  We consider the goodness of fit testing problem for ergodic diffusion
processes. The basic hypothesis is supposed to be simple. The diffusion
coefficient is known and the alternatives are described by the different trend
coefficients. We study the asymptotic distribution of the Cramer-von Mises type
tests based on the empirical distribution function and local time estimator of
the invariant density. At particularly, we propose a transformation which makes
these tests asymptotically distribution free. We discuss the modifications of
this test in the case of composite basic hypothesis.
