Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
7222775 | Nonlinear Analysis: Theory, Methods & Applications | 2014 | 8 Pages |
Abstract
We consider space-time inhomogeneous one-dimensional random walks which move by ±Îx in each time interval Ît with arbitrary transition probabilities depending on position and time. Unlike Donsker's theorem, we study the continuous limit of the random walks as Îx,Îtâ0 under hyperbolic scaling λ1â¥Ît/Îxâ¥Î»0>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 Îx,Îtâ0. 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.
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Authors
Kohei Soga,