产生非马尔可夫正态扩散的薄膜边界条件
Boundary conditions at a thin membrane that generate non--Markovian normal diffusion
摘要：
我们表明，一些边界条件假设在一个薄膜可能会导致正常的扩散不是随机马尔可夫过程。我们认为边界条件定义的拉普拉斯变换，其中有一个线性组合的概率和概率通量上定义的两个膜表面。组合的系数可取决于拉普拉斯变换参数。当考虑膜系统中的扩散时，这种边界条件是最常用的，除非粒子扩散中发生集体或非局部过程。利用Laplace变换找到了BSCK方程，并给出了判断边界条件是否存在满足该方程的扩散方程基本解的判据.如果不满足BSCK方程，则马尔可夫性质被破坏。当概率流在膜上连续时，导出了基本解满足BSCK方程的边界条件的一般形式。给出了半群性质破缺的一个度量。讨论了该测度与非马尔可夫性测度的关系。
英文摘要：
We show that some boundary conditions assumed at a thin membrane may result in normal diffusion not being the stochastic Markov process. We consider boundary conditions defined in terms of the Laplace transform in which there is a linear combination of probabilities and probability fluxes defined on both membrane surfaces. The coefficients of the combination may depend on the Laplace transform parameter. Such boundary conditions are most commonly used when considering diffusion in a membrane system unless collective or non-local processes in particles diffusion occur. We find Bachelier-Smoluchowski-Chapmann-Kolmogorov (BSCK) equation in terms of the Laplace transform and we derive the criterion to check whether the boundary conditions lead to fundamental solutions of diffusion equation satisfying this equation. If the BSCK equation is not met, the Markov property is broken. When a probability flux is continuous at the membrane, the general forms of the boundary conditions for which the fundamental solutions meet the BSCK equation are derived. A measure of broken of semi-group property is also proposed. The relation of this measure to the non-Markovian property measure is discussed.
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