## Relative entropy for diffusion processes

It is possible to define the relative entropy between to diffusion processes using Girsanov theorem only in the case where the diffusion coefficients are the same. Could you explain why it is not posssible to define a relative entropy between the laws of the two following processes: $$dx_1(t)=f(x_1(t))dt +\sigma_1 dW_t$$ and $$dx_2(t)=f(x_2(t))dt +\sigma_2 dW_t$$ when $\sigma_1 \neq \sigma_2$ ?

Beyond the problem of absolute continuity, is not there a way to compare both laws ?

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You noticed that the Wiener measures were singulars I think then that there is nothing more to say. What is entropy between singular measures ? – The Bridge Nov 6 2011 at 22:09
This is precisely the point ! Is it really the end of the story? or the begining of a new one? – unknown (google) Nov 6 2011 at 22:35