Quenched Law

A quenched law is the conditional probability law of a stochastic process with its sampled random environment held fixed. For a random walk in random environment , it is denoted by . The walk may remain random even though its transition probabilities are now fixed. In their study of conditional limits for stationary processes, Volný and Woodroofe (2010, p. 1) call weak convergence of the conditional laws given a sigma-algebra to a fixed nonrandom probability measure for almost every conditioning outcome quenched convergence. They call convergence in probability of those conditional laws weakly -quenched convergence. The corresponding averaged probability law is the annealed law.