{P}(\mathbb{T})$. Following the idea of intrinsic constructions of Brownian motions on the Wasserstein space, we construct a class of particle models whose fluctuating hydrodynamic limits, denoted as $p_t^β$, are solutions to the martingale problems of this class of equations. Specifically, we give a quantitative convergence rate of the particle approximation, which allows us to identify a unique limit distribution depending on $β$. As the inverse temperature rises, the regularizing effect of repulsive interactions becomes stronger. We prove that there exists three thresholds $0<λ_0\leqλ_1<λ_2$ depending on the noise such that, when $β>λ_0$, $p_t^β$ is a non-atomic measure process in $\mathbb{P}(\mathbb{T})$; when $β>λ_1$, $p_t^β$ is absolutely continuous with respect to Lebesgue measure almost surely; when $β>λ_2$, the expectation of the Rényi entropy of $p_t^β$ satisfies an exponential decay estimate. |
| title | On a particle approximation to the Dean-Kawasaki type equation with logarithmic interactions |
| topic | Probability 60H15, 60G57, 60K35, 82B31 |
| url | https://arxiv.org/abs/2204.11309 |