论文标题
在与对数相互作用的Dean-Kawasaki型方程与Dean-Kawasaki型方程式上
On a particle approximation to the Dean-Kawasaki type equation with logarithmic interactions
论文作者
论文摘要
我们考虑$ \ mathbb {t} $上的一类Dean-Kawasaki类型方程，并具有对数的排斥相互作用，具体取决于反向温度$β$以及与噪声部分的新频谱近似，该频谱近似近似于Otto的$ \ Mathbb {p}（\ Mathbbbbbb {t}）$。遵循瓦斯坦斯坦空间上布朗尼运动的内在结构的想法，我们构建了一类粒子模型，这些模型的波动流体动力极限（称为$ p_t^β$）是解决此类方程式的玛格尔问题的解决方案。具体而言，我们给出了粒子近似的定量收敛速率，这使我们能够根据$β$确定唯一的极限分布。 随着逆温度的上升，排斥相互作用的正则化效应变得更强烈。我们证明存在三个阈值$ 0<λ_0\leqλ_1<λ_2$，具体取决于噪声，以至于$β>λ_0$，$ p_t^β$是$ \ mathbb {p}（p}（\ mathbb {t} t}），当$β>λ_1$时，相对于Lebesgue的测量，$ P_T^β$绝对是连续的。当$β>λ_2$时，$ p_t^β$rényi熵的期望满足了指数衰减估计。
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.
