Volume 128, number 8 PHYSICS LETTERS A 18 April 1988

I N T E N S I T Y F L U C T U A T I O N S OF R A N D O M L Y SCATTERED WAVES C O N S I D E R E D AS LOCAL T I M E

J.R. N O R R I S Statistical Laboratory, 16 Mill Lane, Cambridge CB2 1SB, UK

Received 8 February 1988; accepted for publication 12 February 1 988 Communicated by A.R. Bishop

The concept of local time of a stochastic process is available for the study of intensity fluctuations in randomly scattered waves. A result of Jakeman on rays scattered by a corrugated surface of brownian slope is deduced from the Ray-Knight theorem on brownian local time.

A plane o f parallel rays strikes perpendicularly a thin refracting layer. The surface through which the rays emerge is not fiat and so scatters the rays. This induces fluctuations of intensity at unit distance from the layer. How do these relate to the characteristics of the scattering surface? We consider the simple case where the surface is corrugated horizontally, the slope of the vertical cross-section being modelled by a sta- t ionary process X, (t being the vertical space parameter) .

There is a considerable mathematical literature (see for example refs. [ 1,2] and references therein) devoted to the study of the local time or occupation density of a real-valued process ( Y,)t~o. This is a two- parameter non-negative process L = L r, character- ized when it exists by the relation

f L( t ,y) dy= i lA(Ys)ds, A~_~. A 0

L(t, y) measures infinitesimally the "t ime spent at y by Y before t". The following result is proved in ref. [ 3 ], see also ref. [ 4 ].

Theorem. Let (Y,)t~o be a real-valued diffusion starting from 0 with generator

d fq= ½cr2(y) - b ( y ) dy'

where a is continuously differentiable, everywhere

positive, and a, b are such that lim inf Yt = - 0 9 . Take K > 0 and let T denote the first time that Y hits level - K . Then

Zv=L(T ,y ) , y>~-K

is also a diffusion with Z _ ~ = 0 and generator

S=a2(y--- ~ z dz +[l(Y<~°l-b(Y)Zldzz '

y>~ - K .

When a = 1 and b = 0 , this result is known as the Ray-Knight theorem on brownian local time. We are presently interested in the case a = b = l . Take a brownian motion (B~)t>~o starting from 0 and sel Y~=Bt-t. Since Y t ~ - 0 9 as t~09, (Yt),~o has a fi- nal local time

Ly--- lim L Y(t, y)<09 . t ~

Sending K~09, the theorem implies that (Ly)y~ is a diffusion, stationary for y~<0, with generator

The stationary distribution is exponential o f rate 1:

P(Ly~dl)=e-Z dl, y<<.O.

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Volume 128, number 8 PHYSICS LETTERS A 18 April 1988

In the scattering example we have a formal expression

Jr= i ~,,(Xs+s) ds

Thus Ly r is the intensi ty at height y when the top half ( t >~ 0) o f the scattering layer is made opaque. The behav iour of (Jy)y~R predic ted by Jakeman corre- sponds exactly to the s ta t ionary behaviour o f

(Ly)y~O.

for the ray intensity at height y and unit distance from the scatterer. Fo r Xt a b rownian mot ion, Jakeman [ 5,6] ob ta ined a complete descr ip t ion of the behav- iour of ( J y ) y ~ by direct calculation. Assume that Xo=0, set Bt=X_t, for t>~0, and Y,=Bt-t as above, then

oo 0

0 --oc~

References

[ 1 ] D. Williams, Diffusion, Markov processes, and martingales, Vol. 1 (Wiley, New York, 1979).

[2] L.C.G. Rogers and D. Williams, Diffusions, Markov pro- cesses, and martingales, Vol. 2 (Wiley, New York, 1987).

[3] J.R. Norris, L.C.G. Rogers and D. Williams, Probl. Theor. Rel. Fields 74 (1987) 271.

[4] J.R. Norris, L.C.G. Rogers and D. Williams, Phys. Lett. A 112 (1985) 16.

[5] E. Jakeman, J. Phys. A 15 (1982) L55. [6] E. Jakeman, J. Opt. Soc. Am. 72 (1982) 1034.

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