Generalized Space-Time Fractional Stochastic Kinetic Equation

In this paper, we study a class of nonlinear space-time fractional stochastic kinetic equations in R-d with Gaussian noise which is white in time and homogeneous in space. This type of equation constitutes an extension of the nonlinear stochastic heat equation involving fractional derivatives in time and fractional Laplacian in space. We firstly give a necessary condition on the spatial covariance for the existence and uniqueness of the solution. Furthermore, we also study various properties of the solution, such as Holder regularity, the upper bound of second moment, and the stationarity with respect to the spatial variable in the case of linear additive noise.