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July 2014 Semi-Markov approach to continuous time random walk limit processes
Mark M. Meerschaert, Peter Straka
Ann. Probab. 42(4): 1699-1723 (July 2014). DOI: 10.1214/13-AOP905


Continuous time random walks (CTRWs) are versatile models for anomalous diffusion processes that have found widespread application in the quantitative sciences. Their scaling limits are typically non-Markovian, and the computation of their finite-dimensional distributions is an important open problem. This paper develops a general semi-Markov theory for CTRW limit processes in $\mathbb{R}^{d}$ with infinitely many particle jumps (renewals) in finite time intervals. The particle jumps and waiting times can be coupled and vary with space and time. By augmenting the state space to include the scaling limits of renewal times, a CTRW limit process can be embedded in a Markov process. Explicit analytic expressions for the transition kernels of these Markov processes are then derived, which allow the computation of all finite dimensional distributions for CTRW limits. Two examples illustrate the proposed method.


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Mark M. Meerschaert. Peter Straka. "Semi-Markov approach to continuous time random walk limit processes." Ann. Probab. 42 (4) 1699 - 1723, July 2014.


Published: July 2014
First available in Project Euclid: 3 July 2014

zbMATH: 1305.60089
MathSciNet: MR3262490
Digital Object Identifier: 10.1214/13-AOP905

Primary: 60F17 , 60K15
Secondary: 60K20

Keywords: Anomalous diffusion , Continuous time random walk , Functional limit theorem , Lévy process , renewal theory , semi-Markov process , time-change

Rights: Copyright © 2014 Institute of Mathematical Statistics


Vol.42 • No. 4 • July 2014
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