Abstract
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 time-changed with an independent (continuous-time) fractional Poisson process. We call this process ‘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 Mittag-Leffler process. The approach of constructing 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.
Highlights
During the last three decades a vast interest in fractional calculus has emerged due to its success to describe so called ‘anomalous phenomena’ such as anomalous transport and diffusion [1,2,3,4,5,6,7], anomalous relaxation in dielectrics [8], creep models [9], and various complex phenomena [10], just to quote a few examples
Whereas classical continuous-time random walk (CTRW) models are random walks subordinated to an independent Poisson process, their fractional generalizations lead to fat-tailed waiting-time Mittag-Leffler type densities with non-Markovian long memory behavior governed by evolution equations of fractional or generalized fractional types [1,4,6,12,13]
In the context of continuous-time renewal processes a Prabhakar generalization of the fractional Poisson process was introduced by Cahoy and Polito [22], applied to stochastic motions on undirected graphs by Michelitsch and Riascos [23,24,25], and discrete-time versions were developed in our recent paper [26]
Summary
During the last three decades a vast interest in fractional calculus has emerged due to its success to describe so called ‘anomalous phenomena’ such as anomalous transport and diffusion [1,2,3,4,5,6,7], anomalous relaxation in dielectrics [8], creep models [9], and various complex phenomena [10], just to quote a few examples. This process is a strictly increasing walk on the integer line with (discrete) Mittag-Leffler jumps separated by the Mittag-Leffler waiting times of the fractional Poisson process. We obtain a biased (forward) diffusion equation of general space-time fractional type referring to the class (2), which is solved by the continuous-space limit density kernel of the state-probabilities This kernel is derived in explicit form involving so called Prabhakar kernels. The latter as well as its diffusion limit are derived in explicit forms where again Prabhakar kernels and general fractional calculus come into play
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