
Propagation of Uncertainty in a Continuous Time Dynamical System 259
conditions
˙
x = g(t, x), x(0) = X
0
, (7.88)
where x = (x
1
, x
2
, . . . , x
n
)
T
, g = (g
1
, g
2
, . . . , g
n
)
T
is an n-dimensional non-
random vector function of t and x, and X
0
is an n-dimensional random vector.
Equation (7.88) is often referre d to as a crypto-deterministic fo rmulation
(e.g., see [52, 86, 104]) a nd has proved useful in a wide number of appli-
cations, including classical statistical mechanics, statistical thermodynamics,
kinetic theory and bioscienc es (e.g., see [1 04, Chapter 6] for a more detailed
discussion). Depending on the application, the uncertainty in initial condi- ...