## Minimization of the Kullback information of diffusion processes.(English)Zbl 0790.60032

Summary: We compute an explicit expression for the rate function of large deviations for the measure valued empirical process $$\overline X^ N(t)={1\over N}\sum^ N_{i=1}\delta_{X_ i(t)}$$, where the $$X_ i$$’s are independent copies of a diffusion process in $$\mathbb{R}^ d$$. This is done by minimizing the relative entropy (Kullback information) of a probability measure $$Q$$ with respect to the law $$P$$ of $$X_ i$$ when all marginals of $$Q$$ are fixed. The finiteness of the rate function is connected with the existence of conservative diffusions, with a general diffusion matrix. These diffusion processes are constructed in very general cases.

### MSC:

 60F10 Large deviations 60J60 Diffusion processes 60G44 Martingales with continuous parameter 60H05 Stochastic integrals 60J57 Multiplicative functionals and Markov processes
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