Relation between the probability density and other properties of a stationary random process.
We consider the Pope-Ching differential equation [Phys. Fluids A 5, 1529 (1993)] connecting the probability density p(x)(x) of a stationary, homogeneous stochastic process x(t) and the conditional moments of its squared velocity and acceleration. We show that the solution of the Pope-Ching equation can be expressed as n(x)</v(x)/(-1)>, where n(x) is the mean number of crossings of the x level per unit time and </v(x)/(-1)> is the mean inverse velocity of crossing. This result shows that the probability density at x is fully determined by a one-point measurement of crossing velocities, and does not imply knowledge of the x(t) behavior outside of the infinitesimally narrow window near x.
Sokolov
I M
IM
Laboratoire des Milieux Désordonnées et Hétérogènes, Université Pierre et Marie Curie, 4, Place Jussieu, 75252 Paris, France.