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

Consider the invariance principle for a random walk with random environment (denoted by $\mu$) in time on $\text{bfR}$ in a weak quenched sense. We show that a sequence of the random probability measures on $\text{bfR}$ generated by a bounded Lipschitz functional $f$ and $\mu$ will converge in distribution to another random probability measures, which is related to $f$ 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.