Joint Distributions of Random Processes with Non-Gaussian Height Distributions

Appendix D JOINT DISTRIBUTIONS OF RANDOM PROCESSES WITH NON-GAUSSIAN HEIGHT DISTRIBUTIONS A general single-variable Langevin equation takes the form [D.1, D.2],

dx d-T = h(x, r) + g(x, r)~(r),

(D.1)

where ~(r) is a Gaussian-Markov process, satisfying < ~(r) > = 0,

(D.2)

< ~(r)~(~') > - 2~(~- r'). Here we adopt the Stratonovich interpretation of Equation (D.1). The corresponding Fokker-Planck equation for Equation (D.1) is Op(xlxo ; ~)

Or

=

o

o~

Ox [A(x,r)p(x[x0;r)] + ~

[B(x,r)p(xlx0;r)],

(D.3)

where

A(x, r) - h(x, r) + g(x r) Og(x, r) '

(D.4)

(~X

B(x, r) = g2(x, r),

(D.5)

and p(x[xo; r) is the conditional probability density, and x and x0 are separated by distance r. We now consider the solution of Equation (D.3) corresponding to an initial value, p(xjxo; r = o) = 5(x - xo),

(D.6)

and the reflecting barriers' boundary conditions,

0 [B(x, r)p] - A(x, r)p - O, at x - xl, x2. Ox

(D.7)

A further assumption can be made concerning the coefficient A(x, r) and B(x,r): A(x, r) - A(x)F(r), B(x, r) - B(x)F(r).

(D.8)

Then Equation (D.3) can be solved by separation of variables. Letting

p(x[xo ;r) = X (x) T (r), 387

(D.9)

388

JOINT DISTRIBUTIONS OF RANDOM PROCESSES

we have dT =-,~F(r)T(r), dr d2

and

d

dx 2 [B(x)X(x)] - -~x [A(x)X(x)] + A X ( x ) - O.

(D.10)

(D.11)

The solution for Equation (D.10) is obvious: T(r) - T(O) exp[-A

~0r F(r)dr].

(D.12)

The solution for Equation (D.11) is an eigenvalue problem of a secondorder ordinary differential equation. We can give some special form of A(x) and B(x), and Equation (D.11) can be changed to a Sturm-Liouville equation. Letting B(x)

-

/3(cx 2 + dx + e),

(D.13)

A(x)

=

dB(x_____~)+/~(ax + b)

(D.14)

dx

and d W (x) ax + b = W(x). dx cx 2 + dx + c

(D.15)

Equation (D.11) becomes a standard Sturm-Liouville equation, d [ B ( x ) W ( x ) d~X dx

+ A W ( x ) X - O,

(D.16)

with the boundary condition dX B(x)W(x)-~-~x - O , x - xl, x2.

(D.17)

Therefore, the general solution for Equation (D.3) is p(xlxo;t ) - W ( x ) E

e -)~'~ f0~F (t)dtQn(x)Qn(xo),

(D.18)

n

where Qn (x) is the eigenfunction of Equations (D.16) and (D.17), and An is the corresponding eigenvalue. The Q~ satisfy the following orthonormalization relation" f x2 W ( x ) Q n ( x ) Q m ( x ) d x - 5urn. 1

(D.19)

JOINT DISTRIBUTIONSOF RANDOM PROCESSES

389

In fact, Qn (x) are the classic orthogonal polynomials. If the probability density for xo is given as W (x0), then the joint distribution for x and x0 is

p(x, xo;r)

-

W (x) W(xo) ~ R~Q~(x)Qn(xo), n

(D.20)

where

R(r) - exp [- for F(r)drl . The correlation function

(D.21)

R(r) is given as R (r) -

(D.22)

References D.1 H. Risken, The Fokker-Planck Equations: Methods of Solution and Applications (Springer-Verlag, New York, 1984). D.2 J. Honerkamp, Stochastic Dynamical System: Concepts, Numerical Methods, Data Analysis, translated by K. Lindenberg (VCH, New York, 1994).