On a semilinear double fractional heat equation driven by fractional Brownian sheet 

Keywords:Mixed fractional heat equation, fractional Brownian sheet, H\"older regularity, Large Deviation Principle 
Abstract: 
In this paper, we consider the stochastic heat equation of the form
$$
\frac{\partial u}{\partial t}=(\Delta_\alpha+\Delta_\beta)u+\frac{\partial f}{\partial
x}(t,x,u)+\frac{\partial^2W}{\partial t\partial x},
$$
where $1<\beta<\alpha\leq 2$, $W(t,x)$ is a fractional Brownian
sheet, $\Delta_\theta:=(\Delta)^{\theta/2}$ denotes the fractional
Lapalacian operator and $f:[0,T]\times \mathbb{R}\times
\mathbb{R}\rightarrow\mathbb{R}$ is a nonlinear measurable function. We introduce the existence, uniqueness and H\"older regularity of the solution. As a related question, we consider also a large deviation principle associated with the above equation with a small perturbation via an equivalence relationship between Laplace principle and large deviation principle. 



