probability density function of a stochastic process.°3°4 This equation is, however, not 
solved analytically in general, except for a few special cases, so special considerations are 
usually demanded in using it. 
Here the conditional or transitional probability density function p-,(x, Atly) is 
expressed merely as p., y being the value of process X(t) at a preceding time Ar. The 
derivation of this equation is given by Caughey.* 
The Fokker—Planck equation for this transitional probability function is in the form 
apc d las 
— = | —-— p(x) + —— D(x) I p.. 12.13 
at ax ep 2 ax ©) |e Seay 
Here D(x) and Dx)>0 are called the drift coefficient and diffusion coefficient, 
respectively, and are related to the first and second moments of this distribution as 
DOC) = lim Hea [ csp ayyay (12.14) 
Aro At 
1 
D?x) = lim — | (y —x)*pe(x, Atly)dy. (12.15) 
At > 0 At 
Generally these are functions of time t. 
As was stated before, when the general process is inverted into a vector Markov 
process X(r) of N dimensions as X(t) = [X;(t), X2(t) . . . Xy(t)]’, the Fokker—Planck 
equation is more generally in the form 
N N 2 
ezene - Sy 2 Mx 44 » ke) Pe (12.16) 
ot sa OXi 2 721 9x10; v 
where 
(Dyce 1 N 
D; (x)= lim — (yi-x) p(x, Atl y)T1 dy; (12.17) 
Aro At fall 
eR uen a Nv 
Dj (x)= lim — (yi-xi)(Vj—X)Pc( x, Atl y)TI dy;}. (12.18) 
Ar>o At i=1 
—-o 
x, y are the values of process X(r). 
330 
