# Moscow Mathematical Journal

Volume 19, Issue 1, January–March 2019 pp. 51–76.

Regular and Singular Continuous Time Random Walk in Dynamic Random Environment

**Authors**:
C. Boldrighini (1), A. Pellegrinotti (2), and E. A. Zhizhina (3)

**Author institution:**(1) Istituto Nazionale di Alta Matematica (INdAM), GNFM, Unità locale Università Roma Tre, Largo S. Leonardo Murialdo, 1, 00146 Rome, Italy

(2) Dipartimento di Matematica e Fisica, Università di Roma Tre, Largo S. Leonardo Murialdo 1, 00146 Rome, Italy

(3) Institute for Information Transmission Problems, Russian Academy of Sciences

**Summary: **

We consider a homogeneous continuous-time random walk
(CTRW) on the lattice ℤ^{d}, *d* = 1, 2, ..., which is a kind of random trap
model in a time-dependent (“dynamic”) environment. The waiting time
distribution is renewed at each jump, and is given by a general probability density depending on a parameter η>0 such that the average
waiting time is finite for η>1 and infinite for η∈(0, 1]. By applying analytic methods introduced in a previous paper we prove that the
asymptotics of the quenched CTRW and of its annealed version are the
same for all η>0 and *d*>1. We also exhibit explicit formulas for the
correction term to the quenched asymptotics. For the border-line case
η=1 we find an explicit expression for the annealed limiting distribution, which is, to our knowledge, new.

2010 Math. Subj. Class. 60J10, 60K37, 82B41.

**Keywords:**Continuous-time random walk, random traps, dynamic random environment, singular waiting time, random walk in quenched environment.

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