AI 中文总结
研究扩展概率测度空间上的一维扩散模型，通过重排随机热方程并加入熵驱动梯度下降项惩罚动力学，经分裂论证证明方程定义良好，还研究了相关动力学性质及解的密度等，保留了随机热方程的平滑性质。
AI 中文摘要
我们扩展了先前在概率测度空间上引入的一维扩散模型，该模型通过重排随机热方程定义，通过用额外的熵驱动梯度下降项惩罚动力学来实现。通过分裂论证，我们证明尽管重排和熵最小化有相反作用，但得到的惩罚随机热方程是定义良好的。我们研究了相关动力学的几个性质，特别表明解具有满足修正版Dean - Kawasaki方程的密度。随机热方程建立的平滑性质得以保留。
英文摘要
We extend a previously introduced one-dimensional diffusion model on the space of probability measures, defined via the rearranged stochastic heat equation by, penalizing the dynamics with an additional entropy-driven gradient-descent term. By means of a splitting argument, we prove that despite the opposite effects of rearrangement and entropy minimization, the resulting penalized stochastic heat equation is well defined. We study several properties of the associated dynamics and show, in particular, that solutions admit a density satisfying a corrected version of the Dean--Kawasaki equation. The smoothing properties established for the stochastic heat equation are shown to persist.
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