Functional convergence of continuous-time random walks with continuous paths

Author(s):  
Marcin Magdziarz ◽  
Piotr Zebrowski
2014 ◽  
Vol 156 (6) ◽  
pp. 1111-1124 ◽  
Author(s):  
Jun Wang ◽  
Ji Zhou ◽  
Long-Jin Lv ◽  
Wei-Yuan Qiu ◽  
Fu-Yao Ren

Author(s):  
Karina Weron ◽  
Aleksander Stanislavsky ◽  
Agnieszka Jurlewicz ◽  
Mark M. Meerschaert ◽  
Hans-Peter Scheffler

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.


2011 ◽  
Vol 43 (3) ◽  
pp. 782-813 ◽  
Author(s):  
M. Jara ◽  
T. Komorowski

In this paper we consider the scaled limit of a continuous-time random walk (CTRW) based on a Markov chain {Xn,n≥ 0} and two observables, τ(∙) andV(∙), corresponding to the renewal times and jump sizes. Assuming that these observables belong to the domains of attraction of some stable laws, we give sufficient conditions on the chain that guarantee the existence of the scaled limits for CTRWs. An application of the results to a process that arises in quantum transport theory is provided. The results obtained in this paper generalize earlier results contained in Becker-Kern, Meerschaert and Scheffler (2004) and Meerschaert and Scheffler (2008), and the recent results of Henry and Straka (2011) and Jurlewicz, Kern, Meerschaert and Scheffler (2010), where {Xn,n≥ 0} is a sequence of independent and identically distributed random variables.


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