论文标题
退化抛物线方程的熵耗散轨迹方法
A trajectorial approach to entropy dissipation for degenerate parabolic equations
论文作者
论文摘要
我们考虑了一个有界域上的$ \ partial_tp_t =ΔF（p_t）$的脱位扩散方程，并遵守不升华边界条件，对于包括多孔介质方程的一类非线性$ f $。我们为他们得出了熵耗散身份的轨迹类似物，该轨道耗散身份描述了沿着扩散的每个路径的熵耗散速率。我们的方法基于将随机演算应用于基本的概率表示，在我们的上下文中是随机微分方程，在边界上具有正常反射。这种轨迹方法还导致了非线性扩散的Wasserstein梯度流属性的新推导，以及在当前情况下HWI不平等的简单证明。
We consider degenerate diffusion equations of the form $\partial_tp_t = Δf(p_t)$ on a bounded domain and subject to no-flux boundary conditions, for a class of nonlinearities $f$ that includes the porous medium equation. We derive for them a trajectorial analogue of the entropy dissipation identity, which describes the rate of entropy dissipation along every path of the diffusion. Our approach is based on applying stochastic calculus to the underlying probabilistic representations, which in our context are stochastic differential equations with normal reflection on the boundary. This trajectorial approach also leads to a new derivation of the Wasserstein gradient flow property for nonlinear diffusions, as well as to a simple proof of the HWI inequality in the present context.
