Weak Quenched Invariance Principle for Random Walk with Random Environment in Time

You Lü; Wenming Hong

doi:10.1007/s11464-022-0348-z

Frontiers of Mathematics ›› 2025, Vol. 20 ›› Issue (5) : 1109 -1123. DOI: 10.1007/s11464-022-0348-z

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Weak Quenched Invariance Principle for Random Walk with Random Environment in Time

You Lü ¹
, Wenming Hong ²^,^a

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Abstract

Consider the invariance principle for a random walk with a random environment (denoted by μ) in time on ℝ in a weak quenched sense. We show that a sequence of random probability measures on ℝ generated by μ and a bounded Lipschitz functional f will converge in distribution to another random probability measure, which can be represented by f and two independent Brownian motions. The upper bound of the convergence rate has been obtained. We also explain that in general, this convergence can not be strengthened to the almost surely sense.

Keywords

Random environment / invariance principle / weak quenched limits / 60G50

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You Lü,Wenming Hong. Weak Quenched Invariance Principle for Random Walk with Random Environment in Time. Frontiers of Mathematics, 2025, 20(5): 1109-1123 DOI:10.1007/s11464-022-0348-z