Continuous time random walks and the Cauchy problem for the heat equation

2018 ◽  
Vol 136 (1) ◽  
pp. 83-101
Author(s):  
Hugo Aimar ◽  
Gastón Beltritti ◽  
Ivana Gómez
Keyword(s):  
Cauchy Problem ◽  
Heat Equation ◽  
Random Walks ◽  
Continuous Time ◽  
The Cauchy Problem ◽  
Continuous Time Random Walks

scholarly journals Biased Continuous-Time Random Walks with Mittag-Leffler Jumps

2020 ◽  
Vol 4 (4) ◽  
pp. 51 ◽  
Author(s):  
Thomas M. Michelitsch ◽  
Federico Polito ◽  
Alejandro P. Riascos
Keyword(s):  
Cauchy Problem ◽  
Random Walks ◽  
Poisson Process ◽  
Continuous Time ◽  
Laplacian Matrix ◽  
Space Time ◽  
The State ◽  
Diffusion Limit ◽  
Matrix Functions ◽  
Continuous Time Random Walks

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.

scholarly journals Biased Continuous-Time Random Walks with Mittag-Leffler Jumps

2020 ◽  
Author(s):  
Thomas M. Michelitsch ◽  
Federico Polito ◽  
Alejandro P. Riascos
Keyword(s):  
Cauchy Problem ◽  
Random Walks ◽  
Poisson Process ◽  
Continuous Time ◽  
Laplacian Matrix ◽  
Space Time ◽  
State Density ◽  
Diffusion Limit ◽  
Matrix Functions ◽  
Continuous Time Random Walks

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 ‘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.

scholarly journals REVISITING THE DERIVATION OF THE FRACTIONAL DIFFUSION EQUATION

Fractals ◽  
2003 ◽  
Vol 11 (supp01) ◽  
pp. 281-289 ◽  
Author(s):  
ENRICO SCALAS ◽  
RUDOLF GORENFLO ◽  
FRANCESCO MAINARDI ◽  
MARCO RABERTO
Keyword(s):  
Cauchy Problem ◽  
Limit Theorem ◽  
Diffusion Equation ◽  
Continuous Time ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Diffusion Limit ◽  
Straightforward Application ◽  
The Cauchy Problem ◽  
Continuous Time Random Walks

The fractional diffusion equation is derived from the master equation of continuous time random walks (CTRWs) via a straightforward application of the Gnedenko-Kolmogorov limit theorem. The Cauchy problem for the fractional diffusion equation is solved in various important and general cases. The meaning of the proper diffusion limit for CTRWs is discussed.

Heterogeneous Memorized Continuous Time Random Walks in an External Force Fields

2014 ◽  
Vol 156 (6) ◽  
pp. 1111-1124 ◽  
Author(s):  
Jun Wang ◽  
Ji Zhou ◽  
Long-Jin Lv ◽  
Wei-Yuan Qiu ◽  
Fu-Yao Ren
Keyword(s):  
Random Walks ◽  
External Force ◽  
Continuous Time ◽  
Force Fields ◽  
Continuous Time Random Walks

scholarly journals Clustered continuous-time random walks: diffusion and relaxation consequences

2012 ◽  
Vol 468 (2142) ◽  
pp. 1615-1628 ◽  
Author(s):  
Karina Weron ◽  
Aleksander Stanislavsky ◽  
Agnieszka Jurlewicz ◽  
Mark M. Meerschaert ◽  
Hans-Peter Scheffler
Keyword(s):  
Random Walks ◽  
Power Law ◽  
Continuous Time ◽  
Waiting Times ◽  
Scaling Limit ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
The Novel ◽  
Fractional Diffusion Equations ◽  
Continuous Time Random Walks

We present a class of continuous-time random walks (CTRWs), in which random jumps are separated by random waiting times. The novel feature of these CTRWs is that the jumps are clustered. This introduces a coupled effect, with longer waiting times separating larger jump clusters. We show that the CTRW scaling limits are time-changed processes. Their densities solve two different fractional diffusion equations, depending on whether the waiting time is coupled to the preceding jump, or the following one. These fractional diffusion equations can be used to model all types of experimentally observed two power-law relaxation patterns. The parameters of the scaling limit process determine the power-law exponents and loss peak frequencies.

scholarly journals An Application Of The Theory Of Scale Of Banach Spaces

2015 ◽  
Vol 29 (1) ◽  
pp. 51-59
Author(s):  
Łukasz Dawidowski
Keyword(s):  
Banach Space ◽  
Cauchy Problem ◽  
Boundary Conditions ◽  
Heat Equation ◽  
Banach Spaces ◽  
Abstract Cauchy Problem ◽  
Initial Boundary ◽  
Scale Of Banach Spaces ◽  
The Cauchy Problem ◽  
Selection Of

AbstractThe abstract Cauchy problem on scales of Banach space was considered by many authors. The goal of this paper is to show that the choice of the space on scale is significant. We prove a theorem that the selection of the spaces in which the Cauchy problem ut − Δu = u|u|s with initial–boundary conditions is considered has an influence on the selection of index s. For the Cauchy problem connected with the heat equation we will study how the change of the base space influents the regularity of the solutions.

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