The general probability law associated with a Brownian chain

2 The general probability law associated with a Brownian chain 

Let us now consider a chain made of N Gaussian independent links with the same probability law. The position of the point Mj of order j(j = 0,.. ., N) on the chain, can be represented by the vector ry. 

Let us note that, here, we introduce a weight and not a probability distribution, in order to move more easily to the continuous limit. The weight defined above has a continuous limit on the contrary the normalization constant has no limit when the number of variables becomes infinite. Thus, the probability distribution corresponding to the weight has no limit. 

The weight (2.1.14) is used to calculate mean values, and obviously we have 

Let us now consider the chain as a curve made of linear segments Af,-, Mj. The parameter s, with 0 s S — Nl2 will serve to locate a point on this curve. The position vector of a point M corresponding to the coordinate s will be defined by the conditions 

---