ON A SEMILINEAR DOUBLE FRACTIONAL HEAT EQUATION DRIVEN BY FRACTIONAL BROWNIAN SHEET

In this paper, we consider the stochastic heat equation of the form ∂u ∂t = (∆α + ∆β)u+ ∂f ∂x (t, x, u) + ∂W ∂t∂x , where 1 < β < α < 2, W (t, x) is a fractional Brownian sheet, ∆θ := −(−∆) denotes the fractional Lapalacian operator and f : [0, T ] × R × R → R is a nonlinear measurable function. We introduce the existence, uniqueness and Hölder 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.