亚扩散年龄结构更新方程中自相似轮廓的定量收敛性
Quantitative convergence towards a self similar profile in an age-structured renewal equation for subdiffusion
摘要：
连续时间随机游动是随机游动的概括，经常用于解释一致的观察结果，即活细胞中的许多分子经历异常扩散，即亚扩散。在这里，我们描述的subdiffusively连续时间随机游动使用年龄结构的偏微分方程与年龄更新后，每个步行者跳，其中的年龄的步行者是经过的时间，因为它的最后一次跳跃。在空间均匀（零维）的情况下，我们遵循年龄分布的时间演化。受相对熵技术启发的方法使我们能够获得年龄分布收敛到自相似轮廓的定量显式速率，这对应于收敛到重新缩放变量的stationnary轮廓。一个重要的困难来自于这样一个事实，即自相似变量的方程不是自治的，我们没有一个特定的解析解。因此，为了量化后者的收敛，我们估计吸引力的时间依赖性的“伪平衡”，这反过来又收敛到stationnary配置文件。
英文摘要：
Continuous-time random walks are generalisations of random walks frequently used to account for the consistent observations that many molecules in living cells undergo anomalous diffusion, i.e. subdiffusion. Here, we describe the subdiffusive continuous-time random walk using age-structured partial differential equations with age renewal upon each walker jump, where the age of a walker is the time elapsed since its last jump. In the spatially-homogeneous (zero-dimensional) case, we follow the evolution in time of the age distribution. An approach inspired by relative entropy techniques allows us to obtain quantitative explicit rates for the convergence of the age distribution to a self-similar profile, which corresponds to convergence to a stationnary profile for the rescaled variables. An important difficulty arises from the fact that the equation in self-similar variables is not autonomous and we do not have a specific analyitcal solution. Therefore, in order to quantify the latter convergence, we estimate attraction to a time-dependent "pseudo-equilibrium", which in turn converges to the stationnary profile.
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