Open AccessLimiting stochastic processes of shift-periodic dynamical systems
Author(s) -
Julia Stadlmann,
Radek Erban
Publication year - 2019
Publication title -
royal society open science
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.84
H-Index - 51
ISSN - 2054-5703
DOI - 10.1098/rsos.191423
Subject(s) - random walk , brownian motion , limit (mathematics) , limiting , convergence (economics) , algorithm , statistical physics , mathematics , mathematical analysis , physics , statistics , mechanical engineering , engineering , economics , economic growth
A shift-periodic map is a one-dimensional map from the real line to itself which is periodic up to a linear translation and allowed to have singularities. It is shown that iterative sequences x n +1 = F ( x n ) generated by such maps display rich dynamical behaviour. The integer parts ⌊ x n ⌋ give a discrete-time random walk for a suitable initial distribution of x 0 and converge in certain limits to Brownian motion or more general Lévy processes. Furthermore, for certain shift-periodic maps with small holes on [0,1], convergence of trajectories to a continuous-time random walk is shown in a limit.