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# Appendix C

We shall first show that for small n we may write (3.50)

Taking the limit for we then easily check Eq. (3.16). In order to prove Eq. (3.50) we first notice that n being small, all Ri in the exponent in Eq. (3.13) are approximately equal to , from which it follows that  (3.51)

It is now simply a matter of successively performing the integrals. The simplest way to do this is by induction.    (3.52)

The first equals sign is the induction step, and the second one is the final proof.

Now look at the Chapman-Kolmogorov equation  (3.14), and take n to be small. Then differs from zero only if is in the neighbourhood of . We therefore expand the second factor in Eq. (3.14) with respect to .   (3.53)

where denotes the 'th Cartesian component of . Introducing Eqs. (3.50) and (3.53) into ( 3.14) we get (3.54)

where we have used  (3.55)

For small n, we expand the exponent (3.56)

Collecting terms of order n, and using (3.57)

we finally arrive at Eq. (3.15).     Next: Stochastic processes Up: The Gaussian chain Previous: Appendix B
W.J. Briels