stochastic averaging for sdes with hopf drift and polynomial diusion coecients

it is known that a stochastic dierential equation (sde) induces two probabilisticobjects, namely a diusion process and a stochastic ow. while the diusion process isdetermined by the in nitesimal mean and variance given by the coecients of the sde,this is not the case for the stochastic ow induced by the sde. in order to characterize thestochastic ow uniquely the in nitesimal covariance given by the coecients of the sde isneeded in addition. the sdes we consider here are obtained by a weak perturbation of a rigidrotation by random elds which are white in time. in order to obtain information about thestochastic ow induced by this kind of multiscale sdes we use averaging for the in nitesimalcovariance. the main result here is an explicit determination of the coecients of the averagedsde for the case that the diusion coecients of the initial sde are polynomial. to do thiswe develop a complex version of cholesky decomposition algorithm.