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 [Formula: see text]. 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 [Formula: see text] (see Assumption 3.1), which is a subspace of [Formula: see text]. When [Formula: see text] is smooth enough, i.e. the parameter [Formula: see text] 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 [Formula: see text] is smaller, our problem is ill-posed; therefore, we construct a regularization result.