The Weak Coupling Limit Brownian Motion Model and Universality

3 The Weak Coupling Limit Brownian Motion Model and Universality 

This section is expository and it deals with one of the most interesting aspects of the copolymer model. This aspect is however still underdeveloped. 

Let us first introduce a continuous copolymer model. If B = Bt t o and Y = Yt t o are two standard Brownian motions. The law of B is denoted by P and the law of Y is denoted by P. As the notation suggests, B is the continuum analog of S and Y is the continuum analog of the partial sums process associated to w. The measure of the continuous copolymer is defined by 

The free energy is defined in the standard way, however proving the existence of the free energy requires some work. 

The continuous model enjoys the standard properties already verified for the discrete model, notably that 

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