  We prove the well posedness: global existence, uniqueness and regularity of
the solutions, of a class of d-dimensional fractional stochastic active scalar
equations. This class includes the stochastic, dD-quasi-geostrophic equation, $
d\geq 1$, fractional Burgers equation on the circle, fractional nonlocal
transport equation and the 2D-fractional vorticity Navier-Stokes equation. We
consider the multiplicative noise with locally Lipschitz diffusion term in
both, the free and no free divergence modes. The random noise is given by an
$Q-$Wiener process with the covariance $Q$ being either of finite or infinite
trace. In particular, we prove the existence and uniqueness of a global mild
solution for the free divergence mode in the subcritical regime
($\alpha>\alpha_0(d)\geq 1$), martingale solutions in the general regime
($\alpha\in (0, 2)$) and free divergence mode, and a local mild solution for
the general mode and subcritical regime. Different kinds of regularity are also
established for these solutions.
  The method used here is also valid for other equations like fractional
stochastic velocity Navier-Stokes equations (work is in progress). The full
paper will be published in Arxiv after a sufficient progress for these
equations.

Analysis of PDEs