Limits for weighted p-variations and likewise functionals of fractional diffusions with drift

Let Xt be the pathwise solution of a diffusion driven by a fractional Brownian motion with Hurst constant H>1/2 and diffusion coefficient [sigma](t,x). Consider the successive increments of this solution, [Delta]Xi=Xi/n-X(i-1)/n. Using a cylinder approximation for the solution Xt, our main result yields that if 1/2<H<3/4 then, if Z is a standard normal random variable which is independent of BH, the process converges weakly to as n-->[infinity] where W is a Wiener process which is independent of BH and CH,p is a constant which depends on H and on p. In the place of p-variations we may consider functions that satisfy an almost multiplicative structure such as even polynomials or polynomials of absolute values. By considering second order increments of the discrete sample Xi we obtain analogous results for the whole interval 1/2<H<1. Finally, we show convergence is stable in the absence of drift and use this result to discuss weak convergence for weak solutions of the fractional diffusion equation.