### Abstract

We consider space-time inhomogeneous one-dimensional random walks which move by ±Δx in each time interval Δt with arbitrary transition probabilities depending on position and time. Unlike Donsker's theorem, we study the continuous limit of the random walks as Δx,Δt→0 under hyperbolic scaling ^{λ1}≥ Δt/Δx≥^{λ0}>0 with fixed numbers ^{λ1} and ^{λ0}. Our aim is to present explicit formulas and estimates of probabilistic quantities which characterize asymptotics of the random walks as Δx,Δt→0. This provides elementary proofs of several limit theorems on the random walks. In particular, if transition probabilities satisfy a Lipschitz condition, the random walks converge to solutions of ODEs. This is the law of large numbers. The results here will be foundations of a stochastic and variational approach to finite difference approximation of nonlinear PDEs of hyperbolic types.

Original language | English |
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Pages (from-to) | 264-271 |

Number of pages | 8 |

Journal | Nonlinear Analysis, Theory, Methods and Applications |

Volume | 102 |

DOIs | |

Publication status | Published - 2014 Jun |

Externally published | Yes |

### Keywords

- Continuous limit
- Finite difference approximation
- Hyperbolic scaling
- Inhomogeneous random walk
- Law of large numbers

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics