Stochastic Korteweg-de Vries equation driven by fractional Brownian motion

Abstract

We consider the Cauchy problem for the Korteweg-de Vries equation driven by a cylindrical fractional Brownian motion (fBm) in this paper. With Hurst parameter $H\geq\frac{7}{16}$ of the fBm, we obtain the local existence results with initial value in classical Sobolev spaces $H^s$ with $s\geq -\frac{9}{16}$. Furthermore, we give the relation between the Hurst parameter $H$ and the index $s$ to the Sobolev spaces $H^s$, which finds out the regularity between the driven term fBm and the initial value for the stochastic Korteweg-de Vries equation.