Abstract
The goal of this paper is to understand the conditional law of a stochastic process once it has been observed over an interval. To make this precise, we introduce the notion of a continuous disintegration: a regular conditional probability measure which varies continuously in the conditioned parameter. The conditioning is infinite dimensional in character, which leads us to consider the general case of probability measures in Banach spaces. Our main result is that for a certain quantity $M$ based on the covariance structure, $M < \oo$ is a necessary and sufficient condition for a Gaussian measure to have a continuous disintegration. The condition $M < \oo$ is quite reasonable: for the familiar case of stationary processes, $M = 1$.
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