Mixed stochastic heat equation with fractional Laplacian and gradient perturbation

We introduce a new stochastic heat equation with a mixed operator, which is a combination of the standard Laplacian, a fractional Laplacian and the gradient operator, driven by an additive Gaussian noise which is white in time and in space. We establish the existence of the solution and we study its behavior with respect to the time variable. In particular, we establish sharp two-sided interesting estimates of its variance increments, which allow us to get some regularity characteristics of the solution sample paths, among many other possible applications.