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Non-Markovian chains with long-range dependence and their scaling limits

Lorenzo Facciaroni, Costantino Ricciuti, Enrico Scalas

Abstract

There is a well-established theory linking certain semi-Markov chains and continuous-time random walks to time-fractional equations and anomalous diffusion. In this work, we go beyond the semi-Markov framework by considering some non-Markovian chains, which exhibit long-memory behaviour, due to stochastic dependence among their waiting times. Particular attention is devoted to the so-called para-Markov chains. Their waiting times share the same marginal distributions as those of the above mentioned semi-Markov chains, but they are dependent; their joint distribution is of Schur-constant type and is closely related to complete Bernstein functions and De Finetti's theorems. A second model that we focus on is given by time-changed Markov chains, where the random time is the inverse of an increasing stable process. This generalizes well-known semi-Markov models available in the literature, which typically focus solely on the inverse of the Levy stable subordinator. The above mentioned models are unified by a general theory of time change of Markov chains.

Non-Markovian chains with long-range dependence and their scaling limits

Abstract

There is a well-established theory linking certain semi-Markov chains and continuous-time random walks to time-fractional equations and anomalous diffusion. In this work, we go beyond the semi-Markov framework by considering some non-Markovian chains, which exhibit long-memory behaviour, due to stochastic dependence among their waiting times. Particular attention is devoted to the so-called para-Markov chains. Their waiting times share the same marginal distributions as those of the above mentioned semi-Markov chains, but they are dependent; their joint distribution is of Schur-constant type and is closely related to complete Bernstein functions and De Finetti's theorems. A second model that we focus on is given by time-changed Markov chains, where the random time is the inverse of an increasing stable process. This generalizes well-known semi-Markov models available in the literature, which typically focus solely on the inverse of the Levy stable subordinator. The above mentioned models are unified by a general theory of time change of Markov chains.
Paper Structure (14 sections, 10 theorems, 98 equations, 1 figure)

This paper contains 14 sections, 10 theorems, 98 equations, 1 figure.

Table of Contents

  1. Introduction and preliminaries
  2. State of art and vision
  3. Models and main results
  4. Para-Markov chains and para-Markov anomalous diffusive processes
  5. The counting process and related chains
  6. Heuristic derivation of governing equations
  7. Governing equations
  8. Functional convergence to anomalous diffusion
  9. Beyond semi-Markov and para-Markov chains
  10. Chains with dependent waiting times
  11. The case of $\sigma$ with stationary increments
  12. Time change by inverses of increasing stable processes
  13. Self-similar stable processes
  14. Remarkable examples

Key Result

Theorem 2.1

Let $H$ be a strictly increasing subordinator with Laplace exponent $f$ and let $\{L(t),\ t\in\mathbb{R}_+\}$ be the inverse process of $H$. Then

Figures (1)

  • Figure 1: Comparison between different time changes: $\sigma$ is an increasing non-negative process and $(J_1, \ldots, J_n)$ is the vector of the first $n$ waiting times of the time changed process. The survival function of each $J_k$ is $S(\cdot)$ and $f(\cdot)$ is the Bernstein function associated to a subordinator $H$ with inverse $L$, such that $S(t) = \mathbb{E} e^{-\lambda t}$.

Theorems & Definitions (33)

  • Theorem 2.1
  • Example 2.2
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.5
  • proof
  • Corollary 2.6
  • proof
  • Remark 2.7
  • Remark 2.8
  • ...and 23 more