Fractional derivatives non-symmetric and time-dependent Dirichlet forms and the drift form

Using fractional derivatives we show that the drift form integral (infinity )(-infinity) u(x)/dv(x)/dx can be approximated by non-symmetric Dirichlet f orms. A similar result holds for the drift form in R-n with variable coeffi cients if the coefficient functions satisfy certain regularity and commutat or conditions. Since time-dependent Dirichlet forms (in the sense of Y. Osh ima) can be interpreted as sums of a drift form (in tau -direction) and a m ixture of tau -parametrized Dirichlet forms over R-n, our results show that time-dependent Dirichlet forms arise as limits of ordinary non-symmetric D irichlet forms in R x R-n-space. An abstract result on fractional powers of Markov generators allows to extend this observation to generalized Dirichl et forms. Another consequence is that the bilinear form induced by an arbit rary Levy process is the limit of non-symmetric Dirichlet forms.