Quantitative Convergence Towards a Self-Similar Profile in an Age-Structured Renewal Equation for Subdiffusion
Quantitative Convergence Towards a Self-Similar Profile in an Age-Structured Renewal Equation for Subdiffusion
复制标题
年龄结构次扩散更新方程中自相似轮廓的定量收敛
DOI：
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
Álvaro Mateos González
中科院分区：
文献类型：
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作者：
Hugues Berry;T. Lepoutre;Álvaro Mateos González
连续时间随机游走是随机游走的推广，经常被用来解释一致的观察结果，即活细胞中的许多分子经历了异常扩散，即次扩散。在这里，我们使用年龄结构的偏微分方程组来描述次扩散的连续时间随机游动，其中步行者的年龄是自其最后一次跳跃以来经过的时间。在空间均匀(零维)的情况下，我们跟踪年龄分布的时间演变。一种受相对熵技术启发的方法允许我们获得年龄分布收敛到自相似分布的定量显式速率，该自相似分布对应于重新标度变量的收敛到平稳分布。一个重要的困难来自这样一个事实，即自相似变量中的方程不是自治的，我们没有具体的解析解。因此，为了量化后一种收敛，我们估计了依赖于时间的“伪平衡”的吸引性，而伪平衡又收敛于平稳分布。
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 stationary 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 analytical solution. Therefore, in order to quantify the latter convergence, we estimate attraction to a time-dependent “pseudo-equilibrium”, which in turn converges to the stationary profile.