
Propagation of Uncertainty in a Continuous Time Dynamical System 249
By the above equation and (7 .62) we see that the Orn stein–Uhlenbeck process
has the same form of correlation function as dichotomous Markov noise. It
should be noted that the Ornstein–Uhlenbeck process is the only process that
is simultaneou sly Gaussian, Markovian and st ationary (e.g., see [72, p. 172]
or [95, Section 2.5]—this follows from a result due to Doob), and it is often
referred to as the Markov Gaussian colored noise.
7.2.1.1 Evolution of the Probability Density Function of X (t)
Theorem 7.2.2 Assume that X
0
has probability density function p
0
, and
X(t) satisfies (7.63) ...