论文标题
有偏见的连续时间随机步行与Mittag-Leffler跳跃
Biased continuous-time random walks with Mittag-Leffler jumps
论文作者
论文摘要
我们构建可允许的循环laplacian矩阵作为发电机，以严格增加整数上的随机步行。这些Laplacian基质函数是指某种类别的Bernstein函数。该方法与挖掘的有偏见的步行有连接。在此框架内，我们引入了对泊松过程的时空概括，因为它严格增加了步行，而离散的mittag-leffler跳则从属于（连续的）分数泊松过程。我们称此过程为`{\ it时空Mittag-Leffler Process}'。我们为状态概率提供了明确的公式，该状态概率通过kolmogorov-feller（正向）的差异分数方程来解决cauchy问题。我们分析了一个“膨胀”的扩散极限，并通过涉及Mittag-Leffler密度的时空卷积方程获得了库奇问题。我们在此限制的“状态密度内核”解决了这个问题。事实证明，扩散极限与Prabhakar一般分数演算有连接。我们还以这种方式分析了时空分数Mittag-Leffler过程的概括。良好的拉普拉斯发电机功能的构建方法在泊松过程的时空概括以及在挖掘上连续的随机步行领域具有很大的潜力。
We construct admissible circulant Laplacian matrix functions as generators for strictly increasing random walks on the integer line. These Laplacian matrix functions refer to a certain class of Bernstein functions. The approach has connections with biased walks on digraphs. Within this framework, we introduce a space-time generalization of the Poisson process as a strictly increasing walk with discrete Mittag-Leffler jumps subordinated to a (continuous-time) fractional Poisson process. We call this process `{\it space-time Mittag-Leffler process}'. We derive explicit formulae for the state probabilities which solve a Cauchy problem with a Kolmogorov-Feller (forward) difference-differential equation of general fractional type. We analyze a `well-scaled' diffusion limit and obtain a Cauchy problem with a space-time convolution equation involving Mittag-Leffler densities. We deduce in this limit the `state density kernel' solving this Cauchy problem. It turns out that the diffusion limit exhibits connections to Prabhakar general fractional calculus. We also analyze in this way a generalization of the space-time fractional Mittag-Leffler process. The approach of construction of good Laplacian generator functions has a large potential in applications of space-time generalizations of the Poisson process and in the field of continuous-time random walks on digraphs.
