We shall first show that for small *n* we may write

Taking the limit for we then easily check Eq. (3.16). In order to prove Eq. (3.50) we first notice 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
.

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

(3.56) |

Collecting terms of order

(3.57) |

we finally arrive at Eq. (3.15).