The solution of stochastic evolution equation with the fractional derivative

Abstract

In this paper, we consider the stochastic evolution equation driven by the Gaussian noise with white time and colored space, where the noise coefficient is the Marchaud fractional derivative. The key idea is that we transform our model into a stochastic space-fractional equation by taking the Marchaud fractional derivative, and then use Chaos expansion to prove the mild solution. There are three main results in this paper. First, we apply Chaos expansion to obtain the existence, uniqueness and Lyapunove exponent of the solution of the transformed equation. Second, we prove that there exists an unique mild solution of the original equation, the approach is taking the fractional integral operator into the transformed equation. Finally, we explore Hölder continuity of the mild solution.