2010/03/31 by Tom LaGatta · 1 citation
Mathematics · #math.PR #math.FA #msc:41A65 #msc:47A50
published as Teor. Veroyatnost. i Primenen., 57:1 (2012), 192-203
arxiv created 2011/03/28 · arxiv updated 2012/08/24
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, the finiteness of M is a necessary and sufficient condition for a Gaussian measure to have a continuous disintegration. The condition is quite reasonable: for the familiar case of stationary processes, M = 1.