Terminal Value Problem For Stochastic Fractional Equation Within An Operator With Exponential Kernel

Abstract

In this paper, we investigate a terminal value problem for stochastic fractional diffusion equations with Caputo-Fabrizio derivative. The stochastic noise we consider here is the white noise taken value in the Hilbert space W. The main contribution is to investigate the well-posedness and ill-posedness of such problem in two distinct cases of the smoothness of the Hilbert scale space Wν (see Assumption 3.1), which is a subspace of W. When Wν is smooth enough, i.e. the parameter ν is sufficiently large, our problem is well-posed and it has a unique solution in the space of Hölder continuous functions. In contract, in the different case when ν is smaller, our problem is ill-posed; therefore, we construct a regularization result.