TY - JOUR

T1 - Space-time continuous limit of random walks with hyperbolic scaling

AU - Soga, Kohei

N1 - Funding Information:
The work was supported by Grant-in-Aid for Japan Society for the Promotion of Science (JSPS) Fellows ( 20-6856 ).

PY - 2014/6

Y1 - 2014/6

N2 - 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.

AB - 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.

KW - Continuous limit

KW - Finite difference approximation

KW - Hyperbolic scaling

KW - Inhomogeneous random walk

KW - Law of large numbers

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U2 - 10.1016/j.na.2014.02.012

DO - 10.1016/j.na.2014.02.012

M3 - Article

AN - SCOPUS:84896478816

VL - 102

SP - 264

EP - 271

JO - Nonlinear Analysis, Theory, Methods and Applications

JF - Nonlinear Analysis, Theory, Methods and Applications

SN - 0362-546X

ER -