Limit theorems for some continuous-time random walks

Abstract

In this paper we consider the scaled limit of a continuous-time random walk (CTRW) based on a Markov chain {X n , n ≥ 0} and two observables, τ (·) and V (·), 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 andStraka (2011) andJurlewicz, Kern, Meerschaert andScheffler (2010), where {X n , n ≥ 0} is a sequence of independent and identically distributed random variables.

3/4-Fractional Superdiffusion in a System of Harmonic Oscillators Perturbed by a Conservative Noise

“…The latter can be interpreted as the evolution of the density of a Markov process. In [13], [15], the authors study the long time behavior of additive functionals of this Markov process and deduce that the long-time, large-scale limit of the solution of the previous Boltzmann equation converges to the solution of the fractional heat equation:…”

Scaling limits of coupled continuous time random walks and residual order statistics through marked point processes

“…Coupling methods different from forward-/backward-coupling can be found in [6,23], where certain correlations between the waiting times, respectively between the jumps were introduced. A similar approach in a more general setting appears in [13], where coupling is introduced through a Markov chain (Y n ) n∈N and waiting times, respectively jumps are modelled by (J n = τ (Y n ), X n = V (Y n )) for some measurable functions τ, V with values in (0, ∞), respectively R. Note that these couplings in general do not fulfill our i.i.d. assumption on the sequence (J n , X n ) n∈N .…”

Fractional diffusion limit for a kinetic equation with an interface

“…The following result is a simpler version of Proposition 4.1 and Theorem 2.5(i) of [12], see also Theorem 2.4 of [13] and Theorem 3.2 of [5]. Proposition 4.2 Suppose that β > 1 and (y, k) ∈ R * × T * .…”