Generalized space-time fractional stochastic kinetic equation

In this paper, we study a class of nonlinear space-time fractional stochastic kinetic equations in $\mathbb{R}^d$ with Gaussian noise which is white in time and homogeneous in space. This type of equation constitutes an extension of the non-linear stochastic heat equation involving fractional derivative in time and fractional Laplacian in space. We give a necessary condition on the spatial covariance for the existence and uniqueness of the solution. We also study various properties of the solution: path regularity, the behavior of second moment and the stationarity in the case of linear additive noise.