Markov processes and exponential families

We study Markov processes the distribution of which stays for some interval of time in a given exponential family of distributions with one parameter. We show that the generator of such a process can be canonically associated to the generator of a Markov process on , having the same stability property with respect to a natural exponential family of distributions on . Examples of spin systems on a finite set and of Brownian motions on illustrate the results.