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 (1 993)] connecting the probability density p(x)(x) of a stationary, homogeneo us stochastic process x(t) and the conditional moments of its squared veloc ity and acceleration. We show that the solution of the Pope-Ching equation can be expressed as n(x)[\upsilon(x)\(-1)], where n(x) is the mean number o f crossings of the x level per unit time and [\upsilon(x)\(-1)] is the mean inverse velocity of crossing. This result shows that the probability densi ty at x is fully determined by a one-paint measurement of crossing velociti es, and does not imply knowledge of the x(t) behavior outside of the infini tesimally narrow window near x. [S1063-651X(99)06709-4].